Given $g \in N^{\vee}(\mathcal{K})$, there is a unique tuple $(a_i) \in \mathcal{K}^I$ such that

$$g = \prod_{i \in I} x_{\alpha_i^{\vee}}(a_i) \mod (N^{\vee}(\mathcal{K}), N^{\vee}(\mathcal{K}));$$

looking at a specific $i \in I$, we denote by $\mathbf{a}_{i,p}(g)$ the coefficient of $z^p$ in the Laurent series $a_i$. This procedure defines a morphism of groups $\mathbf{a}_{i,p}: N^{\vee}(\mathcal{K}) \to \mathbb{C}$ for each pair $(i, p) \in I \times \mathbb{Z}$.

**Lemma 4.13.** Let $\pi \in \Pi$ and let $(t_1, \dots, t_m)$ be the ordered list of all elements $t \in [0, 1[$ such that $\Phi_a^{\vee,+}(\pi, t) \neq \emptyset$. Set $t_{m+1} = 1$. Let $i \in I$ and set

$$p = \min \{\langle \alpha_i^{\vee}, \pi(t) \rangle \mid t \in [0, 1]\}.$$

Let $r \in \{1, \dots, m+1\}$ and let $(v_\ell) \in \prod_{\ell=r}^m \mathcal{N}^{\vee}(\pi, t_\ell)$.

(i) Let $r^+$ be the smallest element in

$$\{\ell \in \{r, \dots, m\} \mid (\alpha_i^{\vee}, p) \in \Phi_a^{\vee,+}(\pi, t_\ell)\},$$

assuming that this set is nonempty. Then for any $u \in N^{\vee}(f_{\pi(t_r)})$ there exists $(v'_\ell) \in \prod_{\ell=r}^m \mathcal{N}^{\vee}(\pi, t_\ell)$ such that

$$v'_r \cdots v'_m L_{wt}(\pi) = uv_r \cdots v_m L_{wt}(\pi)$$

and

$$\mathbf{a}_{i,p}(v'_{\ell}) = \begin{cases} \mathbf{a}_{i,p}(u) + \mathbf{a}_{i,p}(v_{\ell}) & \text{if } \ell = r^{+}, \\ \mathbf{a}_{i,p}(v_{\ell}) & \text{for all other } \ell \in \{r, \dots, m\}. \end{cases}$$

(ii) For any $c \in 1 + z\mathcal{O}$ and any $\lambda \in \Lambda$, there exists $(v'_\ell) \in \prod_{\ell=r}^m \mathcal{N}^{\vee}(\pi, t_\ell)$ such that

$$v'_r \cdots v'_m L_{wt}(\pi) = c^{\lambda} v_r \cdots v_m L_{wt}(\pi)$$

and $\mathbf{a}_{i,p}(v'_\ell) = \mathbf{a}_{i,p}(v_\ell)$ for all $\ell \in \{r, \dots, m\}$.

(iii) For any $b \in \mathbb{C}$ not in

$$\{0\} \cup \{\mathbf{a}_{i,p}(v_r) + \cdots + \mathbf{a}_{i,p}(v_\ell) \mid \ell \in \{r, \dots, m\}\},$$

there exists $(v'_\ell) \in \prod_{\ell=r}^m \mathcal{N}^{\vee}(\pi, t_\ell)$ such that

$$v'_r \cdots v'_m L_{wt}(\pi) = x_{(-\alpha_i^{\vee}, -p)}(1/b)v_r \cdots v_m L_{wt}(\pi).$$

*Proof.* The lemma is trivial for $r = m + 1$. Proceeding by decreasing induction, we choose $r \in \{1, \dots, m\}$, assume that statements (i)–(iii) hold for $r + 1$, and show that they also hold for $r$. We recall (see the proof of Proposition 4.8) that

$$N^{\vee}(f_{\pi(t_r)}) = \mathcal{N}^{\vee}(\pi, t_r) \bowtie N^{\vee}(f_{\pi(t_r+0)}) \text{ and } N^{\vee}(f_{\pi(t_r+0)}) \subseteq N^{\vee}(f_{\pi(t_{r+1})}).$$

Let $(v_\ell) \in \prod_{\ell=r}^m \mathcal{N}^{\vee}(\pi, t_\ell)$.
Given $g \in N^{\vee}(\mathcal{K})$, there is a unique tuple $(a_i) \in \mathcal{K}^I$ such that

$$g = \prod_{i \in I} x_{\alpha_i^{\vee}}(a_i) \mod (N^{\vee}(\mathcal{K}), N^{\vee}(\mathcal{K}));$$

looking at a specific $i \in I$, we denote by $\mathbf{a}_{i,p}(g)$ the coefficient of $z^p$ in the Laurent series $a_i$. This procedure defines a morphism of groups $\mathbf{a}_{i,p} : N^{\vee}(\mathcal{K}) \to \mathbb{C}$ for each pair $(i, p) \in I \times \mathbb{Z}$.

**Lemma 4.13.** Let $\pi \in \Pi$ and let $(t_1, \dots, t_m)$ be the ordered list of all elements $t \in [0, 1[$ such that $\Phi_a^{\vee,+}(\pi, t) \neq \emptyset$. Set $t_{m+1} = 1$. Let $i \in I$ and set

$$p = \min \{\langle \alpha_i^{\vee}, \pi(t) \rangle \mid t \in [0, 1]\}.$$

Let $r \in \{1, \dots, m+1\}$ and let $(v_\ell) \in \prod_{\ell=r}^m \mathcal{N}^\vee(\pi, t_\ell)$.

(i) Let $r^+$ be the smallest element in

$$\{\ell \in \{r, \dots, m\} \mid (\alpha_i^{\vee}, p) \in \Phi_a^{\vee,+}(\pi, t_\ell)\},$$

assuming that this set is nonempty. Then for any $u \in N^{\vee}(f_{\pi(t_r)})$ there exists $(v'_\ell) \in \prod_{\ell=r}^m \mathcal{N}^{\vee}(\pi, t_\ell)$ such that

$$v'_r \cdots v'_m L_{wt(\pi)} = uv_r \cdots v_m L_{wt(\pi)}$$

and

$$\mathbf{a}_{i,p}(v'_{\ell}) = \begin{cases} \mathbf{a}_{i,p}(u) + \mathbf{a}_{i,p}(v_{\ell}) & \text{if } \ell = r^{+}, \\ \mathbf{a}_{i,p}(v_{\ell}) & \text{for all other } \ell \in \{r, \dots, m\}. \end{cases}$$

(ii) For any $c \in 1 + z\mathcal{O}$ and any $\lambda \in \Lambda$, there exists $(v'_\ell) \in \prod_{\ell=r}^m \mathcal{N}^\vee(\pi, t_\ell)$ such that

$$v'_r \cdots v'_m L_{wt(\pi)} = c^{\lambda} v_r \cdots v_m L_{wt(\pi)}$$

and $\mathbf{a}_{i,p}(v'_\ell) = \mathbf{a}_{i,p}(v_\ell)$ for all $\ell \in \{r, \dots, m\}$.

(iii) For any $b \in \mathbb{C}$ not in

$$\{0\} \cup \{\mathbf{a}_{i,p}(v_r) + \cdots + \mathbf{a}_{i,p}(v_\ell) \mid \ell \in \{r, \dots, m\}\},$$

there exists $(v'_\ell) \in \prod_{\ell=r}^m \mathcal{N}^\vee(\pi, t_\ell)$ such that

$$v'_r \cdots v'_m L_{wt(\pi)} = x_{(-\alpha_i^{\vee}, -p)}(1/b)v_r \cdots v_m L_{wt(\pi)}.$$

*Proof.* The lemma is trivial for $r = m + 1$. Proceeding by decreasing induction, we choose $r \in \{1, \dots, m\}$, assume that statements (i)–(iii) hold for $r + 1$, and show that they also hold for $r$. We recall (see the proof of Proposition 4.8) that

$$N^{\vee}(f_{\pi(t_r)}) = \mathcal{N}^{\vee}(\pi, t_r) \bowtie N^{\vee}(f_{\pi(t_r+0)}) \text{ and } N^{\vee}(f_{\pi(t_r+0)}) \subseteq N^{\vee}(f_{\pi(t_{r+1})}).$$

Let $(v_\ell) \in \prod_{\ell=r}^m \mathcal{N}^{\vee}(\pi, t_\ell)$.
We start with (i). Let $u \in N^{\vee}(\mathfrak{f}_{\pi(t_r)})$. We can write $uv_r \in N^{\vee}(\mathfrak{f}_{\pi(t_r)})$ as a product $v'_r u'$ with $(v'_r, u') \in \mathcal{N}^{\vee}(\pi, t_r) \times N^{\vee}(\mathfrak{f}_{\pi(t_r+0)})$. Then

$$ \mathbf{a}_{i,p}(u) + \mathbf{a}_{i,p}(v_r) = \mathbf{a}_{i,p}(v'_r) + \mathbf{a}_{i,p}(u'). $$

Noting that $u' \in N^{\vee}(\mathfrak{f}_{\pi(t_{r+1})})$, we make use of the inductive assumption: there exists $(v'_\ell) \in \prod_{\ell=r+1}^m \mathcal{N}^{\vee}(\pi, t_\ell)$ such that

$$ v'_{r+1} \cdots v'_m L_{\mathrm{wt}}(\pi) = u' v_{r+1} \cdots v_m L_{\mathrm{wt}}(\pi) $$

and

$$ \mathbf{a}_{i,p}(v'_\ell) = \begin{cases} \mathbf{a}_{i,p}(u') + \mathbf{a}_{i,p}(v_\ell) & \text{if } \ell = (r+1)^+, \\ \mathbf{a}_{i,p}(v_\ell) & \text{for all other } \ell \in \{r+1, \dots, m\}. \end{cases} $$

We distinguish two cases. If $(\alpha_i^\vee, p) \in \Phi_a^{\vee,+}(\pi, t_r)$, then $\mathfrak{f}_{\pi(t_r+0)} \not\subseteq H_{(\alpha_i^\vee, p)}^-$, whence $\mathbf{a}_{i,p}(u') = 0$; also $r^+ = r$ in this case. If $(\alpha_i^\vee, p) \notin \Phi_a^{\vee,+}(\pi, t_r)$, then $\mathbf{a}_{i,p}(v_r) = \mathbf{a}_{i,p}(v'_r) = 0$; here $r^+ = (r+1)^+$. In both cases, routine checks conclude the proof of (i).

We now turn to statement (ii). Let $c \in 1 + z\mathcal{O}$ and let $\lambda \in \Lambda$. One easily checks that any subgroup of the form $N^{\vee}(\mathfrak{f})$, in particular $N^{\vee}(\mathfrak{f}_{\pi(t_r)})$, is stable under conjugation by $c^\lambda$. Additionally, for any $v \in N^{\vee}(\mathfrak{f}_{\pi(t_r)})$, when we write

$$ v \equiv \prod_{i \in I} x_{\alpha_i^\vee}(a_i) \mod (N^{\vee}(\mathcal{K}), N^{\vee}(\mathcal{K})), $$

the Laurent series $a_i$ has valuation at least $p$. This series is multiplied by $c^{(\alpha_i^\vee, \lambda)}$ when one conjugates $v$ by $c^\lambda$. Looking at the coefficient of $z^p$ then gives $\mathbf{a}_{i,p}(v) = \mathbf{a}_{i,p}(c^\lambda v c^{-\lambda})$.

Write $c^\lambda v_r c^{-\lambda} \in N^{\vee}(\mathfrak{f}_{\pi(t_r)})$ as a product $v'_r u$ with $(v'_r, u) \in \mathcal{N}^{\vee}(\pi, t_r) \times N^{\vee}(\mathfrak{f}_{\pi(t_r+0)})$. Then

$$ \mathbf{a}_{i,p}(v_r) = \mathbf{a}_{i,p}(c^\lambda v_r c^{-\lambda}) = \mathbf{a}_{i,p}(v'_r) + \mathbf{a}_{i,p}(u). $$

By induction, there exists $(v'_\ell) \in \prod_{\ell=r+1}^m \mathcal{N}^{\vee}(\pi, t_\ell)$ such that

$$ v'_{r+1} \cdots v'_m L_{\mathrm{wt}}(\pi) = u c^\lambda v_{r+1} \cdots v_m L_{\mathrm{wt}}(\pi) $$

and

$$ \mathbf{a}_{i,p}(v'_\ell) = \begin{cases} \mathbf{a}_{i,p}(u) + \mathbf{a}_{i,p}(v_\ell) & \text{if } \ell = (r+1)^+, \\ \mathbf{a}_{i,p}(v_\ell) & \text{for all other } \ell \in \{r+1, \dots, m\}. \end{cases} $$

Again we distinguish two cases. If $(\alpha_i^\vee, p) \in \Phi_a^{\vee,+}(\pi, t_r)$, then $\mathfrak{f}_{\pi(t_r+0)} \not\subseteq H_{(\alpha_i^\vee, p)}^-$ and therefore $\mathbf{a}_{i,p}(u) = 0$. If $(\alpha_i^\vee, p) \notin \Phi_a^{\vee,+}(\pi, t_r)$, then $\mathbf{a}_{i,p}(v_r) = \mathbf{a}_{i,p}(v'_r) = 0$ and anew $\mathbf{a}_{i,p}(u) = 0$. Thus, $\mathbf{a}_{i,p}(u) = 0$ holds unconditionally, which concludes the proof of (ii).

Lastly, let us deal with statement (iii). We distinguish three cases.

Suppose first that $(\alpha_i^\vee, p) \in \Phi_a^{\vee,+}(\pi, t_r)$. We write $v_r = x_{(\alpha_i^\vee, p)}(a)\tilde{v}_r$ where $a = \mathbf{a}_{i,p}(v_r)$ and $\tilde{v}_r$ is a product of elements $x_\beta(a_\beta)$ with $\beta \in \Phi_a^{\vee,+}(\pi, t_r) \setminus \{(\alpha_i^\vee, p)\}$ and
We start with (i). Let $u \in N^\vee(f_{\pi(t_r)})$. We can write $uv_r \in N^\vee(f_{\pi(t_r)})$ as a product $v'_r u'$ with $(v'_r, u') \in \mathcal{N}^\vee(\pi, t_r) \times N^\vee(f_{\pi(t_r+0)})$. Then

$$ \mathbf{a}_{i,p}(u) + \mathbf{a}_{i,p}(v_r) = \mathbf{a}_{i,p}(v'_r) + \mathbf{a}_{i,p}(u'). $$

Noting that $u' \in N^\vee(f_{\pi(t_{r+1})})$, we make use of the inductive assumption: there exists $(v'_\ell) \in \prod_{\ell=r+1}^m \mathcal{N}^\vee(\pi, t_\ell)$ such that

$$ v'_{r+1} \cdots v'_m L_{\mathrm{wt}}(\pi) = u'v_{r+1} \cdots v_m L_{\mathrm{wt}}(\pi) $$

and

$$ \mathbf{a}_{i,p}(v'_\ell) = \begin{cases} \mathbf{a}_{i,p}(u') + \mathbf{a}_{i,p}(v_\ell) & \text{if } \ell = (r+1)^+, \\ \mathbf{a}_{i,p}(v_\ell) & \text{for all other } \ell \in \{r+1, \dots, m\}. \end{cases} $$

We distinguish two cases. If $(\alpha_i^\vee, p) \in \Phi_a^{\vee,+}(\pi, t_r)$, then $f_{\pi(t_r+0)} \not\subseteq H_{(\alpha_i^\vee, p)}^-$, whence $\mathbf{a}_{i,p}(u') = 0$; also $r^+ = r$ in this case. If $(\alpha_i^\vee, p) \notin \Phi_a^{\vee,+}(\pi, t_r)$, then $\mathbf{a}_{i,p}(v_r) = \mathbf{a}_{i,p}(v'_r) = 0$; here $r^+ = (r+1)^+$. In both cases, routine checks conclude the proof of (i).

We now turn to statement (ii). Let $c \in 1 + z\mathcal{O}$ and let $\lambda \in \Lambda$. One easily checks that any subgroup of the form $N^\vee(f)$, in particular $N^\vee(f_{\pi(t_r)})$, is stable under conjugation by $c^\lambda$. Additionally, for any $v \in N^\vee(f_{\pi(t_r)})$, when we write

$$ v \equiv \prod_{i \in I} x_{\alpha_i^\vee}(a_i) \mod (N^\vee(\mathcal{K}), N^\vee(\mathcal{K})), $$

the Laurent series $a_i$ has valuation at least $p$. This series is multiplied by $c^{(\alpha_i^\vee, \lambda)}$ when one conjugates $v$ by $c^\lambda$. Looking at the coefficient of $z^p$ then gives $\mathbf{a}_{i,p}(v) = \mathbf{a}_{i,p}(c^\lambda vc^{-\lambda})$.

Write $c^\lambda v_r c^{-\lambda} \in N^\vee(f_{\pi(t_r)})$ as a product $v'_r u$ with $(v'_r, u) \in \mathcal{N}^\vee(\pi, t_r) \times N^\vee(f_{\pi(t_r+0)})$. Then

$$ \mathbf{a}_{i,p}(v_r) = \mathbf{a}_{i,p}(c^\lambda v_r c^{-\lambda}) = \mathbf{a}_{i,p}(v'_r) + \mathbf{a}_{i,p}(u). $$

By induction, there exists $(v'_\ell) \in \prod_{\ell=r+1}^m \mathcal{N}^\vee(\pi, t_\ell)$ such that

$$ v'_{r+1} \cdots v'_m L_{\mathrm{wt}}(\pi) = u c^\lambda v_{r+1} \cdots v_m L_{\mathrm{wt}}(\pi) $$

and

$$ \mathbf{a}_{i,p}(v'_\ell) = \begin{cases} \mathbf{a}_{i,p}(u) + \mathbf{a}_{i,p}(v_\ell) & \text{if } \ell = (r+1)^+, \\ \mathbf{a}_{i,p}(v_\ell) & \text{for all other } \ell \in \{r+1, \dots, m\}. \end{cases} $$

Again we distinguish two cases. If $(\alpha_i^\vee, p) \in \Phi_a^{\vee,+}(\pi, t_r)$, then $f_{\pi(t_r+0)} \not\subseteq H_{(\alpha_i^\vee, p)}^-$ and therefore $\mathbf{a}_{i,p}(u) = 0$. If $(\alpha_i^\vee, p) \notin \Phi_a^{\vee,+}(\pi, t_r)$, then $\mathbf{a}_{i,p}(v_r) = \mathbf{a}_{i,p}(v'_r) = 0$ and anew $\mathbf{a}_{i,p}(u) = 0$. Thus, $\mathbf{a}_{i,p}(u) = 0$ holds unconditionally, which concludes the proof of (ii).

Lastly, let us deal with statement (iii). We distinguish three cases.

Suppose first that $(\alpha_i^\vee, p) \in \Phi_a^{\vee,+}(\pi, t_r)$. We write $v_r = x_{(\alpha_i^\vee, p)}(a)\tilde{v}_r$ where $a = \mathbf{a}_{i,p}(v_r)$ and $\tilde{v}_r$ is a product of elements $x_\beta(a_\beta)$ with $\beta \in \Phi_a^{\vee,+}(\pi, t_r) \setminus \{(\alpha_i^\vee, p)\}$ and
$a_β \in \mathbb{C}$. From (18) we get

$$x_{(-\alpha_i^\vee, -p)}(1/b)x_{(\alpha_i^\vee, p)}(a) = (1 - a/b)^{-\alpha_i} x_{(\alpha_i^\vee, p)}(a(1 - a/b))x_{(-\alpha_i^\vee, -p)}(1/(b - a)).$$

By Lemma 4.12,

$$x_{(-\alpha_i^\vee, -p)}(1/(b-a))\tilde{v}_r x_{(-\alpha_i^\vee, -p)}(-1/(b-a))$$

belongs to $N^\vee(\mathfrak{f}_\pi(t_r))$. We write it as a product $\tilde{v}'_r u$ with $(\tilde{v}'_r, u) \in \mathcal{N}^\vee(\pi, t_r) \times N^\vee(\mathfrak{f}_\pi(t_{r+0}))$. By induction, there exists $(v'_\ell) \in \prod_{\ell=r+1}^m \mathcal{N}^\vee(\pi, t_\ell)$ such that

$$v'_{r+1} \cdots v'_m L_{\mathrm{wt}}(\pi) = u x_{(-\alpha_i^\vee, -p)}(1/(b-a)) v_{r+1} \cdots v_m L_{\mathrm{wt}}(\pi).$$

Then

$$x_{(-\alpha_i^\vee, -p)}(1/b)v_r \cdots v_m L_{\mathrm{wt}}(\pi) = (1 - a/b)^{-\alpha_i} [x_{(\alpha_i^\vee, p)}(a(1 - a/b))\tilde{v}'_r] v'_{r+1} \cdots v'_m L_{\mathrm{wt}}(\pi).$$

Denoting the element in square brackets by $v'_r$, we get the desired expression, up to the inconsequential left multiplication by $(1 - a/b)^{-\alpha_i}$.

Suppose now that there exists $q > p$ such that $(\alpha_i^\vee, q) \in \Phi_a^{\vee,+}(\pi, t_r)$; then $a_{i,p}(v_r) = 0$. We write $v_r = x_{(\alpha_i^\vee, q)}(a)\tilde{v}_r$ where $a \in \mathbb{C}$ and $\tilde{v}_r$ is a product of elements $x_\beta(a_\beta)$ with $\beta \in \Phi_a^{\vee,+}(\pi, t_r) \setminus \{(\alpha_i^\vee, q)\}$ and $a_\beta \in \mathbb{C}$. Let $c$ be a square root in $1 + t\mathcal{O}$ of $1 - (a/b)t^{q-p}$. From (18) we get

$$x_{(-\alpha_i^\vee, -p)}(1/b)x_{(\alpha_i^\vee, q)}(a) = c^{-\alpha_i} x_{(\alpha_i^\vee, q)}(a)x_{(-\alpha_i^\vee, -p)}(1/b)c^{-\alpha_i}.$$

By Lemma 4.12,

$$x_{(-\alpha_i^\vee, -p)}(1/b)(c^{-\alpha_i} \tilde{v}_r c^{\alpha_i}) x_{(-\alpha_i^\vee, -p)}(-1/b)$$

belongs to $N^\vee(\mathfrak{f}_\pi(t_r))$; we write it as a product $\tilde{v}'_r u$ with $(\tilde{v}'_r, u) \in \mathcal{N}^\vee(\pi, t_r) \times N^\vee(\mathfrak{f}_\pi(t_{r+0}))$. By induction, there exists $(v'_\ell) \in \prod_{\ell=r+1}^m \mathcal{N}^\vee(\pi, t_\ell)$ such that

$$v'_{r+1} \cdots v'_m L_{\mathrm{wt}}(\pi) = u x_{(-\alpha_i^\vee, -p)}(1/b) c^{-\alpha_i} v_{r+1} \cdots v_m L_{\mathrm{wt}}(\pi).$$

Then

$$x_{(-\alpha_i^\vee, -p)}(1/b)v_r \cdots v_m L_{\mathrm{wt}}(\pi) = c^{-\alpha_i} [x_{(\alpha_i^\vee, q)}(a)\tilde{v}'_r] v'_{r+1} \cdots v'_m L_{\mathrm{wt}}(\pi).$$

Denoting the element in square brackets by $v'_r$, we get the desired expression, up to the inopportune left multiplication by $c^{-\alpha_i}$. The latter can however be wiped off by a further use of the inductive assumption.

Last, suppose that no affine coroot of direction $\alpha_i^\vee$ occurs in $\Phi_a^{\vee,+}(\pi, t_r)$; then $a_{i,p}(v_r) = 0$. By Lemma 4.12,

$$x_{(-\alpha_i^\vee, -p)}(1/b)v_r x_{(-\alpha_i^\vee, -p)}(-1/b)$$

belongs to $N^\vee(\mathfrak{f}_\pi(t_r))$. We write it as a product $v'_r u$ with $(v'_r, u) \in \mathcal{N}^\vee(\pi, t_r) \times N^\vee(\mathfrak{f}_\pi(t_{r+0}))$. By induction, there exists $(v'_\ell) \in \prod_{\ell=r+1}^m \mathcal{N}^\vee(\pi, t_\ell)$ such that

$$v'_{r+1} \cdots v'_m L_{\mathrm{wt}}(\pi) = u x_{(-\alpha_i^\vee, -p)}(1/b) v_{r+1} \cdots v_m L_{\mathrm{wt}}(\pi).$$
$a_β \in \mathbb{C}$. From (18) we get

$$x_{(-\alpha_i^\vee, -p)}(1/b)x_{(\alpha_i^\vee, p)}(a) = (1 - a/b)^{-\alpha_i} x_{(\alpha_i^\vee, p)}(a(1 - a/b))x_{(-\alpha_i^\vee, -p)}(1/(b - a)).$$

By Lemma 4.12,

$$x_{(-\alpha_i^\vee, -p)}(1/(b-a))\tilde{v}_r x_{(-\alpha_i^\vee, -p)}(-1/(b-a))$$

belongs to $N^\vee(\mathfrak{f}_\pi(t_r))$. We write it as a product $\tilde{v}'_r u$ with $(\tilde{v}'_r, u) \in \mathcal{N}^\vee(\pi, t_r) \times N^\vee(\mathfrak{f}_\pi(t_{r+0}))$. By induction, there exists $(v'_\ell) \in \prod_{\ell=r+1}^m \mathcal{N}^\vee(\pi, t_\ell)$ such that

$$v'_{r+1} \cdots v'_m L_{\mathrm{wt}}(\pi) = u x_{(-\alpha_i^\vee, -p)}(1/(b-a)) v_{r+1} \cdots v_m L_{\mathrm{wt}}(\pi).$$

Then

$$x_{(-\alpha_i^\vee, -p)}(1/b)v_r \cdots v_m L_{\mathrm{wt}}(\pi) = (1-a/b)^{-\alpha_i} [x_{(\alpha_i^\vee, p)}(a(1-a/b))\tilde{v}'_r] v'_{r+1} \cdots v'_m L_{\mathrm{wt}}(\pi).$$

Denoting the element in square brackets by $v'_r$, we get the desired expression, up to the inconsequential left multiplication by $(1 - a/b)^{-\alpha_i}$.

Suppose now that there exists $q > p$ such that $(\alpha_i^\vee, q) \in \Phi_a^{\vee,+}(\pi, t_r)$; then $a_{i,p}(v_r) = 0$. We write $v_r = x_{(\alpha_i^\vee, q)}(a)\tilde{v}_r$ where $a \in \mathbb{C}$ and $\tilde{v}_r$ is a product of elements $x_\beta(a_\beta)$ with $\beta \in \Phi_a^{\vee,+}(\pi, t_r) \setminus \{(\alpha_i^\vee, q)\}$ and $a_\beta \in \mathbb{C}$. Let $c$ be a square root in $1 + t\mathcal{O}$ of $1 - (a/b)t^{q-p}$. From (18) we get

$$x_{(-\alpha_i^\vee, -p)}(1/b)x_{(\alpha_i^\vee, q)}(a) = c^{-\alpha_i} x_{(\alpha_i^\vee, q)}(a)x_{(-\alpha_i^\vee, -p)}(1/b)c^{-\alpha_i}.$$

By Lemma 4.12,

$$x_{(-\alpha_i^\vee, -p)}(1/b)(c^{-\alpha_i} \tilde{v}_r c^{\alpha_i}) x_{(-\alpha_i^\vee, -p)}(-1/b)$$

belongs to $N^\vee(\mathfrak{f}_\pi(t_r))$; we write it as a product $\tilde{v}'_r u$ with $(\tilde{v}'_r, u) \in \mathcal{N}^\vee(\pi, t_r) \times N^\vee(\mathfrak{f}_\pi(t_{r+0}))$. By induction, there exists $(v'_\ell) \in \prod_{\ell=r+1}^m \mathcal{N}^\vee(\pi, t_\ell)$ such that

$$v'_{r+1} \cdots v'_m L_{\mathrm{wt}}(\pi) = u x_{(-\alpha_i^\vee, -p)}(1/b) c^{-\alpha_i} v_{r+1} \cdots v_m L_{\mathrm{wt}}(\pi).$$

Then

$$x_{(-\alpha_i^\vee, -p)}(1/b)v_r \cdots v_m L_{\mathrm{wt}}(\pi) = c^{-\alpha_i} [x_{(\alpha_i^\vee, q)}(a)\tilde{v}'_r] v'_{r+1} \cdots v'_m L_{\mathrm{wt}}(\pi).$$

Denoting the element in square brackets by $v'_r$, we get the desired expression, up to the inopportune left multiplication by $c^{-\alpha_i}$. The latter can however be wiped off by a further use of the inductive assumption.

Last, suppose that no affine coroot of direction $\alpha_i^\vee$ occurs in $\Phi_a^{\vee,+}(\pi, t_r)$; then $a_{i,p}(v_r) = 0$. By Lemma 4.12,

$$x_{(-\alpha_i^\vee, -p)}(1/b)v_r x_{(-\alpha_i^\vee, -p)}(-1/b)$$

belongs to $N^\vee(\mathfrak{f}_\pi(t_r))$. We write it as a product $v'_r u$ with $(v'_r, u) \in \mathcal{N}^\vee(\pi, t_r) \times N^\vee(\mathfrak{f}_\pi(t_{r+0}))$. By induction, there exists $(v'_\ell) \in \prod_{\ell=r+1}^m \mathcal{N}^\vee(\pi, t_\ell)$ such that

$$v'_{r+1} \cdots v'_m L_{\mathrm{wt}}(\pi) = u x_{(-\alpha_i^\vee, -p)}(1/b) v_{r+1} \cdots v_m L_{\mathrm{wt}}(\pi).$$
Then

$$x_{(-\alpha_i^{\vee}, -p)}(1/b)v_r \cdots v_m L_{wt}(\pi) = v'_r v'_{r+1} \cdots v'_m L_{wt}(\pi),$$

as desired, which concludes the proof of (iii).

<figure><img src="image_13.png" /></figure>

Let us now consider $i \in I$ and two integral paths $\pi$ and $\eta$ related by the equation $\eta = \tilde{e}_i \pi$. We denote by $p$ the minimum of the function $t \mapsto \langle \alpha_i^{\vee}, \pi(t) \rangle$ over the interval $[0, 1]$ and by $a$ and $b$ the two points in time where $\pi$ is bent to produce $\eta$. Noting that the conditions spelled out in Sect. 4.1 do not uniquely determine $b$, we choose it to be the largest possible: either $b = 1$ or $\langle \alpha_i^{\vee}, \pi(b+h) \rangle > p$ for all small enough $h > 0$.

Let $(t_1, \dots, t_m)$ be the ordered list of all elements in $[0, 1]$ such that $\Phi_a^{\vee,+}(\pi, t) \neq \emptyset$. We set $t_{m+1} = 1$. The set $\Phi_a^{\vee,+}(\pi, a)$ may be empty; if this happens, we insert $a$ in the list $(t_1, \dots, t_m)$, for it will simplify the notation hereafter. On the contrary, the above condition imposed on $b$ ensures that either $b = 1$ or $(\alpha_i^{\vee}, p) \in \Phi_a^{\vee,+}(\pi, b)$, so $b$ automatically appears in the list $(t_1, \dots, t_{m+1})$. We denote by $r$ and $s$ the indices in $\{1, \dots, m+1\}$ such that $a = t_r$ and $b = t_s$. By design $t_r = a < t_{r+1} \le t_s = b$.

**Lemma 4.14.** *Adopt the setting described in the preceding two paragraphs. Choose $(v_\ell) \in \prod_{\ell=1}^m \mathcal{N}^\vee(\pi, t_\ell)$ such that $a_{i,p}(v_s) + \cdots + a_{i,p}(v_\ell) \neq 0$ for each $\ell \in \{s, \dots, m\}$. Then for any $h \in \mathbb{C}^\times$, there exists $(w_\ell) \in \prod_{\ell=1}^m \mathcal{N}^\vee(\eta, t_\ell)$ such that*

$$v_1 \cdots v_{r-1} x_{(-\alpha_i^{\vee}, -p-1)}(h) v_r \cdots v_m L_{wt}(\pi) = w_1 \cdots w_m L_{wt}(\eta).$$

*Proof.* Let $(v_\ell)$ be as in the statement and let $h \in \mathbb{C}^\times$. We set

$$A = v_1 \cdots v_{r-1} \quad \text{and} \quad B = v_r \cdots v_m.$$

We note that $f_{\pi(t_r)} \subseteq H(\alpha_i^{\vee}, p+1)$, so $x(\alpha_i^{\vee}, p+1)(-1/h) \in \mathcal{N}^{\vee}(f_{\pi(t_r)})$.

Using Lemma 4.13 (i), we find $(v'_{r+1}, \dots, v'_m) \in \prod_{\ell=r}^m \mathcal{N}^{\vee}(\pi, t_\ell)$ such that

$$x(\alpha_i^{\vee}, p+1)(-1/h)BL_{wt}(\pi) = v'_r \cdots v'_m L_{wt}(\pi)$$

and $a_{i,p}(v'_\ell) = a_{i,p}(v_\ell)$ for all $\ell \in \{r, \dots, m\}$. We set $c = a_{i,p}(v_s)$ and write $v'_s = x(\alpha_i^{\vee}, p)(c)\tilde{v}'_s$. Then $\tilde{v}'_s \in \mathcal{N}^{\vee}(\pi, t_s)$ and $a_{i,p}(\tilde{v}'_s) = 0$. We also set

$$C = v'_r \cdots v'_{s-1} \quad \text{and} \quad D = \tilde{v}'_s v'_{s+1} \cdots v'_m.$$

Using Lemma 4.13 (iii), we find $(\tilde{v}''_s, v''_{s+1}, \dots, v''_m) \in \prod_{\ell=s}^m \mathcal{N}^{\vee}(\pi, t_\ell)$ such that

$$x_{(-\alpha_i^{\vee}, -p)}(-1/c)DL_{wt}(\pi) = \tilde{v}''_s v''_{s+1} \cdots v''_m L_{wt}(\pi).$$

Last, we set

$$E = x(\alpha_i^{\vee}, p)(c)x(-\alpha_i^{\vee}, -p)(1/c)x(\alpha_i^{\vee}, p)(c),$$

$$F = x(\alpha_i^{\vee}, p)(-c)\tilde{v}''_s v''_{s+1} \cdots v''_m,$$

$$K = x_{(-\alpha_i^{\vee}, -p-1)}(h)x_{(\alpha_i^{\vee}, p+1)}(1/h).$$
Then

$$x_{(-\alpha_i^\vee, -p)}(1/b)v_r \cdots v_m L_{\text{wt}}(\pi) = v'_r v'_{r+1} \cdots v'_m L_{\text{wt}}(\pi),$$

as desired, which concludes the proof of (iii).

<figure><img src="image_12.png" /></figure>

Let us now consider $i \in I$ and two integral paths $\pi$ and $\eta$ related by the equation $\eta = \tilde{e}_i \pi$. We denote by $p$ the minimum of the function $t \mapsto \langle \alpha_i^\vee, \pi(t) \rangle$ over the interval $[0, 1]$ and by $a$ and $b$ the two points in time where $\pi$ is bent to produce $\eta$. Noting that the conditions spelled out in Sect. 4.1 do not uniquely determine $b$, we choose it to be the largest possible: either $b = 1$ or $\langle \alpha_i^\vee, \pi(b+h) \rangle > p$ for all small enough $h > 0$.

Let $(t_1, \dots, t_m)$ be the ordered list of all elements in $[0, 1]$ such that $\Phi_a^{\vee,+}(\pi, t) \neq \emptyset$. We set $t_{m+1} = 1$. The set $\Phi_a^{\vee,+}(\pi, a)$ may be empty; if this happens, we insert $a$ in the list $(t_1, \dots, t_m)$, for it will simplify the notation hereafter. On the contrary, the above condition imposed on $b$ ensures that either $b = 1$ or $(\alpha_i^\vee, p) \in \Phi_a^{\vee,+}(\pi, b)$, so $b$ automatically appears in the list $(t_1, \dots, t_{m+1})$. We denote by $r$ and $s$ the indices in $\{1, \dots, m+1\}$ such that $a = t_r$ and $b = t_s$. By design $t_r = a < t_{r+1} \le t_s = b$.

**Lemma 4.14.** *Adopt the setting described in the preceding two paragraphs. Choose $(v_\ell) \in \prod_{\ell=1}^m \mathcal{N}^\vee(\pi, t_\ell)$ such that $a_{i,p}(v_s) + \cdots + a_{i,p}(v_\ell) \neq 0$ for each $\ell \in \{s, \dots, m\}$. Then for any $h \in \mathbb{C}^\times$, there exists $(w_\ell) \in \prod_{\ell=1}^m \mathcal{N}^\vee(\eta, t_\ell)$ such that*

$$v_1 \cdots v_{r-1} x_{(-\alpha_i^\vee, -p-1)}(h) v_r \cdots v_m L_{\text{wt}}(\pi) = w_1 \cdots w_m L_{\text{wt}}(\eta).$$

*Proof.* Let $(v_\ell)$ be as in the statement and let $h \in \mathbb{C}^\times$. We set

$$A = v_1 \cdots v_{r-1} \quad \text{and} \quad B = v_r \cdots v_m.$$

We note that $f_{\pi(t_r)} \subseteq H(\alpha_i^\vee, p+1)$, so $x(\alpha_i^\vee, p+1)(-1/h) \in \mathcal{N}^\vee(f_{\pi(t_r)})$.

Using Lemma 4.13 (i), we find $(v'_{r+1}, \dots, v'_m) \in \prod_{\ell=r}^m \mathcal{N}^\vee(\pi, t_\ell)$ such that

$$x(\alpha_i^\vee, p+1)(-1/h)BL_{\text{wt}}(\pi) = v'_r \cdots v'_m L_{\text{wt}}(\pi)$$

and $a_{i,p}(v'_\ell) = a_{i,p}(v_\ell)$ for all $\ell \in \{r, \dots, m\}$. We set $c = a_{i,p}(v_s)$ and write $v'_s = x(\alpha_i^\vee, p)(c)\tilde{v}'_s$. Then $\tilde{v}'_s \in \mathcal{N}^\vee(\pi, t_s)$ and $a_{i,p}(\tilde{v}'_s) = 0$. We also set

$$C = v'_r \cdots v'_{s-1} \quad \text{and} \quad D = \tilde{v}'_s v'_{s+1} \cdots v'_m.$$

Using Lemma 4.13 (iii), we find $(\tilde{v}''_s, v''_{s+1}, \dots, v''_m) \in \prod_{\ell=s}^m \mathcal{N}^\vee(\pi, t_\ell)$ such that

$$x_{(-\alpha_i^\vee, -p)}(-1/c)DL_{\text{wt}}(\pi) = \tilde{v}''_s v''_{s+1} \cdots v''_m L_{\text{wt}}(\pi).$$

Last, we set

$$E = x(\alpha_i^\vee, p)(c)x(-\alpha_i^\vee, -p)(1/c)x(\alpha_i^\vee, p)(c),$$

$$F = x(\alpha_i^\vee, p)(-c)\tilde{v}''_s v''_{s+1} \cdots v''_m,$$

$$K = x_{(-\alpha_i^\vee, -p-1)}(h)x_{(\alpha_i^\vee, p+1)}(1/h).$$
Then

$$Ax_{(-\alpha_i^\vee, -p-1)}(h)BL_{wt(\pi)} = AKCEFL_{wt(\pi)}. \quad (21)$$

Observing that

$$\Phi_a^{\vee,+}(\eta, t_\ell) = \begin{cases} \Phi_a^{\vee,+}(\pi, t_\ell) & \text{if } 1 \le \ell < r, \\ \{(\alpha_i^\vee, p+1)\} \sqcup s_{(\alpha_i^\vee, p+1)}(\Phi_a^{\vee,+}(\pi, t_r)) & \text{if } \ell = r, \\ s_{(\alpha_i^\vee, p+1)}(\Phi_a^{\vee,+}(\pi, t_\ell)) & \text{if } r < \ell < s, \\ \tau_{\alpha_i}(\Phi_a^{\vee,+}(\pi, t_\ell)) & \text{if } s \le \ell \le m, \end{cases}$$

we check that the sequence

$$(v_1, \dots, v_{r-1}, x_{(\alpha_i^\vee, p+1)}(-h)(z^{(p+1)\alpha_i} \overline{s_i}) v_r'(z^{(p+1)\alpha_i} \overline{s_i})^{-1}, \\ (z^{(p+1)\alpha_i} \overline{s_i}) v_{r+1}'(z^{(p+1)\alpha_i} \overline{s_i})^{-1}, \dots, (z^{(p+1)\alpha_i} \overline{s_i}) v_{s-1}'(z^{(p+1)\alpha_i} \overline{s_i})^{-1}, \\ z^{\alpha_i} x_{(\alpha_i^\vee, p)}(-c) \tilde{v}_s'' z^{-\alpha_i}, z^{\alpha_i} v_{s+1}'' z^{-\alpha_i}, \dots, z^{\alpha_i} v_m'' z^{-\alpha_i} \quad (22)$$

belongs to $\prod_{\ell=1}^m \mathcal{N}^\vee(\eta, t_\ell)$. In addition, the product of the elements in this sequence is

$$Ax_{(\alpha_i^\vee, p+1)}(-h)(z^{(p+1)\alpha_i} \overline{s_i}) C(z^{(p+1)\alpha_i} \overline{s_i})^{-1} z^{\alpha_i} F z^{-\alpha_i}.$$

We now apply two transformations to the sequence (22): we conjugate the last $m-s+1$ terms by $(-c)^{-\alpha_i}$, and we conjugate the last $m-r+1$ by $h^{-\alpha_i}$. The resulting sequence, denoted by $(w_\ell)$, still belongs to $\prod_{\ell=1}^m \mathcal{N}^\vee(\eta, t_\ell)$, because all our constructions are $T^\vee(\mathbb{C})$-equivariant.

Observing that

$$K = h^{-\alpha_i} x_{(\alpha_i^\vee, p+1)}(-h)(z^{(p+1)\alpha_i} \overline{s_i}) \quad \text{and} \quad E = (z^{(p+1)\alpha_i} \overline{s_i})^{-1} (-c)^{-\alpha_i} z^{\alpha_i}$$

(see (19)), we obtain

$$w_1 \cdots w_m = AKCEF z^{-\alpha_i} (-ch)^{\alpha_i},$$

and a comparison with (21) yields

$$Ax_{(-\alpha_i^\vee, -p-1)}(h)BL_{wt(\pi)} = AKCEF z^{-\alpha_i} L_{wt(\eta)} = w_1 \cdots w_m L_{wt(\eta)},$$

as desired.

We can now prove Proposition 4.5 (iv). We consider the situation

$$\eta_1 \otimes \cdots \otimes \eta_n = \tilde{e}_i (\pi_1 \otimes \cdots \otimes \pi_n)$$

in the crystal $\Pi^{\otimes n}$, and our aim is to show that $\mathring{\mathbb{Z}}(\pi_1 \otimes \cdots \otimes \pi_n)$ is contained in the closure of $\mathring{\mathbb{Z}}(\eta_1 \otimes \cdots \otimes \eta_n)$ in $\mathrm{Gr}_n$.

As in the proof of Proposition 4.5 (iii), we regard the concatenation $\pi = \pi_1 * \cdots * \pi_n$ as a map from $[0, n]$ to $\Lambda_{\mathbb{R}}$, each path $\pi_1, \dots, \pi_n$ being travelled at nominal speed, and
Then

$$Ax_{(-\alpha_i^\vee, -p-1)}(h)BL_{wt(\pi)} = AKCEFL_{wt(\pi)}. \quad (21)$$

Observing that

$$\Phi_a^{\vee,+}(\eta, t_\ell) = \begin{cases} \Phi_a^{\vee,+}(\pi, t_\ell) & \text{if } 1 \le \ell < r, \\ \{(\alpha_i^\vee, p+1)\} \sqcup s_{(\alpha_i^\vee, p+1)}(\Phi_a^{\vee,+}(\pi, t_r)) & \text{if } \ell = r, \\ s_{(\alpha_i^\vee, p+1)}(\Phi_a^{\vee,+}(\pi, t_\ell)) & \text{if } r < \ell < s, \\ \tau_{\alpha_i}(\Phi_a^{\vee,+}(\pi, t_\ell)) & \text{if } s \le \ell \le m, \end{cases}$$

we check that the sequence

$$(v_1, \dots, v_{r-1}, x_{(\alpha_i^\vee, p+1)}(-h)(z^{(p+1)\alpha_i} \overline{s_i}) v_r'(z^{(p+1)\alpha_i} \overline{s_i})^{-1}, \\ (z^{(p+1)\alpha_i} \overline{s_i}) v_{r+1}'(z^{(p+1)\alpha_i} \overline{s_i})^{-1}, \dots, (z^{(p+1)\alpha_i} \overline{s_i}) v_{s-1}'(z^{(p+1)\alpha_i} \overline{s_i})^{-1}, \\ z^{\alpha_i} x_{(\alpha_i^\vee, p)}(-c) \tilde{v}_s'' z^{-\alpha_i}, z^{\alpha_i} v_{s+1}'' z^{-\alpha_i}, \dots, z^{\alpha_i} v_m'' z^{-\alpha_i} \quad (22)$$

belongs to $\prod_{\ell=1}^m \mathcal{N}^\vee(\eta, t_\ell)$. In addition, the product of the elements in this sequence is

$$Ax_{(\alpha_i^\vee, p+1)}(-h)(z^{(p+1)\alpha_i} \overline{s_i}) C(z^{(p+1)\alpha_i} \overline{s_i})^{-1} z^{\alpha_i} F z^{-\alpha_i}.$$

We now apply two transformations to the sequence (22): we conjugate the last $m-s+1$ terms by $(-c)^{-\alpha_i}$, and we conjugate the last $m-r+1$ by $h^{-\alpha_i}$. The resulting sequence, denoted by $(w_\ell)$, still belongs to $\prod_{\ell=1}^m \mathcal{N}^\vee(\eta, t_\ell)$, because all our constructions are $T^\vee(\mathbb{C})$-equivariant.

Observing that

$$K = h^{-\alpha_i} x_{(\alpha_i^\vee, p+1)}(-h)(z^{(p+1)\alpha_i} \overline{s_i}) \quad \text{and} \quad E = (z^{(p+1)\alpha_i} \overline{s_i})^{-1} (-c)^{-\alpha_i} z^{\alpha_i}$$

(see (19)), we obtain

$$w_1 \cdots w_m = AKCEF z^{-\alpha_i} (-ch)^{\alpha_i},$$

and a comparison with (21) yields

$$Ax_{(-\alpha_i^\vee, -p-1)}(h)BL_{wt(\pi)} = AKCEF z^{-\alpha_i} L_{wt(\eta)} = w_1 \cdots w_m L_{wt(\eta)},$$

as desired.

We can now prove Proposition 4.5 (iv). We consider the situation

$$\eta_1 \otimes \cdots \otimes \eta_n = \tilde{e}_i (\pi_1 \otimes \cdots \otimes \pi_n)$$

in the crystal $\Pi^{\otimes n}$, and our aim is to show that $\mathring{\mathbb{Z}}(\pi_1 \otimes \cdots \otimes \pi_n)$ is contained in the closure of $\mathring{\mathbb{Z}}(\eta_1 \otimes \cdots \otimes \eta_n)$ in $\mathrm{Gr}_n$.

As in the proof of Proposition 4.5 (iii), we regard the concatenation $\pi = \pi_1 * \cdots * \pi_n$ as a map from $[0, n]$ to $\Lambda_{\mathbb{R}}$, each path $\pi_1, \dots, \pi_n$ being travelled at nominal speed, and
the same for $\eta = \eta_1 * \cdots * \eta_n$. Thus, for each $j \in \{1, \dots, n\}$ the restriction of $\pi$ to the interval $[j-1, j]$ is $\pi_j$, up to the obvious shifts in time and space.

By Proposition 4.2 (ii), we have $\eta = \tilde{\epsilon}_i \pi$. We denote by $a$ and $b$ the two points in time where $\pi$ is bent to produce $\eta$. Let $(t_1, \dots, t_m)$ be the ordered list of all elements in $[0, n]$ such that $\Phi_a^{\vee,+}(\pi, t) \neq \emptyset$. We insert $a$ in this list if it does not already appear there. We set $t_0 = 0$ and $t_{m+1} = n$. We denote by $r$ and $s$ the indices in $\{1, \dots, m+1\}$ such that $a = t_r$ and $b = t_s$.

There is a unique integer $k \in \{1, \dots, n\}$ such that $a$ and $b$ both belong to $[k-1, k]$. Plainly, $\eta_k = \tilde{\epsilon}_i \pi_k$ and $\eta_j = \pi_j$ for all $j \in \{1, \dots, n\} \setminus \{k\}$. We record that $\eta_1 * \cdots * \eta_j = \tilde{\epsilon}_i (\pi_1 * \cdots * \pi_j)$ if $j \in \{k, \dots, n\}$.

For $j \in \{1, \dots, n\}$, we set $v_j = \mathrm{wt}(\pi_j) + \cdots + \mathrm{wt}(\pi_j)$ and denote by $m_j$ the largest $\ell \in \{0, \dots, m\}$ such that $t_\ell \in [0, j]$. Then $\mathring{\mathbb{Z}}(\pi_1 \otimes \cdots \otimes \pi_n)$ is the set of all elements

$$ \left[ \left( \prod_{\ell=1}^{m_1} v_\ell \right) z^{v_1}, z^{-v_1} \left( \prod_{\ell=m_1+1}^{m_2} v_\ell \right) z^{v_2}, \dots, z^{-v_{n-1}} \left( \prod_{\ell=m_{n-1}+1}^{m_n} v_\ell \right) z^{v_n} \right] \quad (23) $$

with $(v_\ell) \in \prod_{\ell=1}^m \mathcal{N}^\vee(\pi, t_\ell)$.

Now assume that $(v_\ell)$ is chosen so that $\mathbf{a}_{i,p}(v_s) + \cdots + \mathbf{a}_{i,p}(v_\ell) \neq 0$ for each $\ell \in \{s, \dots, m\}$ and pick $h \in \mathbb{C}^\times$. Lemma 4.14 provides a sequence $(w_\ell) \in \prod_{\ell=1}^m \mathcal{N}^\vee(\eta, t_\ell)$ such that

$$ v_1 \cdots v_{r-1} x(-\alpha_i^{\vee}, -p-1)(h) v_r \cdots v_m L_{\mathrm{wt}(\pi)} = w_1 \cdots w_m L_{\mathrm{wt}(\eta)}. $$

However, $(w_\ell)$ satisfies more equations: for $j \in \{1, \dots, n\}$, we have

$$ \left\{ \begin{array}{ll} v_1 \cdots v_{m_j} L_{v_j} = w_1 \cdots w_{m_j} L_{v_j} & \text{if } j < k, \\ v_1 \cdots v_{r-1} x(-\alpha_i^{\vee}, -p-1)(h) v_r \cdots v_{m_j} L_{v_j} = w_1 \cdots w_{m_j} L_{v_j + \alpha_i} & \text{if } j \ge k, \end{array} \right. \quad (24) $$

in the first case because $w_\ell = v_\ell$ for all $\ell \in \{1, \dots, m_{k-1}\}$, in the second case because Lemma 4.14 would have returned the subsequence $(w_\ell)_{1 \le \ell \le m_j}$ if we had fed it with the paths $\pi_1 * \cdots * \pi_j$ and $\eta_1 * \cdots * \eta_j$ and the datum $(v_\ell)_{1 \le \ell \le m_j}$ and $h$.

The system (24) translates to a single equation in $\mathrm{Gr}_n$, which demonstrates that the element obtained by inserting $x(-\alpha_i^{\vee}, -p-1)(h)$ just before $v_r$ in (23) belongs to the set $\mathring{\mathbb{Z}}(\eta_1 \otimes \cdots \otimes \eta_n)$. Letting $h$ tend to 0, we conclude that (23) lies in the closure of this set. To be sure, this conclusion has been reached under the assumption that $\mathbf{a}_{i,p}(v_s) + \cdots + \mathbf{a}_{i,p}(v_\ell) \neq 0$ for each $\ell \in \{s, \dots, m\}$, but this restriction can be removed by a small perturbation of $\mathbf{a}_{i,p}(v_s)$.

Thus, Proposition 4.5 (iv) is, at last, fully proven.

## 5. Comparison with the tensor product basis

We keep the notation from Sect. 2. Let $\lambda = (\lambda_1, \dots, \lambda_n)$ in $(\Lambda^+)^n$. The tensor product $V(\lambda)$ can be endowed on the one hand with its MV basis (Sect. 2.3), on the other hand
the same for $\eta = \eta_1 * \cdots * \eta_n$. Thus, for each $j \in \{1, \dots, n\}$ the restriction of $\pi$ to the interval $[j-1, j]$ is $\pi_j$, up to the obvious shifts in time and space.

By Proposition 4.2 (ii), we have $\eta = \tilde{e}_i \pi$. We denote by $a$ and $b$ the two points in time where $\pi$ is bent to produce $\eta$. Let $(t_1, \dots, t_m)$ be the ordered list of all elements in $[0, n]$ such that $\Phi_a^{\vee,+}(\pi, t) \neq \emptyset$. We insert $a$ in this list if it does not already appear there. We set $t_0 = 0$ and $t_{m+1} = n$. We denote by $r$ and $s$ the indices in $\{1, \dots, m+1\}$ such that $a = t_r$ and $b = t_s$.

There is a unique integer $k \in \{1, \dots, n\}$ such that $a$ and $b$ both belong to $[k-1, k]$. Plainly, $\eta_k = \tilde{e}_i \pi_k$ and $\eta_j = \pi_j$ for all $j \in \{1, \dots, n\} \setminus \{k\}$. We record that $\eta_1 * \cdots * \eta_j = \tilde{e}_i (\pi_1 * \cdots * \pi_j)$ if $j \in \{k, \dots, n\}$.

For $j \in \{1, \dots, n\}$, we set $v_j = \mathrm{wt}(\pi_j) + \dots + \mathrm{wt}(\pi_j)$ and denote by $m_j$ the largest $\ell \in \{0, \dots, m\}$ such that $t_\ell \in [0, j]$. Then $\mathring{\mathbb{Z}}(\pi_1 \otimes \dots \otimes \pi_n)$ is the set of all elements

$$ \left[ \left( \prod_{\ell=1}^{m_1} v_\ell \right) z^{v_1}, z^{-v_1} \left( \prod_{\ell=m_1+1}^{m_2} v_\ell \right) z^{v_2}, \dots, z^{-v_{n-1}} \left( \prod_{\ell=m_{n-1}+1}^{m_n} v_\ell \right) z^{v_n} \right] \quad (23) $$

with $(v_\ell) \in \prod_{\ell=1}^m \mathcal{N}^\vee(\pi, t_\ell)$.

Now assume that $(v_\ell)$ is chosen so that $\mathbf{a}_{i,p}(v_s) + \dots + \mathbf{a}_{i,p}(v_\ell) \neq 0$ for each $\ell \in \{s, \dots, m\}$ and pick $h \in \mathbb{C}^\times$. Lemma 4.14 provides a sequence $(w_\ell) \in \prod_{\ell=1}^m \mathcal{N}^\vee(\eta, t_\ell)$ such that

$$ v_1 \cdots v_{r-1} x(-\alpha_i^{\vee}, -p-1)(h) v_r \cdots v_m L_{\mathrm{wt}(\pi)} = w_1 \cdots w_m L_{\mathrm{wt}(\eta)}. $$

However, $(w_\ell)$ satisfies more equations: for $j \in \{1, \dots, n\}$, we have

$$ \left\{ \begin{array}{ll} v_1 \cdots v_{m_j} L_{v_j} = w_1 \cdots w_{m_j} L_{v_j} & \text{if } j < k, \\ v_1 \cdots v_{r-1} x(-\alpha_i^{\vee}, -p-1)(h) v_r \cdots v_{m_j} L_{v_j} = w_1 \cdots w_{m_j} L_{v_j} + \alpha_i & \text{if } j \ge k, \end{array} \right. \quad (24) $$

in the first case because $w_\ell = v_\ell$ for all $\ell \in \{1, \dots, m_{k-1}\}$, in the second case because Lemma 4.14 would have returned the subsequence $(w_\ell)_{1 \le \ell \le m_j}$ if we had fed it with the paths $\pi_1 * \cdots * \pi_j$ and $\eta_1 * \cdots * \eta_j$ and the datum $(v_\ell)_{1 \le \ell \le m_j}$ and $h$.

The system (24) translates to a single equation in $\mathrm{Gr}_n$, which demonstrates that the element obtained by inserting $x(-\alpha_i^{\vee}, -p-1)(h)$ just before $v_r$ in (23) belongs to the set $\mathring{\mathbb{Z}}(\eta_1 \otimes \cdots \otimes \eta_n)$. Letting $h$ tend to 0, we conclude that (23) lies in the closure of this set. To be sure, this conclusion has been reached under the assumption that $\mathbf{a}_{i,p}(v_s) + \dots + \mathbf{a}_{i,p}(v_\ell) \neq 0$ for each $\ell \in \{s, \dots, m\}$, but this restriction can be removed by a small perturbation of $\mathbf{a}_{i,p}(v_s)$.

Thus, Proposition 4.5 (iv) is, at last, fully proven.

## 5. Comparison with the tensor product basis

We keep the notation from Sect. 2. Let $\lambda = (\lambda_1, \dots, \lambda_n)$ in $(\Lambda^+)^n$. The tensor product $V(\lambda)$ can be endowed on the one hand with its MV basis (Sect. 2.3), on the other hand
with the tensor product of the MV bases of the factors $V(\lambda_1), \dots, V(\lambda_n)$. In this section, we compare these two bases through the explicit identification

$$F(\mathcal{I}_{\lambda_1} * \dots * \mathcal{I}_{\lambda_n}) \cong F(\mathcal{I}_{\lambda_1}) \otimes \dots \otimes F(\mathcal{I}_{\lambda_n})$$

afforded by Beilinson and Drinfeld's fusion product. We show that the transition matrix is upper unitriangular and that its entries are intersection multiplicities. The order relation needed to convey the triangularity involves the inclusion of cycles.

### 5.1. Deformations

The Beilinson–Drinfeld Grassmannian $\mathcal{G}r^{BD}$ is a relative version of the affine Grassmannian where the base is the space of effective divisors on a smooth curve. The choice of the affine line amply satisfies our needs and offers three advantages: there is a natural global coordinate on $\mathbb{A}^1$, every $G$-torsor on $\mathbb{A}^1$ is trivializable, and the monodromy of any local system is trivial. Rather than looking for more generality, we will pragmatically stick with this choice. Consistent with Sect. 2, the coordinate on $\mathbb{A}^1$ is denoted by $z$.

Formally, the Beilinson–Drinfeld Grassmannian $\mathcal{G}r_n^{BD}$ is defined as the functor on the category of commutative $\mathbb{C}$-algebras that assigns to an algebra $R$ the set of isomorphism classes of triples $(x_1, \dots, x_n; \mathcal{F}, \beta)$, where $(x_1, \dots, x_n) \in \mathbb{A}^n(R)$, $\mathcal{F}$ is a $G^\vee$-torsor over $\mathbb{A}_R^1$ and $\beta$ is a trivialization of $\mathcal{F}$ away from the points $x_1, \dots, x_n$ ([6, Sect. 5.3.10]; [43, Definition 3.3]; [46, Definition 3.1.1]). We denote by $\pi: \mathcal{G}r_n^{BD} \to \mathbb{A}^n$ the morphism to the base which forgets $\mathcal{F}$ and $\beta$. It is known that $\mathcal{G}r_n^{BD}$ is representable by an ind-scheme and $\pi$ is ind-proper.

We are only interested in the set of $\mathbb{C}$-points, endowed with its ind-variety structure. Using a trivialization of $\mathcal{F}$, we can thus adopt the following simplified definition: $\mathcal{G}r_n^{BD}$ is the set of pairs $(x_1, \dots, x_n; [\beta])$, where $(x_1, \dots, x_n) \in \mathbb{C}^n$ and $[\beta]$ belongs to the homogeneous space

$$G^{\vee}(\mathbb{C}[z, (z-x_1)^{-1}, \dots, (z-x_n)^{-1}]) / G^{\vee}(\mathbb{C}[z]).$$

This set is endowed with the structure of an ind-variety.

**Example 5.1** ([6, Remark in Sect. 5.3.10]). We consider the case $G^\vee = \text{GL}_N$. Here the datum of $[\beta]$ is equivalent to the datum of the $\mathbb{C}[z]$-lattice $\beta(L_0)$ in $\mathbb{C}(z)^N$, where $L_0 = \mathbb{C}[z]^N$ is the standard lattice. Let us write $\mathbf{x}$ for the point $(x_1, \dots, x_n)$ and set $f_{\mathbf{x}} = (z-x_1)\cdots(z-x_n)$. Then a lattice $L$ is of this form $\beta(L_0)$ if and only if there exists a positive integer $k$ such that $f_{\mathbf{x}}^k L_0 \subseteq L \subseteq f_{\mathbf{x}}^{-k} L_0$. For each positive integer $k$, define $(\mathcal{G}r_n^{BD})_k$ to be the subset of $\mathcal{G}r_n^{BD}$ consisting of all pairs $(\mathbf{x}; L)$ with $f_{\mathbf{x}}^k L_0 \subseteq L \subseteq f_{\mathbf{x}}^{-k} L_0$. We identify $\mathbb{C}[z]/(f_{\mathbf{x}}^{2k})$ with the vector space $V$ of polynomials of degree strictly less than $2kn$, and subsequently identify $L_0/f_{\mathbf{x}}^{2k} L_0$ with $V^N$. The space $(\mathcal{G}r_n^{BD})_k$ can then be realized as a Zariski-closed subset of

$$\mathbb{C}^n \times \bigcup_{d=0}^{2knN} \mathbb{G}_d(V^N)$$
with the tensor product of the MV bases of the factors $V(\lambda_1), \dots, V(\lambda_n)$. In this section, we compare these two bases through the explicit identification

$$F(\mathcal{I}_{\lambda_1} * \dots * \mathcal{I}_{\lambda_n}) \cong F(\mathcal{I}_{\lambda_1}) \otimes \dots \otimes F(\mathcal{I}_{\lambda_n})$$

afforded by Beilinson and Drinfeld's fusion product. We show that the transition matrix is upper unitriangular and that its entries are intersection multiplicities. The order relation needed to convey the triangularity involves the inclusion of cycles.

### 5.1. Deformations

The Beilinson–Drinfeld Grassmannian $\mathcal{G}r^{\mathrm{BD}}$ is a relative version of the affine Grassmannian where the base is the space of effective divisors on a smooth curve. The choice of the affine line amply satisfies our needs and offers three advantages: there is a natural global coordinate on $\mathbb{A}^1$, every $G$-torsor on $\mathbb{A}^1$ is trivializable, and the monodromy of any local system is trivial. Rather than looking for more generality, we will pragmatically stick with this choice. Consistent with Sect. 2, the coordinate on $\mathbb{A}^1$ is denoted by $z$.

Formally, the Beilinson–Drinfeld Grassmannian $\mathcal{G}r_n^{\mathrm{BD}}$ is defined as the functor on the category of commutative $\mathbb{C}$-algebras that assigns to an algebra $R$ the set of isomorphism classes of triples $(x_1, \dots, x_n; \mathcal{F}, \beta)$, where $(x_1, \dots, x_n) \in \mathbb{A}^n(R)$, $\mathcal{F}$ is a $G^\vee$-torsor over $\mathbb{A}_R^1$ and $\beta$ is a trivialization of $\mathcal{F}$ away from the points $x_1, \dots, x_n$ ([6, Sect. 5.3.10]; [43, Definition 3.3]; [46, Definition 3.1.1]). We denote by $\pi : \mathcal{G}r_n^{\mathrm{BD}} \to \mathbb{A}^n$ the morphism to the base which forgets $\mathcal{F}$ and $\beta$. It is known that $\mathcal{G}r_n^{\mathrm{BD}}$ is representable by an ind-scheme and $\pi$ is ind-proper.

We are only interested in the set of $\mathbb{C}$-points, endowed with its ind-variety structure. Using a trivialization of $\mathcal{F}$, we can thus adopt the following simplified definition: $\mathcal{G}r_n^{\mathrm{BD}}$ is the set of pairs $(x_1, \dots, x_n; [\beta])$, where $(x_1, \dots, x_n) \in \mathbb{C}^n$ and $[\beta]$ belongs to the homogeneous space

$$G^{\vee}(\mathbb{C}[z, (z-x_1)^{-1}, \dots, (z-x_n)^{-1}]) / G^{\vee}(\mathbb{C}[z]).$$

This set is endowed with the structure of an ind-variety.

**Example 5.1** ([6, Remark in Sect. 5.3.10]). We consider the case $G^{\vee} = \mathrm{GL}_N$. Here the datum of $[\beta]$ is equivalent to the datum of the $\mathbb{C}[z]$-lattice $\beta(L_0)$ in $\mathbb{C}(z)^N$, where $L_0 = \mathbb{C}[z]^N$ is the standard lattice. Let us write $\mathbf{x}$ for the point $(x_1, \dots, x_n)$ and set $f_{\mathbf{x}} = (z-x_1)\cdots(z-x_n)$. Then a lattice $L$ is of this form $\beta(L_0)$ if and only if there exists a positive integer $k$ such that $f_{\mathbf{x}}^k L_0 \subseteq L \subseteq f_{\mathbf{x}}^{-k} L_0$. For each positive integer $k$, define $(\mathcal{G}r_n^{\mathrm{BD}})_k$ to be the subset of $\mathcal{G}r_n^{\mathrm{BD}}$ consisting of all pairs $(\mathbf{x}; L)$ with $f_{\mathbf{x}}^k L_0 \subseteq L \subseteq f_{\mathbf{x}}^{-k} L_0$. We identify $\mathbb{C}[z]/(f_{\mathbf{x}}^{2k})$ with the vector space $V$ of polynomials of degree strictly less than $2kn$, and subsequently identify $L_0/f_{\mathbf{x}}^{2k} L_0$ with $V^N$. The space $(\mathcal{G}r_n^{\mathrm{BD}})_k$ can then be realized as a Zariski-closed subset of

$$\mathbb{C}^n \times \bigcup_{d=0}^{2knN} \mathrm{G}_d(V^N)$$
where $\mathbb{G}_d(V^N)$ denotes the Grassmannian of $d$-planes in $V^N$. In this way, $\mathcal{G}r_n^{\text{BD}}$ is the inductive limit of a system of algebraic varieties and closed embeddings, in other words, an ind-variety.

We also want to deform the $n$-fold convolution variety $\text{Gr}_n$. Accordingly, we define $\mathcal{G}r_n$ as the set of pairs $(x_1, \dots, x_n; [\beta_1, \dots, \beta_n])$, where $(x_1, \dots, x_n) \in \mathbb{C}^n$ and $[\beta_1, \dots, \beta_n]$ belongs to

$$G^{\vee}(\mathbb{C}[z, (z-x_1)^{-1}]) \times^{G^{\vee}(\mathbb{C}[z])} \dots \times^{G^{\vee}(\mathbb{C}[z])} G^{\vee}(\mathbb{C}[z, (z-x_n)^{-1}]) / G^{\vee}(\mathbb{C}[z])$$

(see [43, Definition 3.8] or [46, (3.1.21)]). The set $\mathcal{G}r_n$ is endowed with the structure of an ind-variety; it comes with a map $m_n: \mathcal{G}r_n \to \mathcal{G}r_n^{\text{BD}}$ defined by

$$m_n(x_1, \dots, x_n; [\beta_1, \dots, \beta_n]) = (x_1, \dots, x_n; [\beta_1 \cdots \beta_n]).$$

**Example 5.2.** We again consider the case $G^{\vee} = \text{GL}_N$. Then an element in $\mathcal{G}r_n$ is the datum of a point $(x_1, \dots, x_n) \in \mathbb{C}^n$ and a sequence $(L_1, \dots, L_n)$ of $\mathbb{C}[z]$-lattices in $\mathbb{C}(z)^N$ for which there exists a positive integer $k$ such that

$$(z - x_j)^k L_{j-1} \subseteq L_j \subseteq (z - x_j)^{-k} L_{j-1}$$

for all $j \in \{1, \dots, n\}$; here again $L_0 = \mathbb{C}[z]^N$ is the standard lattice and $L_j = (\beta_1 \cdots \beta_j)(L_0)$.

In the above example, we can partition $\mathcal{G}r_n$ into cells by specifying the relative positions of the pairs of lattices $(L_{j-1}, L_j)$ in terms of invariant factors. This construction can be generalized to an arbitrary group $G$ as follows: given $\lambda = (\lambda_1, \dots, \lambda_n)$ in $(\Lambda^+)^n$, we define $\mathcal{G}r_n^{\lambda}$ as the subset of $\mathcal{G}r_n$ consisting of all pairs $(x_1, \dots, x_n; [\beta_1, \dots, \beta_n])$ with

$$\beta_j \in G^{\vee}(\mathbb{C}[z]) (z - x_j)^{\lambda_j} G^{\vee}(\mathbb{C}[z])$$

for $j \in \{1, \dots, n\}$. The Cartan decomposition

$$G^{\vee}(\mathbb{C}[z, (z-x_j)^{-1}]) = \bigsqcup_{\lambda_j \in \Lambda^{+}} G^{\vee}(\mathbb{C}[z]) (z-x_j)^{\lambda_j} G^{\vee}(\mathbb{C}[z])$$

yields

$$\mathcal{G}r_n = \bigsqcup_{\lambda \in (\Lambda^+)^n} \mathcal{G}r_n^{\lambda}$$

and it can be checked that

$$\overline{\mathcal{G}r_n^{\lambda}} = \bigsqcup_{\substack{\mu \in (\Lambda^+)^n \\ \mu_1 \le \lambda_1, \dots, \mu_n \le \lambda_n}} \mathcal{G}r_n^{\mu}. \qquad (25)$$

In addition, the maps $(x_1, \dots, x_j; [\beta_1, \dots, \beta_j]) \mapsto (x_1, \dots, x_{j-1}; [\beta_1, \dots, \beta_{j-1}])$ exhibit $\mathcal{G}r_n^{\lambda}$ as the total space of an iterated fibration with base $\mathcal{G}r_1^{\lambda_1}$ and successive
where $\mathbb{G}_d(V^N)$ denotes the Grassmannian of $d$-planes in $V^N$. In this way, $\mathcal{G}r_n^{BD}$ is the inductive limit of a system of algebraic varieties and closed embeddings, in other words, an ind-variety.

We also want to deform the $n$-fold convolution variety $Gr_n$. Accordingly, we define $\mathcal{G}r_n$ as the set of pairs $(x_1, \dots, x_n; [\beta_1, \dots, \beta_n])$, where $(x_1, \dots, x_n) \in \mathbb{C}^n$ and $[\beta_1, \dots, \beta_n]$ belongs to

$$G^{\vee}(\mathbb{C}[z, (z-x_1)^{-1}]) \times^{G^{\vee}(\mathbb{C}[z])} \dots \times^{G^{\vee}(\mathbb{C}[z])} G^{\vee}(\mathbb{C}[z, (z-x_n)^{-1}]) / G^{\vee}(\mathbb{C}[z])$$

(see [43, Definition 3.8] or [46, (3.1.21)]). The set $\mathcal{G}r_n$ is endowed with the structure of an ind-variety; it comes with a map $m_n: \mathcal{G}r_n \to \mathcal{G}r_n^{BD}$ defined by

$$m_n(x_1, \dots, x_n; [\beta_1, \dots, \beta_n]) = (x_1, \dots, x_n; [\beta_1 \cdots \beta_n]).$$

**Example 5.2.** We again consider the case $G^{\vee} = \text{GL}_N$. Then an element in $\mathcal{G}r_n$ is the datum of a point $(x_1, \dots, x_n) \in \mathbb{C}^n$ and a sequence $(L_1, \dots, L_n)$ of $\mathbb{C}[z]$-lattices in $\mathbb{C}(z)^N$ for which there exists a positive integer $k$ such that

$$(z - x_j)^k L_{j-1} \subseteq L_j \subseteq (z - x_j)^{-k} L_{j-1}$$

for all $j \in \{1, \dots, n\}$; here again $L_0 = \mathbb{C}[z]^N$ is the standard lattice and $L_j = (\beta_1 \cdots \beta_j)(L_0)$.

In the above example, we can partition $\mathcal{G}r_n$ into cells by specifying the relative positions of the pairs of lattices $(L_{j-1}, L_j)$ in terms of invariant factors. This construction can be generalized to an arbitrary group $G$ as follows: given $\lambda = (\lambda_1, \dots, \lambda_n)$ in $(\Lambda^+)^n$, we define $\mathcal{G}r_n^{\lambda}$ as the subset of $\mathcal{G}r_n$ consisting of all pairs $(x_1, \dots, x_n; [\beta_1, \dots, \beta_n])$ with

$$\beta_j \in G^{\vee}(\mathbb{C}[z]) (z - x_j)^{\lambda_j} G^{\vee}(\mathbb{C}[z])$$

for $j \in \{1, \dots, n\}$. The Cartan decomposition

$$G^{\vee}(\mathbb{C}[z, (z-x_j)^{-1}]) = \bigsqcup_{\lambda_j \in \Lambda^{+}} G^{\vee}(\mathbb{C}[z]) (z-x_j)^{\lambda_j} G^{\vee}(\mathbb{C}[z])$$

yields

$$\mathcal{G}r_n = \bigsqcup_{\lambda \in (\Lambda^+)^n} \mathcal{G}r_n^{\lambda}$$

and it can be checked that

$$\overline{\mathcal{G}r_n^{\lambda}} = \bigsqcup_{\substack{\mu \in (\Lambda^+)^n \\ \mu_1 \le \lambda_1, \dots, \mu_n \le \lambda_n}} \mathcal{G}r_n^{\mu}. \qquad (25)$$

In addition, the maps $(x_1, \dots, x_j; [\beta_1, \dots, \beta_j]) \mapsto (x_1, \dots, x_{j-1}; [\beta_1, \dots, \beta_{j-1}])$ exhibit $\mathcal{G}r_n^{\lambda}$ as the total space of an iterated fibration with base $\mathcal{G}r_1^{\lambda_1}$ and successive
fibers $\mathcal{G}r_1^{\lambda_2}, \dots, \mathcal{G}r_1^{\lambda_n}$. It follows that $\mathcal{G}r_n^{\lambda}$ is a smooth connected variety of dimension $2\rho(|\lambda|) + n$.

Let us now investigate the fibers of the map $\pi \circ m_n : \mathcal{G}r_n \to \mathbb{C}^n$. Given $x \in \mathbb{C}$, we set $\mathcal{O}_x = \mathbb{C}[z-x]$ and $\mathcal{K}_x = \mathbb{C}((z-x))$. We identify $\mathcal{O}$ and $\mathcal{K}$ with $\mathcal{O}_x$ and $\mathcal{K}_x$ by means of the map $z \mapsto z-x$.

We fix $\mathbf{x} = (x_1, \dots, x_n)$ in $\mathbb{C}^n$. Let $\mathrm{supp}(\mathbf{x})$ be the set of values $y \in \mathbb{C}$ that appear in the tuple $\mathbf{x}$. For $y \in \mathrm{supp}(\mathbf{x})$, denote by $m_y$ the number of indices $j \in \{1, \dots, n\}$ such that $x_j = y$ and choose an increasing sequence $(p_0 = 0, p_1, p_2, \dots, p_{m_y} = n)$ in such a way that each interval $[p_{k-1} + 1, p_k]$ contains exactly one index $j$ such that $x_j = y$. For $\boldsymbol{\beta} = [\beta_1, \dots, \beta_n]$ in the fiber

$$ (\mathcal{G}r_n)_x = G^\vee(\mathbb{C}[z, (z-x_1)^{-1}]) \times^{G^\vee(\mathbb{C}[z])} \dots \times^{G^\vee(\mathbb{C}[z])} G^\vee(\mathbb{C}[z, (z-x_n)^{-1}]) / G^\vee(\mathbb{C}[z]), $$
we define $\Theta(\boldsymbol{\beta})_y$ as the point $[(\beta_1 \cdots \beta_{p_1}), (\beta_{p_1+1} \cdots \beta_{p_2}), \dots, (\beta_{p_{m_y-1}+1} \cdots \beta_n)]$ in

$$ \underbrace{G^\vee(\mathcal{K}_y) \times^{G^\vee(\mathcal{O}_y)} \dots \times^{G^\vee(\mathcal{O}_y)} G^\vee(\mathcal{K}_y)}_{m_y \text{ factors } G^\vee(\mathcal{K}_y)} / G^\vee(\mathcal{O}_y) \cong \mathrm{Gr}_{m_y} $$

(note that $\Theta(\boldsymbol{\beta})_y$ does not depend on this choice, because $\beta_j \in G^\vee(\mathcal{O}_y)$ if $x_j \neq y$).

**Proposition 5.3.** *The map $\boldsymbol{\beta} \mapsto (\Theta(\boldsymbol{\beta})_y)$ is a bijection*

$$ (\mathcal{G}r_n)_x \xrightarrow{\approx} \prod_{y \in \mathrm{supp}(\mathbf{x})} \mathrm{Gr}_{m_y}. $$

*Proof.* Combining the Iwasawa decomposition (1) with the easily proven equality

$$ N^{\vee}(\mathcal{K}_x) = N^{\vee}(\mathbb{C}[z, (z-x)^{-1}]) N^{\vee}(\mathcal{O}_x), \qquad (26) $$

we obtain the well-known equality

$$ G^{\vee}(\mathcal{K}_x) = G^{\vee}(\mathbb{C}[z, (z-x)^{-1}]) G^{\vee}(\mathcal{O}_x), $$

for each $x \in \mathbb{C}$.

The case $n = 1$ of the proposition is banal. Assume that $n \ge 2$, and for $y \in \mathrm{supp}(\mathbf{x})$, pick $\boldsymbol{\gamma}_y \in \mathrm{Gr}_{m_y}$. Set $\mathbf{x}' = (x_1, \dots, x_{n-1})$ and $m = m_{x_n}$, and write $\boldsymbol{\gamma}_{x_n} = [\gamma_1, \dots, \gamma_m]$. Reasoning by induction on $n$, we know that there is a unique $\boldsymbol{\beta}' = [\beta_1, \dots, \beta_{n-1}]$ in $(\mathcal{G}r_{n-1})_{\mathbf{x}'}$ such that

$$ \Theta(\boldsymbol{\beta}')_y = \begin{cases} \boldsymbol{\gamma}_y & \text{if } x_n \neq y, \\ [\gamma_1, \dots, \gamma_{m-1}] & \text{if } x_n = y. \end{cases} $$

The elements $\gamma_1, \dots, \gamma_m$ belong to $G^\vee(\mathcal{K})$, which we identify with $G^\vee(\mathcal{K}_{x_n})$. We choose $\beta_n \in G^\vee(\mathbb{C}[z, (z-x_n)^{-1}])$ such that

$$ (\beta_1 \dots \beta_{n-1})^{-1} (\gamma_1 \dots \gamma_m) \in \beta_n G^\vee(\mathcal{O}_{x_n}). $$

Then $[\beta_1, \dots, \beta_{n-1}, \beta_n]$ is the unique element $\boldsymbol{\beta}$ in $(\mathcal{G}r_n)_x$ such that $\Theta(\boldsymbol{\beta})_y = \boldsymbol{\gamma}_y$ for all $y$. ■
fibers $\mathcal{G}r_1^{\lambda_2}, \dots, \mathcal{G}r_1^{\lambda_n}$. It follows that $\mathcal{G}r_n^{\lambda}$ is a smooth connected variety of dimension $2\rho(|\lambda|) + n$.

Let us now investigate the fibers of the map $\pi \circ m_n : \mathcal{G}r_n \to \mathbb{C}^n$. Given $x \in \mathbb{C}$, we set $\mathcal{O}_x = \mathbb{C}[z-x]$ and $\mathcal{K}_x = \mathbb{C}((z-x))$. We identify $\mathcal{O}$ and $\mathcal{K}$ with $\mathcal{O}_x$ and $\mathcal{K}_x$ by means of the map $z \mapsto z-x$.

We fix $\mathbf{x} = (x_1, \dots, x_n)$ in $\mathbb{C}^n$. Let $\text{supp}(\mathbf{x})$ be the set of values $y \in \mathbb{C}$ that appear in the tuple $\mathbf{x}$. For $y \in \text{supp}(\mathbf{x})$, denote by $m_y$ the number of indices $j \in \{1, \dots, n\}$ such that $x_j = y$ and choose an increasing sequence $(p_0 = 0, p_1, p_2, \dots, p_{m_y} = n)$ in such a way that each interval $[p_{k-1} + 1, p_k]$ contains exactly one index $j$ such that $x_j = y$. For $\boldsymbol{\beta} = [\beta_1, \dots, \beta_n]$ in the fiber

$$ (\mathcal{G}r_n)_x = G^\vee(\mathbb{C}[z, (z-x_1)^{-1}]) \times^{G^\vee(\mathbb{C}[z])} \dots \times^{G^\vee(\mathbb{C}[z])} G^\vee(\mathbb{C}[z, (z-x_n)^{-1}]) / G^\vee(\mathbb{C}[z]), $$
we define $\Theta(\boldsymbol{\beta})_y$ as the point $[(\beta_1 \cdots \beta_{p_1}), (\beta_{p_1+1} \cdots \beta_{p_2}), \dots, (\beta_{p_{m_y-1}+1} \cdots \beta_n)]$ in

$$ \underbrace{G^\vee(\mathcal{K}_y) \times^{G^\vee(\mathcal{O}_y)} \dots \times^{G^\vee(\mathcal{O}_y)} G^\vee(\mathcal{K}_y)}_{m_y \text{ factors } G^\vee(\mathcal{K}_y)} / G^\vee(\mathcal{O}_y) \cong \text{Gr}_{m_y} $$

(note that $\Theta(\boldsymbol{\beta})_y$ does not depend on this choice, because $\beta_j \in G^\vee(\mathcal{O}_y)$ if $x_j \neq y$).

**Proposition 5.3.** *The map $\boldsymbol{\beta} \mapsto (\Theta(\boldsymbol{\beta})_y)$ is a bijection*

$$ (\mathcal{G}r_n)_x \xrightarrow{\approx} \prod_{y \in \text{supp}(\mathbf{x})} \text{Gr}_{m_y}. $$

*Proof.* Combining the Iwasawa decomposition (1) with the easily proven equality

$$ N^{\vee}(\mathcal{K}_x) = N^{\vee}(\mathbb{C}[z, (z-x)^{-1}])N^{\vee}(\mathcal{O}_x), \qquad (26) $$

we obtain the well-known equality

$$ G^{\vee}(\mathcal{K}_x) = G^{\vee}(\mathbb{C}[z, (z-x)^{-1}])G^{\vee}(\mathcal{O}_x), $$

for each $x \in \mathbb{C}$.

The case $n = 1$ of the proposition is banal. Assume that $n \ge 2$, and for $y \in \text{supp}(\mathbf{x})$, pick $\boldsymbol{\gamma}_y \in \text{Gr}_{m_y}$. Set $\mathbf{x}' = (x_1, \dots, x_{n-1})$ and $m = m_{x_n}$, and write $\boldsymbol{\gamma}_{x_n} = [\gamma_1, \dots, \gamma_m]$. Reasoning by induction on $n$, we know that there is a unique $\boldsymbol{\beta}' = [\beta_1, \dots, \beta_{n-1}]$ in $(\mathcal{G}r_{n-1})_{\mathbf{x}'}$ such that

$$ \Theta(\boldsymbol{\beta}')_y = \begin{cases} \boldsymbol{\gamma}_y & \text{if } x_n \neq y, \\ [\gamma_1, \dots, \gamma_{m-1}] & \text{if } x_n = y. \end{cases} $$

The elements $\gamma_1, \dots, \gamma_m$ belong to $G^\vee(\mathcal{K})$, which we identify with $G^\vee(\mathcal{K}_{x_n})$. We choose $\beta_n \in G^\vee(\mathbb{C}[z, (z-x_n)^{-1}])$ such that

$$ (\beta_1 \dots \beta_{n-1})^{-1} (\gamma_1 \dots \gamma_m) \in \beta_n G^\vee(\mathcal{O}_{x_n}). $$

Then $[\beta_1, \dots, \beta_{n-1}, \beta_n]$ is the unique element $\boldsymbol{\beta}$ in $(\mathcal{G}r_n)_x$ such that $\Theta(\boldsymbol{\beta})_y = \boldsymbol{\gamma}_y$ for all $y$. ■
Keep the notation above for **x** and the integers $m_y$ and let $\lambda = (\lambda_1, \dots, \lambda_n)$ in $(\Lambda^+)^n$. For each $y \in \text{supp}(\mathbf{x})$, define $\lambda_y \in (\Lambda^+)^{m_y}$ as the ordered tuple formed by the weights $\lambda_j$ for $j \in \{1, \dots, n\}$ such that $x_j = y$. Then, under the bijection given in Proposition 5.3, the fiber $(\mathfrak{g}\mathbf{r}_n^\lambda)_x$ identifies with

$$ \prod_{y \in \text{supp}(\mathbf{x})} \text{Gr}_{m_y}^{\lambda_y}. $$

## 5.2. Global cycles

Recall our notation $N^\vee$ for the unipotent radical of $B^\vee$. For $\mu \in \Lambda$ and $x \in \mathbb{C}$, we define

$$ \tilde{S}_{\mu|\mathbf{x}} = (z-x)^{\mu} N^{\vee}(\mathbb{C}[z, (z-x)^{-1}]) = N^{\vee}(\mathbb{C}[z, (z-x)^{-1}]) (z-x)^{\mu}. $$

Equation (26) expresses that the natural map

$$ N^{\vee}(\mathbb{C}[z, (z-x)^{-1}]) / N^{\vee}(\mathbb{C}[z]) \to N^{\vee}(\mathcal{K}_{\mathbf{x}}) / N^{\vee}(\mathcal{O}_{\mathbf{x}}) $$

is bijective; composing with the natural map $N^\vee(\mathcal{K})/N^\vee(\mathcal{O}) \to \text{Gr}$, we obtain, after left multiplication by $(z-x)^\mu$, a bijection

$$ \tilde{S}_{\mu|\mathbf{x}} / N^{\vee}(\mathbb{C}[z]) \xrightarrow{\sim} S_{\mu}. $$

For $(\mu_1, \dots, \mu_n) \in \Lambda^n$, let $S_{\mu_1} \alpha \dots \alpha S_{\mu_n}$ be the set of all pairs $(x_1, \dots, x_n; [\beta_1, \dots, \beta_n])$ with $(x_1, \dots, x_n) \in \mathbb{C}^n$ and $[\beta_1, \dots, \beta_n]$ in

$$ \tilde{S}_{\mu_1|x_1} \times^{N^{\vee}(\mathbb{C}[z])} \dots \times^{N^{\vee}(\mathbb{C}[z])} \tilde{S}_{\mu_n|x_n} / N^{\vee}(\mathbb{C}[z]). $$

Rewriting the Iwasawa decomposition as

$$ G^{\vee}(\mathbb{C}[z, (z-x)^{-1}]) = \bigsqcup_{\mu \in \Lambda} N^{\vee}(\mathbb{C}[z, (z-x)^{-1}]) (z-x)^{\mu} G^{\vee}(\mathbb{C}[z]), $$

we then see that the natural map

$$ \Psi: \bigsqcup_{(\mu_1, \dots, \mu_n) \in \Lambda^n} S_{\mu_1} \alpha \dots \alpha S_{\mu_n} \to \mathfrak{g}\mathbf{r}_n $$

is bijective. Here $\Psi$ is regarded as the calligraphic variant of the letter $\Psi$ used in Sect. 2.2; these two glyphs may be hard to distinguish, but hopefully this choice will not lead to any confusion.

More generally, given $(\mu_1, \dots, \mu_n) \in \Lambda^n$ and $N^\vee(\mathcal{O})$-stable subsets $Z_1 \subseteq S_{\mu_1}, \dots, Z_n \subseteq S_{\mu_n}$, we define $Z_1 \alpha \dots \alpha Z_n$ to be the subset of all pairs $(x_1, \dots, x_n; [\beta_1, \dots, \beta_n])$ with $(x_1, \dots, x_n) \in \mathbb{C}^n$ and

$$ [\beta_1, \dots, \beta_n] \in \tilde{Z}_{1|x_1} \times^{N^{\vee}(\mathbb{C}[z])} \dots \times^{N^{\vee}(\mathbb{C}[z])} \tilde{Z}_{n|x_n} / N^{\vee}(\mathbb{C}[z]) $$

where each $\tilde{Z}_{j|x_j}$ is the preimage of $Z_j$ under the map $\tilde{S}_{\mu_j|x_j} \to S_{\mu_j}$.

For $\mu \in \Lambda$, we define

$$ \dot{S}_{\mu} = \bigcup_{(\mu_1, \dots, \mu_n) \in \Lambda^n \atop \mu_1 + \dots + \mu_n = \mu} \Psi(S_{\mu_1} \alpha \dots \alpha S_{\mu_n}). $$
Keep the notation above for **x** and the integers $m_y$ and let $\lambda = (\lambda_1, \dots, \lambda_n)$ in $(\Lambda^+)^n$. For each $y \in \text{supp}(\mathbf{x})$, define $\lambda_y \in (\Lambda^+)^{m_y}$ as the ordered tuple formed by the weights $\lambda_j$ for $j \in \{1, \dots, n\}$ such that $x_j = y$. Then, under the bijection given in Proposition 5.3, the fiber $(\mathfrak{g}\mathbf{r}_n^\lambda)_x$ identifies with

$$ \prod_{y \in \text{supp}(\mathbf{x})} \text{Gr}_{m_y}^{\lambda_y}. $$

## 5.2. Global cycles

Recall our notation $N^\vee$ for the unipotent radical of $B^\vee$. For $\mu \in \Lambda$ and $x \in \mathbb{C}$, we define

$$ \tilde{S}_{\mu|x} = (z-x)^{\mu} N^{\vee}(\mathbb{C}[z, (z-x)^{-1}]) = N^{\vee}(\mathbb{C}[z, (z-x)^{-1}]) (z-x)^{\mu}. $$

Equation (26) expresses that the natural map

$$ N^{\vee}(\mathbb{C}[z, (z-x)^{-1}]) / N^{\vee}(\mathbb{C}[z]) \to N^{\vee}(\mathcal{K}_x) / N^{\vee}(\mathcal{O}_x) $$

is bijective; composing with the natural map $N^\vee(\mathcal{K})/N^\vee(\mathcal{O}) \to \text{Gr}$, we obtain, after left multiplication by $(z-x)^\mu$, a bijection

$$ \tilde{S}_{\mu|x} / N^{\vee}(\mathbb{C}[z]) \xrightarrow{\sim} S_{\mu}. $$

For $(\mu_1, \dots, \mu_n) \in \Lambda^n$, let $S_{\mu_1} \alpha \cdots \alpha S_{\mu_n}$ be the set of all pairs $(x_1, \dots, x_n; [\beta_1, \dots, \beta_n])$ with $(x_1, \dots, x_n) \in \mathbb{C}^n$ and $[\beta_1, \dots, \beta_n]$ in

$$ \tilde{S}_{\mu_1|x_1} \times^{N^{\vee}(\mathbb{C}[z])} \dots \times^{N^{\vee}(\mathbb{C}[z])} \tilde{S}_{\mu_n|x_n} / N^{\vee}(\mathbb{C}[z]). $$

Rewriting the Iwasawa decomposition as

$$ G^{\vee}(\mathbb{C}[z, (z-x)^{-1}]) = \bigsqcup_{\mu \in \Lambda} N^{\vee}(\mathbb{C}[z, (z-x)^{-1}]) (z-x)^{\mu} G^{\vee}(\mathbb{C}[z]), $$

we then see that the natural map

$$ \Psi: \bigsqcup_{(\mu_1, \dots, \mu_n) \in \Lambda^n} S_{\mu_1} \alpha \cdots \alpha S_{\mu_n} \to \mathfrak{g}\mathbf{r}_n $$

is bijective. Here $\Psi$ is regarded as the calligraphic variant of the letter $\Psi$ used in Sect. 2.2; these two glyphs may be hard to distinguish, but hopefully this choice will not lead to any confusion.

More generally, given $(\mu_1, \dots, \mu_n) \in \Lambda^n$ and $N^\vee(\mathcal{O})$-stable subsets $Z_1 \subseteq S_{\mu_1}, \dots, Z_n \subseteq S_{\mu_n}$, we define $Z_1 \alpha \cdots \alpha Z_n$ to be the subset of all pairs $(x_1, \dots, x_n; [\beta_1, \dots, \beta_n])$ with $(x_1, \dots, x_n) \in \mathbb{C}^n$ and

$$ [\beta_1, \dots, \beta_n] \in \tilde{Z}_{1|x_1} \times^{N^{\vee}(\mathbb{C}[z])} \dots \times^{N^{\vee}(\mathbb{C}[z])} \tilde{Z}_{n|x_n} / N^{\vee}(\mathbb{C}[z]) $$

where each $\tilde{Z}_{j|x_j}$ is the preimage of $Z_j$ under the map $\tilde{S}_{\mu_j|x_j} \to S_{\mu_j}$.

For $\mu \in \Lambda$, we define

$$ \dot{S}_{\mu} = \bigcup_{(\mu_1, \dots, \mu_n) \in \Lambda^n \atop \mu_1 + \dots + \mu_n = \mu} \Psi(S_{\mu_1} \alpha \cdots \alpha S_{\mu_n}). $$
**Proposition 5.4.** Let $\lambda = (\lambda_1, \dots, \lambda_n)$ in $(\Lambda^+)^n$ and let $\mu \in \Lambda$.

(i) All the irreducible components of $\overline{\mathfrak{g}r_n^\lambda} \cap \dot{S}_\mu$ have dimension $\rho(|\lambda| + \mu) + n$.

(ii) The map $(Z_1, \dots, Z_n) \mapsto \overline{\Psi(Z_1 \propto \dots \propto Z_n)}$ induces a bijection

$$ \bigcup_{\substack{(\mu_1, \dots, \mu_n) \in \Lambda^n \\ \mu_1 + \dots + \mu_n = \mu}} *\mathcal{L}(\lambda_1)_{\mu_1} \times \dots \times *\mathcal{L}(\lambda_n)_{\mu_n} \xrightarrow{\cong} \text{Irr}(\overline{\mathfrak{g}r_n^\lambda} \cap \dot{S}_\mu). $$

(The bar above $\Psi(Z_1 \propto \dots \propto Z_n)$ means closure in $\dot{S}_\mu$.)

*Proof.* Let $(\mu_1, \dots, \mu_n) \in \Lambda^n$ be such that $\mu_1 + \dots + \mu_n = \mu$ and let $(Z_1, \dots, Z_n) \in *\mathcal{L}(\lambda_1)_{\mu_1} \times \dots \times *\mathcal{L}(\lambda_n)_{\mu_n}$. Then the set $\Psi(Z_1 \propto \dots \propto Z_n)$ is irreducible. By Proposition 5.3 and its proof, the fiber of this set over a point $\mathbf{x} \in \mathbb{C}^n$ is isomorphic to the product, over all $\mathbf{y} \in \text{supp}(\mathbf{x})$, of cycles

$$ \Psi(Z_{j_1} \times \dots \times Z_{j_m}) \subseteq \text{Gr}_m $$

where $j_1, \dots, j_m$ are the indices $j \in \{1, \dots, n\}$ such that $x_j = \mathbf{y}$. We remark that if we set $\lambda_y = (\lambda_{j_1}, \dots, \lambda_{j_m})$ and $\mu_y = \mu_{j_1} + \dots + \mu_{j_m}$, then this cycle belongs to $*\mathcal{L}(\lambda_y)_{\mu_y}$. By Proposition 2.2 (i), the dimension of the fiber of $\Psi(Z_1 \propto \dots \propto Z_n)$ over $\mathbf{x}$ is therefore

$$ \sum_{\mathbf{y} \in \text{supp}(\mathbf{x})} \rho(|\lambda_y| + \mu_y) = \rho(|\lambda| + \mu) $$

and we conclude that $\Psi(Z_1 \propto \dots \propto Z_n)$ has dimension $\rho(|\lambda| + \mu) + n$.

To finish the proof, we observe that the sets $\Psi(Z_1 \propto \dots \propto Z_n)$ cover $\overline{\mathfrak{g}r_n^\lambda} \cap \dot{S}_\mu$ and are not redundant. ■

Our MV bases are defined with the help of the unstable subsets $T_\mu$ instead of the stable subsets $S_\mu$. We can easily adapt the constructions of this subsection to this case by replacing the Borel subgroup $B^\vee$ with its opposite with respect to $T^\vee$, and replacing similarly its unipotent radical $N^\vee$. We shall do this while keeping the notation $\propto$ and $\Psi$. Note that when we replace $\dot{S}_\mu$ by

$$ \dot{T}_\mu = \bigcup_{\substack{(\mu_1, \dots, \mu_n) \in \Lambda^n \\ \mu_1 + \dots + \mu_n = \mu}} \Psi(T_{\mu_1} \propto \dots \propto T_{\mu_n}) $$

in Proposition 5.4, $\rho(|\lambda| + \mu) + n$ must be replaced by $\rho(|\lambda| - \mu) + n$ and the sets $*\mathcal{L}(\lambda_j)_{\mu_j}$ must be replaced by their unstarred counterparts.

### 5.3. The fusion product

For any $\mathbf{x} \in \mathbb{C}$, the fibers of $\mathfrak{g}r_n^{\text{BD}}$ and $\mathfrak{g}r_n$ over $(x, \dots, x)$ are isomorphic to $\text{Gr}$ and $\text{Gr}_n$, respectively. Thus,

$$ \mathfrak{g}r_n^{\text{BD}}|_{\Delta} \xrightarrow{\cong} \Delta \times \text{Gr} \quad \text{and} \quad \mathfrak{g}r_n|_{\Delta} \xrightarrow{\cong} \Delta \times \text{Gr}_n, $$
**Proposition 5.4.** Let $\lambda = (\lambda_1, \dots, \lambda_n)$ in $(\Lambda^+)^n$ and let $\mu \in \Lambda$.

(i) All the irreducible components of $\overline{\mathfrak{g}r_n^\lambda} \cap \dot{S}_\mu$ have dimension $\rho(|\lambda| + \mu) + n$.

(ii) The map $(Z_1, \dots, Z_n) \mapsto \overline{\Psi(Z_1 \propto \dots \propto Z_n)}$ induces a bijection

$$ \bigcup_{\substack{(\mu_1, \dots, \mu_n) \in \Lambda^n \\ \mu_1 + \dots + \mu_n = \mu}} *\mathcal{L}(\lambda_1)_{\mu_1} \times \dots \times *\mathcal{L}(\lambda_n)_{\mu_n} \xrightarrow{\approx} \text{Irr}(\overline{\mathfrak{g}r_n^\lambda} \cap \dot{S}_\mu). $$

(The bar above $\Psi(Z_1 \propto \dots \propto Z_n)$ means closure in $\dot{S}_\mu$.)

*Proof.* Let $(\mu_1, \dots, \mu_n) \in \Lambda^n$ be such that $\mu_1 + \dots + \mu_n = \mu$ and let $(Z_1, \dots, Z_n) \in *\mathcal{L}(\lambda_1)_{\mu_1} \times \dots \times *\mathcal{L}(\lambda_n)_{\mu_n}$. Then the set $\Psi(Z_1 \propto \dots \propto Z_n)$ is irreducible. By Proposition 5.3 and its proof, the fiber of this set over a point $\mathbf{x} \in \mathbb{C}^n$ is isomorphic to the product, over all $\mathbf{y} \in \text{supp}(\mathbf{x})$, of cycles

$$ \Psi(Z_{j_1} \times \dots \times Z_{j_m}) \subseteq \text{Gr}_m $$

where $j_1, \dots, j_m$ are the indices $j \in \{1, \dots, n\}$ such that $x_j = \mathbf{y}$. We remark that if we set $\lambda_y = (\lambda_{j_1}, \dots, \lambda_{j_m})$ and $\mu_y = \mu_{j_1} + \dots + \mu_{j_m}$, then this cycle belongs to $*\mathcal{L}(\lambda_y)_{\mu_y}$. By Proposition 2.2 (i), the dimension of the fiber of $\Psi(Z_1 \propto \dots \propto Z_n)$ over $\mathbf{x}$ is therefore

$$ \sum_{\mathbf{y} \in \text{supp}(\mathbf{x})} \rho(|\lambda_y| + \mu_y) = \rho(|\lambda| + \mu) $$

and we conclude that $\Psi(Z_1 \propto \dots \propto Z_n)$ has dimension $\rho(|\lambda| + \mu) + n$.

To finish the proof, we observe that the sets $\Psi(Z_1 \propto \dots \propto Z_n)$ cover $\overline{\mathfrak{g}r_n^\lambda} \cap \dot{S}_\mu$ and are not redundant. ■

Our MV bases are defined with the help of the unstable subsets $T_\mu$ instead of the stable subsets $S_\mu$. We can easily adapt the constructions of this subsection to this case by replacing the Borel subgroup $B^\vee$ with its opposite with respect to $T^\vee$, and replacing similarly its unipotent radical $N^\vee$. We shall do this while keeping the notation $\propto$ and $\Psi$. Note that when we replace $\dot{S}_\mu$ by

$$ \dot{T}_\mu = \bigcup_{\substack{(\mu_1, \dots, \mu_n) \in \Lambda^n \\ \mu_1 + \dots + \mu_n = \mu}} \Psi(T_{\mu_1} \propto \dots \propto T_{\mu_n}) $$

in Proposition 5.4, $\rho(|\lambda| + \mu) + n$ must be replaced by $\rho(|\lambda| - \mu) + n$ and the sets $*\mathcal{L}(\lambda_j)_{\mu_j}$ must be replaced by their unstarred counterparts.

### 5.3. The fusion product

For any $\mathbf{x} \in \mathbb{C}$, the fibers of $\mathfrak{g}r_n^{\text{BD}}$ and $\mathfrak{g}r_n$ over $(x, \dots, x)$ are isomorphic to $\text{Gr}$ and $\text{Gr}_n$, respectively. Thus,

$$ \mathfrak{g}r_n^{\text{BD}}|_{\Delta} \xrightarrow{\approx} \Delta \times \text{Gr} \quad \text{and} \quad \mathfrak{g}r_n|_{\Delta} \xrightarrow{\approx} \Delta \times \text{Gr}_n, $$
where $\Delta$ is the small diagonal, defined as the image of the map $x \mapsto (x, \dots, x)$ from $\mathbb{C}$ to $\mathbb{C}^n$. At the other extreme, the morphism $m_n : \mathcal{G}r_n \to \mathcal{G}r_n^{\text{BD}}$ is an isomorphism after restriction to the open locus $U \subseteq \mathbb{C}^n$ of points with pairwise different coordinates (see [46, Lemma 3.1.23]), and by Proposition 5.3, $\mathcal{G}r_n|_U$ is isomorphic to $U \times (\text{Gr})^n$. We define maps $\tau, i, j$ and $\zeta$ according to the diagram

<figure><img src="image_14.png" /></figure>

Let $\lambda \in (\Lambda^+)^n$ and $\mu \in \Lambda$, set

$$ \mathcal{B}(\lambda) = \text{IC}(\overline{\mathcal{G}r_n^\lambda}, \mathcal{C}), \quad d = \dim \mathcal{G}r_n^\lambda = 2\rho(|\lambda|) + n, \quad k = 2\rho(\mu) - n $$

and denote the inclusion $\dot{T}_\mu \to \mathcal{G}r_n$ by $\dot{i}_\mu$. The next statement is due to Mirković and Vilonen.

**Proposition 5.5.** (i) There are natural isomorphisms

$$ i^1 \mathcal{B}(\lambda)[n] \cong \tau^! \text{IC}(\overline{\text{Gr}_n^\lambda}, \mathcal{C}) $$

and

$$ j^1 \mathcal{B}(\lambda)[n] \cong \xi^! (\text{IC}(\overline{\text{Gr}_{\lambda^1}}), \mathcal{C}) \boxtimes \dots \boxtimes (\text{IC}(\overline{\text{Gr}_{\lambda n}}), \mathcal{C}). $$

(ii) Each cohomology sheaf of $(\pi \circ m_n)_* \mathcal{B}(\lambda)$ is a local system on $\mathbb{C}^n$.

(iii) The complex of sheaves $(\pi \circ m_n \circ \dot{i}_\mu)_* (\dot{i}_\mu)^! \mathcal{B}(\lambda)$ is concentrated in degree $k$ and its $k$-th cohomology sheaf is a local system on $\mathbb{C}^n$.

*Proof.* To prove statement (i), one follows the reasoning in [4, Sect. 1.7.5], noting that $\mathcal{B}(\lambda)$ and $\text{IC}(\overline{\text{Gr}_n^\lambda}, \mathcal{C})$ are the sheaves denoted by $(\tau^\circ \mathcal{I}_{\lambda_1}) \boxtimes \dots \boxtimes (\tau^\circ \mathcal{I}_{\lambda_n})$ and $\mathcal{I}_{\lambda_1} \boxtimes \dots \boxtimes \mathcal{I}_{\lambda_n}$ in loc. cit. Statement (ii) is [39, (6.4)]. Statement (iii) is contained in the proof of [39, Proposition 6.4], up to a base change in the Cartesian square

$$ \begin{array}{ccc} \dot{T}_{\mu} & \xrightarrow{i_{\mu}} & \mathcal{G}r_n \\ \bigdownarrow & & \bigdownarrow^{m_n} \\ T_{\mu}(\mathbb{A}^n) & \xrightarrow{k_{\mu}} & \mathcal{G}r_n^{\text{BD}} \end{array} $$

■

Combining Propositions 2.1 and 5.5 (i), we see that the total cohomology of the stalk of the complex $(\pi \circ m_n)_* \mathcal{B}(\lambda)$ identifies with $F(\mathcal{I}_\lambda)$ at any point in $\Delta$, and with $F(\mathcal{I}_{\lambda_1}) \otimes \dots \otimes F(\mathcal{I}_{\lambda_n})$ at any point in $U$. Statement (ii) in Proposition 5.5 thus provides the identification

$$ F(\mathcal{I}_{\lambda}) \cong F(\mathcal{I}_{\lambda_1}) \otimes \dots \otimes F(\mathcal{I}_{\lambda_n}) $$
where $\Delta$ is the small diagonal, defined as the image of the map $x \mapsto (x, \dots, x)$ from $\mathbb{C}$ to $\mathbb{C}^n$. At the other extreme, the morphism $m_n : \mathcal{G}r_n \to \mathcal{G}r_n^{\text{BD}}$ is an isomorphism after restriction to the open locus $U \subseteq \mathbb{C}^n$ of points with pairwise different coordinates (see [46, Lemma 3.1.23]), and by Proposition 5.3, $\mathcal{G}r_n|_U$ is isomorphic to $U \times (\text{Gr})^n$. We define maps $\tau, i, j$ and $\zeta$ according to the diagram

<figure><img src="image_13.png" /></figure>

Let $\lambda \in (\Lambda^+)^n$ and $\mu \in \Lambda$, set

$$ \mathcal{B}(\lambda) = \text{IC}(\overline{\mathcal{G}r_n^{\lambda}}, \mathcal{C}), \quad d = \dim \mathcal{G}r_n^{\lambda} = 2\rho(|\lambda|) + n, \quad k = 2\rho(\mu) - n $$

and denote the inclusion $\dot{T}_{\mu} \to \mathcal{G}r_n$ by $\dot{i}_{\mu}$. The next statement is due to Mirković and Vilonen.

**Proposition 5.5.** (i) There are natural isomorphisms

$$ i^{!} \mathcal{B}(\lambda)[n] \cong \tau^{!} \text{IC}(\overline{\text{Gr}_n^{\lambda}}, \mathcal{C}) $$

and

$$ j^{!} \mathcal{B}(\lambda)[n] \cong \xi^{!} (\text{IC}(\overline{\text{Gr}_{\lambda}^{1}}, \mathcal{C}) \boxtimes \dots \boxtimes \text{IC}(\overline{\text{Gr}_{\lambda n}}, \mathcal{C})). $$

(ii) Each cohomology sheaf of $(\pi \circ m_n)_* \mathcal{B}(\lambda)$ is a local system on $\mathbb{C}^n$.

(iii) The complex of sheaves $(\pi \circ m_n \circ \dot{i}_{\mu})_* (\dot{i}_{\mu})^! \mathcal{B}(\lambda)$ is concentrated in degree $k$ and its $k$-th cohomology sheaf is a local system on $\mathbb{C}^n$.

*Proof.* To prove statement (i), one follows the reasoning in [4, Sect. 1.7.5], noting that $\mathcal{B}(\lambda)$ and $\text{IC}(\overline{\text{Gr}_n^{\lambda}}, \mathcal{C})$ are the sheaves denoted by $(\tau^\circ \mathcal{I}_{\lambda_1}) \boxtimes \dots \boxtimes (\tau^\circ \mathcal{I}_{\lambda_n})$ and $\mathcal{I}_{\lambda_1} \boxtimes \dots \boxtimes \mathcal{I}_{\lambda_n}$ in loc. cit. Statement (ii) is [39, (6.4)]. Statement (iii) is contained in the proof of [39, Proposition 6.4], up to a base change in the Cartesian square

$$ \begin{array}{ccc} \dot{T}_{\mu} & \xrightarrow{i_{\mu}} & \mathcal{G}r_n \\ \bigdownarrow & & \bigdownarrow^{m_n} \\ T_{\mu}(\mathbb{A}^n) & \xrightarrow{k_{\mu}} & \mathcal{G}r_n^{\text{BD}} \end{array} $$

■

Combining Propositions 2.1 and 5.5 (i), we see that the total cohomology of the stalk of the complex $(\pi \circ m_n)_* \mathcal{B}(\lambda)$ identifies with $F(\mathcal{I}_{\lambda})$ at any point in $\Delta$, and with $F(\mathcal{I}_{\lambda_1}) \otimes \dots \otimes F(\mathcal{I}_{\lambda_n})$ at any point in $U$. Statement (ii) in Proposition 5.5 thus provides the identification

$$ F(\mathcal{I}_{\lambda}) \cong F(\mathcal{I}_{\lambda_1}) \otimes \dots \otimes F(\mathcal{I}_{\lambda_n}) $$
required to compare the two bases of $V(\lambda)$. Statement (iii) further identifies the weight spaces

$$F_{\mu}(\mathcal{I}_{\lambda}) \cong \bigoplus_{\substack{(\mu_1, \dots, \mu_n) \in \Delta^n \\ \mu_1 + \dots + \mu_n = \mu}} F_{\mu_1}(\mathcal{I}_{\lambda_1}) \otimes \dots \otimes F_{\mu_n}(\mathcal{I}_{\lambda_n}).$$

### 5.4. Intersection multiplicities

We keep the setup introduced in the previous section. In addition, we denote by

$$\mathcal{L}_{\mu}(\lambda) = \mathcal{H}^k(\pi \circ m_n \circ \dot{i}_{\mu})_* (\dot{i}_{\mu})^! \mathcal{B}(\lambda)$$

the local system appearing in Proposition 5.5 (iii).

For each point $\mathbf{x} \in \mathbb{C}^n$, we define maps as indicated below:

<figure><img src="image_15.png" /></figure>

where for instance $(\mathcal{G}r_n^\lambda)_\mathbf{x}$ is the fiber of $\mathcal{G}r_n^\lambda$ over $\mathbf{x}$. (The notation $i$ and $j$ does not designate the same maps as in the previous subsection.) We then construct the following diagram, referred to as (愛) below:

<figure><img src="image_16.png" /></figure>

The left vertical arrow in (愛) is the restriction of the cohomology with support in $\dot{T}_{\mu}$ from $\mathcal{G}r_n$ to $(\mathcal{G}r_n)_\mathbf{x}$. In other words, it is the image by the functor $H^k(\dot{T}_{\mu}, (\dot{i}_{\mu})^!)$ of the adjunction morphism $\mathcal{B}(\lambda) \to i_*i^*\mathcal{B}(\lambda)$. Lemma 5.7 below implies that it is an isomorphism. Likewise, the middle vertical arrow is the restriction from $\mathcal{G}r_n^\lambda$ to $(\mathcal{G}r_n^\lambda)_\mathbf{x}$, afforded by the adjunction morphism $j^*\mathcal{B}(\lambda) \to h_*h^*j^*\mathcal{B}(\lambda)$.

On the top line, the left arrow is the restriction from $\mathcal{G}r_n$ to $\mathcal{G}r_n^\lambda$, fulfilled by the adjunction morphism $\mathcal{B}(\lambda) \to j_*j^*\mathcal{B}(\lambda) = j_*\mathcal{C}_{\mathcal{G}r_n}[d]$. On the bottom line, it is the restriction from $(\mathcal{G}r_n)_\mathbf{x}$ to $(\mathcal{G}r_n^\lambda)_\mathbf{x}$, achieved by $i^*\mathcal{B}(\lambda) \to (j')_* (j')^*i^*\mathcal{B}(\lambda)$. Mirković and Vilonen's argument (reproduced in Sect. 2.3) shows that these two arrows are isomorphisms.
required to compare the two bases of $V(\lambda)$. Statement (iii) further identifies the weight spaces

$$F_{\mu}(\mathcal{I}_{\lambda}) \cong \bigoplus_{\substack{(\mu_1, \dots, \mu_n) \in \Delta^n \\ \mu_1 + \dots + \mu_n = \mu}} F_{\mu_1}(\mathcal{I}_{\lambda_1}) \otimes \dots \otimes F_{\mu_n}(\mathcal{I}_{\lambda_n}).$$

### 5.4. Intersection multiplicities

We keep the setup introduced in the previous section. In addition, we denote by

$$\mathcal{L}_{\mu}(\lambda) = \mathcal{H}^k(\pi \circ m_n \circ \dot{i}_{\mu})_* (\dot{i}_{\mu})^! \mathcal{B}(\lambda)$$

the local system appearing in Proposition 5.5 (iii).

For each point $\mathbf{x} \in \mathbb{C}^n$, we define maps as indicated below:

<figure><img src="image_14.png" /></figure>

where for instance $(\mathcal{G}r_n^\lambda)_\mathbf{x}$ is the fiber of $\mathcal{G}r_n^\lambda$ over $\mathbf{x}$. (The notation $i$ and $j$ does not designate the same maps as in the previous subsection.) We then construct the following diagram, referred to as (愛) below:

<figure><img src="image_15.png" /></figure>

The left vertical arrow in (愛) is the restriction of the cohomology with support in $\dot{T}_{\mu}$ from $\mathcal{G}r_n$ to $(\mathcal{G}r_n)_\mathbf{x}$. In other words, it is the image by the functor $H^k(\dot{T}_{\mu}, (\dot{i}_{\mu})^!)$ of the adjunction morphism $\mathcal{B}(\lambda) \to i_*i^*\mathcal{B}(\lambda)$. Lemma 5.7 below implies that it is an isomorphism. Likewise, the middle vertical arrow is the restriction from $\mathcal{G}r_n^\lambda$ to $(\mathcal{G}r_n^\lambda)_\mathbf{x}$, afforded by the adjunction morphism $j^*\mathcal{B}(\lambda) \to h_*h^*j^*\mathcal{B}(\lambda)$.

On the top line, the left arrow is the restriction from $\mathcal{G}r_n$ to $\mathcal{G}r_n^\lambda$, fulfilled by the adjunction morphism $\mathcal{B}(\lambda) \to j_*j^*\mathcal{B}(\lambda) = j_*\mathcal{C}_{\mathcal{G}r_n}[d]$. On the bottom line, it is the restriction from $(\mathcal{G}r_n)_\mathbf{x}$ to $(\mathcal{G}r_n^\lambda)_\mathbf{x}$, achieved by $i^*\mathcal{B}(\lambda) \to (j')_* (j')^*i^*\mathcal{B}(\lambda)$. Mirković and Vilonen's argument (reproduced in Sect. 2.3) shows that these two arrows are isomorphisms.
The two paths around the left square in (H) are two different expressions for the restriction from $\mathfrak{S}r_n$ to $(\mathfrak{S}r_n^\lambda)_x$; therefore this square commutes.

In both lines of (H) the right arrow is Alexander duality. We note that $H_{d-k}^{\text{BM}}(\mathfrak{S}r_n^\lambda \cap \dot{T}_\mu)$ and $H_{d-k-2n}^{\text{BM}}((\mathfrak{S}r_n^\lambda \cap \dot{T}_\mu)_x)$ are the top-dimensional Borel-Moore homology groups.

The map $h$ is a regular embedding of codimension $n$. Its orientation class (generalized Thom class) is an element

$$u_x \in H^{2n}((\mathfrak{S}r_n^\lambda)_x, h^1\underline{\mathbb{C}}_{\mathfrak{S}r_n^\lambda}).$$

The right vertical arrow in (H) is the cap product with

$$g^*u_x \in H^{2n}((\mathfrak{S}r_n^\lambda \cap \dot{T}_\mu)_x, (h')^!\underline{\mathbb{C}}_{\mathfrak{S}r_n^\lambda \cap \dot{T}_\mu}),$$

the restriction of $u_x$ to $\mathfrak{S}r_n^\lambda \cap \dot{T}_\mu$.

**Lemma 5.6.** In the diagram (H), the square on the right commutes.

*Proof.* Applying formula IX.4.9 in [24], we get $u_x \cap [\mathfrak{S}r_n^\lambda] = [(\mathfrak{S}r_n^\lambda)_x]$.

Formula (8) in [15, Sect. 19.1] (or formula IX.3.4 in [24]) asserts that given a topological manifold $X$ and inclusions of closed subsets $a: A \to X$ and $b: B \to X$, for any

$$\alpha \in H^*(A, a^!\underline{\mathbb{C}}_X), \quad \beta \in H^*(B, b^!\underline{\mathbb{C}}_X), \quad C \in H_{\bullet}^{\text{BM}}(X)$$

one has

$$(b^*\alpha) \cap (\beta \cap C) = (\alpha \cup \beta) \cap C. \qquad (27)$$

Using the six operations formalism, one checks without much trouble that this result is also valid if $A$ and $B$ are only locally closed.

Now pick

$$\xi \in H^{k+d}(\mathfrak{S}r_n^\lambda \cap \dot{T}_\mu, g^!\underline{\mathbb{C}}_{\mathfrak{S}r_n^\lambda}).$$

Applying (27) twice and using the fact that $u_x$ has even degree, we compute

$$(h^*\xi) \cap (u_x \cap [\mathfrak{S}r_n^\lambda]) = (\xi \cup u_x) \cap [\mathfrak{S}r_n^\lambda] = (u_x \cup \xi) \cap [\mathfrak{S}r_n^\lambda] = (g^*u_x) \cap (\xi \cap [\mathfrak{S}r_n^\lambda]).$$

This equality means precisely that $\xi$ has the same image under the two paths in (H) that circumscribe the square on the right. ■

**Lemma 5.7.** There are natural isomorphisms

$$H^k(\dot{T}_\mu, (i_\mu)^!\mathcal{B}(\lambda)) \cong H^0(\mathbb{C}^n, \mathcal{L}_\mu(\lambda)) \quad \text{and} \quad H^k((\dot{T}_\mu)_x, (i'_\mu)^!i^*\mathcal{B}(\lambda)) \cong (\mathcal{L}_\mu(\lambda))_x$$
and the left vertical arrow in (H) is the stalk map $H^0(\mathbb{C}^n, \mathcal{L}_\mu(\lambda)) \to (\mathcal{L}_\mu(\lambda))_x$.

*Proof.* The first isomorphism is

$$H^0(\mathbb{C}^n, \mathcal{L}_\mu(\lambda)) = H^k(\mathbb{C}^n, (\pi \circ m_n)_*(i_\mu)_*(i'_\mu)^!\mathcal{B}(\lambda)) = H^k(\dot{T}_\mu, (i_\mu)^!\mathcal{B}(\lambda)).$$
The two paths around the left square in (H) are two different expressions for the restriction from $\mathfrak{g}r_n$ to $(\mathfrak{g}r_n^\lambda)_x$; therefore this square commutes.

In both lines of (H) the right arrow is Alexander duality. We note that $H_{d-k}^{\text{BM}}(\mathfrak{g}r_n^\lambda \cap \dot{T}_\mu)$ and $H_{d-k-2n}^{\text{BM}}((\mathfrak{g}r_n^\lambda \cap \dot{T}_\mu)_x)$ are the top-dimensional Borel-Moore homology groups.

The map $h$ is a regular embedding of codimension $n$. Its orientation class (generalized Thom class) is an element

$$u_x \in H^{2n}((\mathfrak{g}r_n^\lambda)_x, h^1\underline{\mathbb{C}}_{\mathfrak{g}r_n^\lambda}).$$

The right vertical arrow in (H) is the cap product with

$$g^*u_x \in H^{2n}((\mathfrak{g}r_n^\lambda \cap \dot{T}_\mu)_x, (h')^!\underline{\mathbb{C}}_{\mathfrak{g}r_n^\lambda \cap \dot{T}_\mu}),$$

the restriction of $u_x$ to $\mathfrak{g}r_n^\lambda \cap \dot{T}_\mu$.

**Lemma 5.6.** In the diagram (H), the square on the right commutes.

*Proof.* Applying formula IX.4.9 in [24], we get $u_x \cap [\mathfrak{g}r_n^\lambda] = [(\mathfrak{g}r_n^\lambda)_x]$.

Formula (8) in [15, Sect. 19.1] (or formula IX.3.4 in [24]) asserts that given a topological manifold $X$ and inclusions of closed subsets $a: A \to X$ and $b: B \to X$, for any

$$\alpha \in H^\bullet(A, a^!\underline{\mathbb{C}}_X), \quad \beta \in H^\bullet(B, b^!\underline{\mathbb{C}}_X), \quad C \in H_{\bullet}^{\text{BM}}(X)$$

one has

$$(b^*\alpha) \cap (\beta \cap C) = (\alpha \cup \beta) \cap C. \qquad (27)$$

Using the six operations formalism, one checks without much trouble that this result is also valid if $A$ and $B$ are only locally closed.

Now pick

$$\xi \in H^{k+d}(\mathfrak{g}r_n^\lambda \cap \dot{T}_\mu, g^!\underline{\mathbb{C}}_{\mathfrak{g}r_n^\lambda}).$$

Applying (27) twice and using the fact that $u_x$ has even degree, we compute

$$(h^*\xi) \cap (u_x \cap [\mathfrak{g}r_n^\lambda]) = (\xi \cup u_x) \cap [\mathfrak{g}r_n^\lambda] = (u_x \cup \xi) \cap [\mathfrak{g}r_n^\lambda] = (g^*u_x) \cap (\xi \cap [\mathfrak{g}r_n^\lambda]).$$

This equality means precisely that $\xi$ has the same image under the two paths in (H) that circumscribe the square on the right. ■

**Lemma 5.7.** There are natural isomorphisms

$$H^k(\dot{T}_\mu, (i_\mu)^!\mathcal{B}(\lambda)) \cong H^0(\mathbb{C}^n, \mathcal{L}_\mu(\lambda)) \quad \text{and} \quad H^k((\dot{T}_\mu)_x, (i'_\mu)^!i^*\mathcal{B}(\lambda)) \cong (\mathcal{L}_\mu(\lambda))_x$$
and the left vertical arrow in (H) is the stalk map $H^0(\mathbb{C}^n, \mathcal{L}_\mu(\lambda)) \to (\mathcal{L}_\mu(\lambda))_x$.

*Proof.* The first isomorphism is

$$H^0(\mathbb{C}^n, \mathcal{L}_\mu(\lambda)) = H^k(\mathbb{C}^n, (\pi \circ m_n)_*(i_\mu)_*(i'_\mu)^!\mathcal{B}(\lambda)) = H^k(\dot{T}_\mu, (i'_\mu)^!\mathcal{B}(\lambda)).$$
The second one requires the notion of a universally locally acyclic complex (see [10, Sect. 5.1]). Specifically, $\mathcal{B}(\lambda)$ is $(\pi \circ m_n)$-ULA ([42, proof of Proposition IV.3.4] or [43, Lemma 3.20]), so there is an isomorphism

$$i^*\mathcal{B}(\lambda) \to i^!\mathcal{B}(\lambda)[2n].$$

Then

$$
\begin{align*}
H^k((\dot{T}_\mu)_x, (i'_\mu)^! i^* \mathcal{B}(\lambda)) &= H^k((\dot{T}_\mu)_x, (i'_\mu)^! i^! \mathcal{B}(\lambda)[2n]) \\
&= H^k(\{\mathbf{x}\}, (\pi \circ m_n)_* (i'_\mu)_* (i'_\mu)^! i^! \mathcal{B}(\lambda)[2n]) \\
&= H^k(\{\mathbf{x}\}, (\pi \circ m_n)_* (i'_\mu)_* i^! (i_\mu)^! \mathcal{B}(\lambda)[2n]) \\
&= H^k(\{\mathbf{x}\}, (i_0)^!(\pi \circ m_n)_* (i'_\mu)_* (i_\mu)^! \mathcal{B}(\lambda)[2n]),
\end{align*}
$$

the last step being proper base change. Now $(\pi \circ m_n)_*(i'_\mu)_*(i_\mu)^! \mathcal{B}(\lambda)$ is the local system $\mathcal{L}_\mu(\lambda)$ shifted by $-k$, and therefore

$$
\begin{align*}
H^k((\dot{T}_\mu)_x, (i'_\mu)^! i^* \mathcal{B}(\lambda)) &= H^0(\{\mathbf{x}\}, (i_0)^! \mathcal{L}_\mu(\lambda)[2n]) = H^0(\{\mathbf{x}\}, (i_0)^* \mathcal{L}_\mu(\lambda)) \\
&= (\mathcal{L}_\mu(\lambda))_x
\end{align*}
$$

as desired.

<figure><img src="image_17.png" /></figure>

By Proposition 5.4, the irreducible components of $\mathcal{G}r_n^\lambda \cap \dot{T}_\mu$ are all top-dimensional and can be indexed by

$$ \bigcup_{\substack{(\mu_1, \dots, \mu_n) \in \Lambda^n \\ \mu_1 + \dots + \mu_n = \mu}} \mathcal{L}(\lambda_1)_{\mu_1} \times \dots \times \mathcal{L}(\lambda_n)_{\mu_n}; \qquad (28) $$

namely, to a tuple $\mathbf{Z} = (Z_1, \dots, Z_n)$ is assigned the component

$$\mathcal{X}(\mathbf{Z}) = \overline{\Psi(Z_1 \propto \dots \propto Z_n)} \cap \mathcal{G}r_n^\lambda,$$

the bar denoting closure in $\dot{T}_\mu$. From now on, to simplify the writing, we will substitute $\mathcal{L}(\lambda)_\mu$ for the cumbersome compound (28), using implicitly the bijection (2).

The proof of Proposition 5.4 shows that for any $\mathbf{x} \in \mathbb{C}^n$, the irreducible components of the fiber $(\mathcal{G}r_n^\lambda \cap \dot{T}_\mu)_\mathbf{x}$ all have the same dimension and can be indexed by $\mathcal{L}(\lambda)_\mu$. Let us look more closely at two particular cases.

If $\mathbf{x} \in \mathbb{C}^n$ lies in the open locus $U$ of points with pairwise different coordinates, then, under the bijection $(\mathcal{G}r_n)_\mathbf{x} \cong (\mathrm{Gr})^n$ from Proposition 5.3, the irreducible components of $(\mathcal{G}r_n^\lambda \cap \dot{T}_\mu)_\mathbf{x}$ are identified with the sets

$$\mathcal{X}(\mathbf{Z})_\mathbf{x} \cong (Z_1 \cap \mathrm{Gr}^{\lambda_1}) \times \dots \times (Z_n \cap \mathrm{Gr}^{\lambda_n}) \qquad (29)$$

with $\mathbf{Z} = (Z_1, \dots, Z_n)$ in $\mathcal{L}(\lambda)_\mu$.

On the other hand, recalling that an element $\mathbf{Z} \in \mathcal{L}(\lambda)_\mu$ is a subset of $\overline{\mathrm{Gr}}_n^\lambda$, we may consider the preimage $\mathcal{Y}(\mathbf{Z})$ of $\Delta \times (\mathbf{Z} \cap \mathrm{Gr}_n^\lambda)$ under the isomorphism $\mathcal{G}r_n|_{\Delta} \cong \Delta \times \mathrm{Gr}_n$.
The second one requires the notion of a universally locally acyclic complex (see [10, Sect. 5.1]). Specifically, $\mathcal{B}(\lambda)$ is $(\pi \circ m_n)$-ULA ([42, proof of Proposition IV.3.4] or [43, Lemma 3.20]), so there is an isomorphism

$$i^*\mathcal{B}(\lambda) \to i^!\mathcal{B}(\lambda)[2n].$$

Then

$$
\begin{align*}
H^k((\dot{T}_\mu)_x, (i'_\mu)^! i^* \mathcal{B}(\lambda)) &= H^k((\dot{T}_\mu)_x, (i'_\mu)^! i^! \mathcal{B}(\lambda)[2n]) \\
&= H^k(\{\mathbf{x}\}, (\pi \circ m_n)_* (i'_\mu)_* (i'_\mu)^! i^! \mathcal{B}(\lambda)[2n]) \\
&= H^k(\{\mathbf{x}\}, (\pi \circ m_n)_* (i'_\mu)_* i^! (i_\mu)^! \mathcal{B}(\lambda)[2n]) \\
&= H^k(\{\mathbf{x}\}, (i_0)^!(\pi \circ m_n)_* (i_\mu)_* (i'_\mu)^! \mathcal{B}(\lambda)[2n]),
\end{align*}
$$

the last step being proper base change. Now $(\pi \circ m_n)_*(i'_\mu)_*(i_\mu)^! \mathcal{B}(\lambda)$ is the local system $\mathcal{L}_\mu(\lambda)$ shifted by $-k$, and therefore

$$
\begin{align*}
H^k((\dot{T}_\mu)_x, (i'_\mu)^! i^* \mathcal{B}(\lambda)) &= H^0(\{\mathbf{x}\}, (i_0)^! \mathcal{L}_\mu(\lambda)[2n]) = H^0(\{\mathbf{x}\}, (i_0)^* \mathcal{L}_\mu(\lambda)) \\
&= (\mathcal{L}_\mu(\lambda))_x
\end{align*}
$$

as desired.

<figure><img src="image_16.png" /></figure>

By Proposition 5.4, the irreducible components of $\mathcal{Gr}_n^\lambda \cap \dot{T}_\mu$ are all top-dimensional and can be indexed by

$$ \bigcup_{\substack{(\mu_1, \dots, \mu_n) \in \Lambda^n \\ \mu_1 + \dots + \mu_n = \mu}} \mathcal{L}(\lambda_1)_{\mu_1} \times \dots \times \mathcal{L}(\lambda_n)_{\mu_n}; \qquad (28) $$

namely, to a tuple $\mathbf{Z} = (Z_1, \dots, Z_n)$ is assigned the component

$$ \mathcal{X}(\mathbf{Z}) = \overline{\Psi(Z_1 \propto \dots \propto Z_n)} \cap \mathcal{Gr}_n^\lambda, $$

the bar denoting closure in $\dot{T}_\mu$. From now on, to simplify the writing, we will substitute $\mathcal{L}(\lambda)_\mu$ for the cumbersome compound (28), using implicitly the bijection (2).

The proof of Proposition 5.4 shows that for any $\mathbf{x} \in \mathbb{C}^n$, the irreducible components of the fiber $(\mathcal{Gr}_n^\lambda \cap \dot{T}_\mu)_\mathbf{x}$ all have the same dimension and can be indexed by $\mathcal{L}(\lambda)_\mu$. Let us look more closely at two particular cases.

If $\mathbf{x} \in \mathbb{C}^n$ lies in the open locus $U$ of points with pairwise different coordinates, then, under the bijection $(\mathcal{Gr}_n)_\mathbf{x} \cong (\mathrm{Gr})^n$ from Proposition 5.3, the irreducible components of $(\mathcal{Gr}_n^\lambda \cap \dot{T}_\mu)_\mathbf{x}$ are identified with the sets

$$ \mathcal{X}(\mathbf{Z})_\mathbf{x} \cong (Z_1 \cap \mathrm{Gr}^{\lambda_1}) \times \dots \times (Z_n \cap \mathrm{Gr}^{\lambda_n}) \qquad (29) $$

with $\mathbf{Z} = (Z_1, \dots, Z_n)$ in $\mathcal{L}(\lambda)_\mu$.

On the other hand, recalling that an element $\mathbf{Z} \in \mathcal{L}(\lambda)_\mu$ is a subset of $\overline{\mathrm{Gr}_n^\lambda}$, we may consider the preimage $\mathcal{Y}(\mathbf{Z})$ of $\Delta \times (\mathbf{Z} \cap \mathrm{Gr}_n^\lambda)$ under the isomorphism $\mathcal{Gr}_n|_{\Delta} \cong \Delta \times \mathrm{Gr}_n$.
Then for any $\mathbf{x} \in \Delta$, the irreducible components of the fiber $(\mathcal{G}r_n^\lambda \cap \dot{T}_\mu)_\mathbf{x}$ are the sets $\mathcal{Y}(\mathbf{Z})_\mathbf{x}$ for $\mathbf{Z} \in \mathcal{Z}(\lambda)_\mu$.

Let us introduce a last piece of notation before stating the next theorem. In Sect. 2.3, we explained the construction of the MV basis of the $\mu$-weight space of $V(\lambda)$. This basis is in bijection with $\mathcal{Z}(\lambda)_\mu$ and we denote by $\langle \mathbf{Z} \rangle$ the element indexed by $\mathbf{Z}$. On the other hand, given $\mathbf{Z} = (Z_1, \dots, Z_n)$ in $\mathcal{Z}(\lambda_1) \times \dots \times \mathcal{Z}(\lambda_n)$, we can look at $\langle\langle \mathbf{Z} \rangle\rangle = \langle Z_1 \rangle \otimes \dots \otimes \langle Z_n \rangle$, another element in $V(\lambda)$.

**Theorem 5.8.** Let $(\mathbf{Z}', \mathbf{Z}'') \in (\mathcal{Z}(\lambda))_\mu^2$. The coefficient $a_{\mathbf{Z}', \mathbf{Z}''}$ in the expansion

$$ \langle\langle \mathbf{Z}'' \rangle\rangle = \sum_{\mathbf{Z} \in \mathcal{Z}(\lambda)_\mu} a_{\mathbf{Z}, \mathbf{Z}''} \langle \mathbf{Z} \rangle $$

is the multiplicity of $\mathcal{Y}(\mathbf{Z}')$ in the intersection product $\mathcal{X}(\mathbf{Z}'') \cdot (\mathcal{G}r_n^\lambda)|_\Delta$ computed in the ambient space $\mathcal{G}r_n^\lambda$.

*Proof.* Taking into account Lemma 5.7, the diagram (愛) can be rewritten as follows:

$$ \begin{aligned} H^0(\mathbb{C}^n, \mathcal{L}_\mu(\lambda)) &\xrightarrow{\approx} H_{\text{top}}^{\text{BM}}(\mathcal{G}r_n^\lambda \cap \dot{T}_\mu) \\ \xrightarrow{\approx} \langle\langle \mathcal{L}_\mu(\lambda) \rangle\rangle_\mathbf{x} &\xrightarrow{\approx} H_{\text{top}}^{\text{BM}}(\langle\langle \mathcal{G}r_n^\lambda \cap \dot{T}_\mu \rangle\rangle_\mathbf{x}) \end{aligned} $$

The fundamental classes of the irreducible components of $\mathcal{G}r_n^\lambda \cap \dot{T}_\mu$ and $(\mathcal{G}r_n^\lambda \cap \dot{T}_\mu)_\mathbf{x}$ provide bases of the two Borel-Moore homology groups, both indexed by $\mathcal{Z}(\lambda)_\mu$. In these bases, the right vertical arrow can be regarded as a matrix, say $Q_\mathbf{x}$. This matrix can be computed by intersection theory: applying [15, Theorem 19.2], we see that if $\mathbf{x} \in U$ (respectively, $\mathbf{x} \in \Delta$), then the entry in $Q_\mathbf{x}$ at position $(\mathbf{Z}', \mathbf{Z}'')$ is the multiplicity of $\mathcal{X}(\mathbf{Z}')_\mathbf{x}$ (respectively, $\mathcal{Y}(\mathbf{Z}')_\mathbf{x}$) in the intersection product

$$ \mathcal{X}(\mathbf{Z}'') \cdot (\mathcal{G}r_n^\lambda)_\mathbf{x} $$

computed in the ambient space $\mathcal{G}r_n^\lambda$. Identifying $\mathcal{G}r_n|_U$ with $U \times (\text{Gr})^n$ by virtue of Proposition 5.3 and using the description over $U$ of $\mathcal{X}(\mathbf{Z}')$ and $\mathcal{X}(\mathbf{Z}'')$ afforded by (29), we see that $Q_\mathbf{x}$ is the identity matrix for each point $\mathbf{x} \in U$.

According to the discussion at the end of Sect. 5.3, the geometric Satake correspondence identifies $V(\lambda)_\mu$ with each fiber of the local system $\mathcal{L}_\mu(\lambda)$. The basis element $\langle \mathbf{Z} \rangle$ is the fundamental class of $\mathcal{X}(\mathbf{Z})_\mathbf{x}$ when $\mathbf{x} \in \Delta$, and the basis element $\langle\langle \mathbf{Z} \rangle\rangle$ is the fundamental class of $\mathcal{X}(\mathbf{Z})_\mathbf{x}$ when $\mathbf{x} \in U$. Therefore, the coefficient $a_{\mathbf{Z}', \mathbf{Z}''}$ in the statement of the theorem is the entry at position $(\mathbf{Z}', \mathbf{Z}'')$ in the product $Q_{x_\Delta} \times (Q_{x_U})^{-1}$, for any choice of $(x_\Delta, x_U) \in \Delta \times U$. ■

In particular, the entries $a_{\mathbf{Z}', \mathbf{Z}''}$ of the transition matrix between our two bases are nonnegative integers.

**Proposition 5.9.** In the setup of Theorem 5.8, the diagonal entry $a_{\mathbf{Z}', \mathbf{Z}''}$ is equal to 1.
Then for any $\mathbf{x} \in \Delta$, the irreducible components of the fiber $(\mathcal{G}r_n^\lambda \cap \dot{T}_\mu)_\mathbf{x}$ are the sets $\mathcal{Y}(\mathbf{Z})_\mathbf{x}$ for $\mathbf{Z} \in \mathcal{Z}(\lambda)_\mu$.

Let us introduce a last piece of notation before stating the next theorem. In Sect. 2.3, we explained the construction of the MV basis of the $\mu$-weight space of $V(\lambda)$. This basis is in bijection with $\mathcal{L}(\lambda)_\mu$ and we denote by $\langle \mathbf{Z} \rangle$ the element indexed by $\mathbf{Z}$. On the other hand, given $\mathbf{Z} = (Z_1, \dots, Z_n)$ in $\mathcal{L}(\lambda_1) \times \dots \times \mathcal{L}(\lambda_n)$, we can look at $\langle\langle \mathbf{Z} \rangle\rangle = \langle Z_1 \rangle \otimes \dots \otimes \langle Z_n \rangle$, another element in $V(\lambda)$.

**Theorem 5.8.** Let $(\mathbf{Z}', \mathbf{Z}'') \in (\mathcal{Z}(\lambda))_\mu)^2$. The coefficient $a_{\mathbf{Z}', \mathbf{Z}''}$ in the expansion

$$ \langle\langle \mathbf{Z}'' \rangle\rangle = \sum_{\mathbf{Z} \in \mathcal{Z}(\lambda)_\mu} a_{\mathbf{Z}, \mathbf{Z}''} \langle \mathbf{Z} \rangle $$

is the multiplicity of $\mathcal{Y}(\mathbf{Z}')$ in the intersection product $\mathcal{X}(\mathbf{Z}'') \cdot (\mathcal{G}r_n^\lambda)|_\Delta$ computed in the ambient space $\mathcal{G}r_n^\lambda$.

*Proof.* Taking into account Lemma 5.7, the diagram (愛) can be rewritten as follows:

$$ \begin{aligned} H^0(\mathbb{C}^n, \mathcal{L}_\mu(\lambda)) &\xrightarrow{\approx} H_{\text{top}}^{\text{BM}}(\mathcal{G}r_n^\lambda \cap \dot{T}_\mu) \\ \xrightarrow{\approx} (\mathcal{L}_\mu(\lambda))_\mathbf{x} &\xrightarrow{\approx} H_{\text{top}}^{\text{BM}}((\mathcal{G}r_n^\lambda \cap \dot{T}_\mu)_\mathbf{x}) \end{aligned} $$

The fundamental classes of the irreducible components of $\mathcal{G}r_n^\lambda \cap \dot{T}_\mu$ and $(\mathcal{G}r_n^\lambda \cap \dot{T}_\mu)_\mathbf{x}$ provide bases of the two Borel-Moore homology groups, both indexed by $\mathcal{L}(\lambda)_\mu$. In these bases, the right vertical arrow can be regarded as a matrix, say $Q_\mathbf{x}$. This matrix can be computed by intersection theory: applying [15, Theorem 19.2], we see that if $\mathbf{x} \in U$ (respectively, $\mathbf{x} \in \Delta$), then the entry in $Q_\mathbf{x}$ at position $(\mathbf{Z}', \mathbf{Z}'')$ is the multiplicity of $\mathcal{X}(\mathbf{Z}')_\mathbf{x}$ (respectively, $\mathcal{Y}(\mathbf{Z}')_\mathbf{x}$) in the intersection product

$$ \mathcal{X}(\mathbf{Z}'') \cdot (\mathcal{G}r_n^\lambda)_\mathbf{x} $$

computed in the ambient space $\mathcal{G}r_n^\lambda$. Identifying $\mathcal{G}r_n|_U$ with $U \times (\text{Gr})^n$ by virtue of Proposition 5.3 and using the description over $U$ of $\mathcal{X}(\mathbf{Z}')$ and $\mathcal{X}(\mathbf{Z}'')$ afforded by (29), we see that $Q_\mathbf{x}$ is the identity matrix for each point $\mathbf{x} \in U$.

According to the discussion at the end of Sect. 5.3, the geometric Satake correspondence identifies $V(\lambda)_\mu$ with each fiber of the local system $\mathcal{L}_\mu(\lambda)$. The basis element $\langle \mathbf{Z} \rangle$ is the fundamental class of $\mathcal{X}(\mathbf{Z})_\mathbf{x}$ when $\mathbf{x} \in \Delta$, and the basis element $\langle\langle \mathbf{Z} \rangle\rangle$ is the fundamental class of $\mathcal{X}(\mathbf{Z})_\mathbf{x}$ when $\mathbf{x} \in U$. Therefore, the coefficient $a_{\mathbf{Z}', \mathbf{Z}''}$ in the statement of the theorem is the entry at position $(\mathbf{Z}', \mathbf{Z}'')$ in the product $Q_{x_\Delta} \times (Q_{x_U})^{-1}$, for any choice of $(x_\Delta, x_U) \in \Delta \times U$. ■

In particular, the entries $a_{\mathbf{Z}', \mathbf{Z}''}$ of the transition matrix between our two bases are nonnegative integers.

**Proposition 5.9.** In the setup of Theorem 5.8, the diagonal entry $a_{\mathbf{Z}', \mathbf{Z}''}$ is equal to 1.
*Proof.* Write $\mathbb{Z}'' = (Z_1, \dots, Z_n)$ in $\mathcal{Z}(\lambda_1) \times \dots \times \mathcal{Z}(\lambda_n)$. By the slice theorem applied to the quotient map $G^\vee(\mathbb{C}[z, z^{-1}]) \to \text{Gr}$ (or, in this concrete situation, using [17, Remark 15 and Corollary 5]), we can find, for each $j \in \{1, \dots, n\}$, an affine variety $U_j$ and a map $\phi_j : U_j \to G^\vee(\mathbb{C}[z, z^{-1}])$ such that $u \mapsto [\phi_j(u)]$ sends $U_j$ isomorphically to an open subset of $\text{Gr}^{\lambda_j}$ which meets $Z_j$.

For $x \in \mathbb{C}$ and $u \in U_j$, let $\phi_j(u)|_x$ denote the result of substituting $z-x$ for $z$ in $\phi_j(u)$. We can then define an open embedding $\phi$ as in the diagram

<figure><img src="image_18.png" /></figure>

by setting

$$ \phi(x_1, \dots, x_n; u_1, \dots, u_n) = (x_1, \dots, x_n; [\phi_1(u_1)|_{x_1}, \dots, \phi_n(u_n)|_{x_n}]). $$

Since intersection multiplicities are of local nature, $a_{\mathbb{Z}''\mathbb{Z}''}$ can be computed after restriction to the image of $\phi$, where the situation is that of a trivial bundle. ■

### 5.5. An example

It is possible to put coordinates on $\mathcal{G}r_n^\lambda$ and to effectively compute the intersection multiplicities mentioned in Theorem 5.8. In this section, we look at the case of the group $G = \text{SL}_3$. We adopt the usual description

$$ \Lambda = (\mathbb{Z}\varepsilon_1 \oplus \mathbb{Z}\varepsilon_2 \oplus \mathbb{Z}\varepsilon_3) / \mathbb{Z}(\varepsilon_1 + \varepsilon_2 + \varepsilon_3) $$

of the weight lattice, so that $V(\varepsilon_1)$ is the defining representation of $G$ and $V(-\varepsilon_3)$ is its dual.

We consider the sequence of dominant weights $\lambda = (\varepsilon_1, -\varepsilon_3)$. The basic MV cycles are $Z_i = \text{Gr}^{\varepsilon_1} \cap T_{\varepsilon_i}$ and $Z_{-i} = \text{Gr}^{-\varepsilon_3} \cap T_{-\varepsilon_i}$ for $i \in \{1, 2, 3\}$, and with this notation

$$ \mathcal{Z}(\lambda) = \{(Z_i, Z_{-j}) \mid (i, j) \in \{1, 2, 3\}^2\}. $$

To abbreviate, we set $Z_{i,-j} = (Z_i, Z_{-j})$. For weight reasons, $\langle\langle Z_{i,-j} \rangle\rangle = \langle Z_{i,-j} \rangle$ if $i \neq j$. The rest of the transition matrix between the two bases is given as follows:

$$ \langle\langle Z_{1,-1} \rangle\rangle = \langle Z_{1,-1} \rangle, $$

$$ \langle\langle Z_{2,-2} \rangle\rangle = \langle Z_{2,-2} \rangle + \langle Z_{1,-1} \rangle, $$

$$ \langle\langle Z_{3,-3} \rangle\rangle = \langle Z_{3,-3} \rangle + \langle Z_{2,-2} \rangle. $$

From these relations, we get $\langle Z_{3,-3} \rangle = \langle\langle Z_{3,-3} \rangle\rangle - \langle\langle Z_{2,-2} \rangle\rangle + \langle\langle Z_{1,-1} \rangle\rangle$. This allows one to check that $\langle Z_{3,-3} \rangle$ is $G$-invariant, which in truth is a consequence of the compatibility of the MV basis of $V(\lambda)$ with the isotypic filtration (Theorem 3.4).

As an example, let us sketch out a computation which justifies that $\langle Z_{1,-1} \rangle$ appears with coefficient 1 in $\langle\langle Z_{2,-2} \rangle\rangle$. We consider two charts on $\mathcal{G}r_2^\lambda$, both with $\mathbb{C}^6$ as domain:
*Proof.* Write $\mathbb{Z}'' = (Z_1, \dots, Z_n)$ in $\mathcal{Z}(\lambda_1) \times \dots \times \mathcal{Z}(\lambda_n)$. By the slice theorem applied to the quotient map $G^\vee(\mathbb{C}[z, z^{-1}]) \to \mathrm{Gr}$ (or, in this concrete situation, using [17, Remark 15 and Corollary 5]), we can find, for each $j \in \{1, \dots, n\}$, an affine variety $U_j$ and a map $\phi_j : U_j \to G^\vee(\mathbb{C}[z, z^{-1}])$ such that $u \mapsto [\phi_j(u)]$ sends $U_j$ isomorphically to an open subset of $\mathrm{Gr}^{\lambda_j}$ which meets $Z_j$.

For $x \in \mathbb{C}$ and $u \in U_j$, let $\phi_j(u)|_x$ denote the result of substituting $z-x$ for $z$ in $\phi_j(u)$. We can then define an open embedding $\phi$ as in the diagram

$$ \begin{tikzcd} \mathbb{C}^n \times (U_1 \times \dots \times U_n) \arrow[r, "\phi"] & \mathcal{G}r_n^{\lambda} \arrow[d, "\pi \circ m_n"] \\ \mathbb{C}^n \arrow[d, "\cong"] & \mathbb{C}^n \arrow[d, "\cong"'] \arrow[r, "\phi(x_1, \dots, x_n; u_1, \dots, u_n)"] \arrow[d, "\cong"'] \end{tikzcd} $$

by setting

$$ \phi(x_1, \dots, x_n; u_1, \dots, u_n) = (x_1, \dots, x_n; [\phi_1(u_1)|_{x_1}, \dots, \phi_n(u_n)|_{x_n}]). $$

Since intersection multiplicities are of local nature, $a_{\mathbb{Z}''\mathbb{Z}''}$ can be computed after restriction to the image of $\phi$, where the situation is that of a trivial bundle. ■

### 5.5. An example

It is possible to put coordinates on $\mathcal{G}r_n^{\lambda}$ and to effectively compute the intersection multiplicities mentioned in Theorem 5.8. In this section, we look at the case of the group $G = \mathrm{SL}_3$. We adopt the usual description

$$ \Lambda = (\mathbb{Z}\varepsilon_1 \oplus \mathbb{Z}\varepsilon_2 \oplus \mathbb{Z}\varepsilon_3) / \mathbb{Z}(\varepsilon_1 + \varepsilon_2 + \varepsilon_3) $$

of the weight lattice, so that $V(\varepsilon_1)$ is the defining representation of $G$ and $V(-\varepsilon_3)$ is its dual.

We consider the sequence of dominant weights $\lambda = (\varepsilon_1, -\varepsilon_3)$. The basic MV cycles are $Z_i = \mathrm{Gr}^{\varepsilon_1} \cap T_{\varepsilon_i}$ and $Z_{-i} = \mathrm{Gr}^{-\varepsilon_3} \cap T_{-\varepsilon_i}$ for $i \in \{1, 2, 3\}$, and with this notation

$$ \mathcal{Z}(\lambda) = \{(Z_i, Z_{-j}) \mid (i, j) \in \{1, 2, 3\}^2\}. $$

To abbreviate, we set $Z_{i,-j} = (Z_i, Z_{-j})$. For weight reasons, $\langle\langle Z_{i,-j} \rangle\rangle = \langle Z_{i,-j} \rangle$ if $i \neq j$. The rest of the transition matrix between the two bases is given as follows:

$$ \begin{align*} \langle\langle \mathbf{Z}_{1,-1} \rangle\rangle &= \langle\langle \mathbf{Z}_{1,-1} \rangle\rangle, \\ \langle\langle \mathbf{Z}_{2,-2} \rangle\rangle &= \langle\langle \mathbf{Z}_{2,-2} \rangle\rangle + \langle\langle \mathbf{Z}_{1,-1} \rangle\rangle, \\ \langle\langle \mathbf{Z}_{3,-3} \rangle\rangle &= \langle\langle \mathbf{Z}_{3,-3} \rangle\rangle + \langle\langle \mathbf{Z}_{2,-2} \rangle\rangle. \end{align*} $$

From these relations, we get $\langle\langle \mathbf{Z}_{3,-3} \rangle\rangle = \langle\langle \mathbf{Z}_{3,-3} \rangle\rangle - \langle\langle \mathbf{Z}_{2,-2} \rangle\rangle + \langle\langle \mathbf{Z}_{1,-1} \rangle\rangle$. This allows one to check that $\langle\langle \mathbf{Z}_{3,-3} \rangle\rangle$ is $G$-invariant, which in truth is a consequence of the compatibility of the MV basis of $V(\lambda)$ with the isotypic filtration (Theorem 3.4).

As an example, let us sketch out a computation which justifies that $\langle\langle \mathbf{Z}_{1,-1} \rangle\rangle$ appears with coefficient 1 in $\langle\langle \mathbf{Z}_{2,-2} \rangle\rangle$. We consider two charts on $\mathcal{G}r_2^{\lambda}$, both with $\mathbb{C}^6$ as domain:
$$
\begin{align*}
\phi_1 : (x_1, x_2, a, b, c, d) &\mapsto \left( x_1, x_2; \begin{bmatrix} z-x_1 & a & b \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{bmatrix}, \begin{pmatrix} 1 & 0 & 0 \\ c & z-x_2 & 0 \\ d & 0 & z-x_2 \end{pmatrix} \right), \\
\phi_2 : (x_1, x_2, a', b', c', d') &\mapsto \left( x_1, x_2; \begin{bmatrix} 1 & 0 & 0 \\ a' & z-x_1 & b' \\ 0 & 0 & 1 \end{bmatrix}, \begin{pmatrix} z-x_2 & c' & 0 \\ 0 & 1 & 0 \\ 0 & d' & z-x_2 \end{pmatrix} \right).
\end{align*}
$$

(The matrices here belong to the group PGL$_3(\mathbb{C}[z - x_1, z - x_2])$. One easily computes the transition map between these two charts:

$$
a' = 1/a, \quad b' = -b/a, \quad c' = -a(ac + bd + x_2 - x_1), \quad d' = -ad.
$$

In the chart $\phi_1$, the cycle $\mathcal{Y}(\mathbf{Z}_{1,-1})$ is defined by the equations $a = b = x_2 - x_1 = 0$.
In the chart $\phi_2$, the cycle $\mathcal{X}(\mathbf{Z}_{2,-2})$ is defined by the equations $b' = c' = 0$.
Thus, the
ideals in $R = \mathbb{C}[x_1, x_2, a, b, c, d]$ of the subvarieties

$$
V = \phi_1^{-1}(\mathcal{Y}(\mathbf{Z}_{1,-1})) \quad \text{and} \quad X = \phi_1^{-1}(\mathcal{X}(\mathbf{Z}_{2,-2}))
$$

are respectively

$$
\mathfrak{p} = (a, b, x_2 - x_1) \quad \text{and} \quad \mathfrak{q} = (b, ac + x_2 - x_1).
$$

Since $\mathfrak{q} \subseteq \mathfrak{p}$, we have $V \subseteq X$; in fact, $V$ is a subvariety of $X$ of codimension 1. The local
ring $A = \mathcal{O}_{V,X}$ of $X$ along $V$ is the localization of $R/\mathfrak{q}$ at the ideal $\mathfrak{p}/\mathfrak{q}$. Observing that
$c$ is not in $\mathfrak{p}$, we see that its image in $A$ is invertible, and then that $x_2 - x_1$ generates the
maximal ideal of $A$. As a consequence, the order of vanishing of $x_2 - x_1$ along $V$ (see
[15, Sect. 1.2]) is 1. By definition, this is the multiplicity of $\mathcal{Y}(\mathbf{Z}_{1,-1})$ in the intersection
product $\mathcal{X}(\mathbf{Z}_{2,-2}) \cdot \mathfrak{g} r_2^{\lambda}|_{\Delta}$.

5.6. Factorizations

A nice feature of the Beilinson–Drinfeld Grassmannian is its factorizable structure (see for
instance [42, Proposition II.1.13]). On the other side of the geometric Satake equivalence,
this corresponds to associativity properties of partial tensor products.

Let $\mathbf{n} = (n_1, \dots, n_r)$ be a composition of $n$ with $r$ parts. We define the partial diagonal

$$
\Delta_{\mathbf{n}} = \{ \underbrace{(x_1, \dots, x_1)}_{n_1 \text{ times}} , \dots , \underbrace{(x_r, \dots, x_r)}_{n_r \text{ times}} \mid (x_1, \dots, x_r) \in \mathbb{C}^r \}.
$$

We write $\boldsymbol{\lambda}$ as a concatenation $(\boldsymbol{\lambda}_{(1)}, \dots, \boldsymbol{\lambda}_{(r)})$, where each $\boldsymbol{\lambda}_{(j)}$ belongs to $(\Lambda^+)^{n_j}$, and similarly we write each $\boldsymbol{Z} \in \mathcal{Z}(\boldsymbol{\lambda})_{\mu}$ as $(\boldsymbol{Z}_{(1)}, \dots, \boldsymbol{Z}_{(r)})$ with $\boldsymbol{Z}_{(j)} \in \mathcal{Z}(\boldsymbol{\lambda}_{(j)})$. Then

$$
V(\boldsymbol{\lambda}) = V(\boldsymbol{\lambda}_{(1)}) \otimes \cdots \otimes V(\boldsymbol{\lambda}_{(r)}) \quad \text{and} \quad \langle \boldsymbol{Z}_{(j)} \rangle \in V(\boldsymbol{\lambda}_{(j)}).
$$

Further, define

$$
\mathcal{X}(\mathbf{Z}, \mathbf{n}) = \overline{\Psi(\overline{Z_1 \alpha \cdots \alpha Z_n})|_{\Delta_n} \cap \mathfrak{g} r_n^{\lambda}}
$$
$$
\begin{align*}
\phi_1 : (x_1, x_2, a, b, c, d) &\mapsto \left( x_1, x_2; \begin{bmatrix} z-x_1 & a & b \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{bmatrix}, \begin{pmatrix} 1 & 0 & 0 \\ c & z-x_2 & 0 \\ d & 0 & z-x_2 \end{pmatrix} \right), \\
\phi_2 : (x_1, x_2, a', b', c', d') &\mapsto \left( x_1, x_2; \begin{bmatrix} 1 & 0 & 0 \\ a' & z-x_1 & b' \\ 0 & 0 & 1 \end{bmatrix}, \begin{pmatrix} z-x_2 & c' & 0 \\ 0 & 1 & 0 \\ 0 & d' & z-x_2 \end{pmatrix} \right).
\end{align*}
$$

(The matrices here belong to the group PGL$_3(\mathbb{C}[z - x_1, z - x_2])$. One easily computes
the transition map between these two charts:

$$
a' = 1/a, \quad b' = -b/a, \quad c' = -a(ac + bd + x_2 - x_1), \quad d' = -ad.
$$

In the chart $\phi_1$, the cycle $\mathcal{Y}(\mathbf{Z}_{1,-1})$ is defined by the equations $a = b = x_2 - x_1 = 0$.
In the chart $\phi_2$, the cycle $\mathcal{X}(\mathbf{Z}_{2,-2})$ is defined by the equations $b' = c' = 0$.
Thus, the
ideals in $R = \mathbb{C}[x_1, x_2, a, b, c, d]$ of the subvarieties

$$
V = \phi_1^{-1}(\mathcal{Y}(\mathbf{Z}_{1,-1})) \quad \text{and} \quad X = \phi_1^{-1}(\mathcal{X}(\mathbf{Z}_{2,-2}))
$$

are respectively

$$
\mathfrak{p} = (a, b, x_2 - x_1) \quad \text{and} \quad \mathfrak{q} = (b, ac + x_2 - x_1).
$$

Since $\mathfrak{q} \subseteq \mathfrak{p}$, we have $V \subseteq X$; in fact, $V$ is a subvariety of $X$ of codimension 1. The local
ring $A = \mathcal{O}_{V,X}$ of $X$ along $V$ is the localization of $R/\mathfrak{q}$ at the ideal $\mathfrak{p}/\mathfrak{q}$. Observing that
$c$ is not in $\mathfrak{p}$, we see that its image in $A$ is invertible, and then that $x_2 - x_1$ generates the
maximal ideal of $A$. As a consequence, the order of vanishing of $x_2 - x_1$ along $V$ (see
[15, Sect. 1.2]) is 1. By definition, this is the multiplicity of $\mathcal{Y}(\mathbf{Z}_{1,-1})$ in the intersection
product $\mathcal{X}(\mathbf{Z}_{2,-2}) \cdot \mathfrak{g} r_2^{\lambda}|_{\Delta}$.

5.6. Factorizations

A nice feature of the Beilinson–Drinfeld Grassmannian is its factorizable structure (see for
instance [42, Proposition II.1.13]). On the other side of the geometric Satake equivalence,
this corresponds to associativity properties of partial tensor products.

Let $\mathbf{n} = (n_1, \dots, n_r)$ be a composition of $n$ with $r$ parts. We define the partial diagonal

$$
\Delta_{\mathbf{n}} = \{ (x_1, \dots, x_1, \dots, x_r, \dots, x_r) \mid (x_1, \dots, x_r) \in \mathbb{C}^r \}.
$$

We write $\boldsymbol{\lambda}$ as a concatenation $(\boldsymbol{\lambda}_{(1)}, \dots, \boldsymbol{\lambda}_{(r)})$, where each $\boldsymbol{\lambda}_{(j)}$ belongs to $(\Lambda^+)^{n_j}$, and similarly we write each $\boldsymbol{Z} \in \mathcal{Z}(\boldsymbol{\lambda})_{\mu}$ as $(\boldsymbol{Z}_{(1)}, \dots, \boldsymbol{Z}_{(r)})$ with $\boldsymbol{Z}_{(j)} \in \mathcal{Z}(\boldsymbol{\lambda}_{(j)})$. Then

$$
V(\boldsymbol{\lambda}) = V(\boldsymbol{\lambda}_{(1)}) \otimes \cdots \otimes V(\boldsymbol{\lambda}_{(r)}) \quad \text{and} \quad \langle \boldsymbol{Z}_{(j)} \rangle \in V(\boldsymbol{\lambda}_{(j)}).
$$

Further, define

$$
\mathcal{X}(\mathbf{Z}, \mathbf{n}) = \overline{\Psi(\overline{Z_1 \propto \dots \propto Z_n})|_{\Delta_n} \cap \mathfrak{g} r_n^{\lambda}}
$$
where the bar means closure in $(\dot{T}_\mu)|_{\Delta_n}$. The $\mathcal{X}(\mathbf{Z}, \mathbf{n})$ generalize the set $\mathcal{X}(\mathbf{Z})$ defined in Sect. 5.4, as the latter corresponds to the composition $(1, \dots, 1)$.

Theorem 5.8 can then be extended to this context in a straightforward fashion, as demonstrated by the following statement.

**Proposition 5.10.** Let $(\mathbf{Z}', \mathbf{Z}'') \in (\mathcal{Z}(\lambda))_{\mu}^2$. The coefficient $b_{\mathbf{Z}', \mathbf{Z}''}$ in the expansion

$$ \langle \mathbf{Z}''_{(1)} \rangle \otimes \dots \otimes \langle \mathbf{Z}''_{(r)} \rangle = \sum_{\mathbf{Z} \in \mathcal{Z}(\lambda)_{\mu}} b_{\mathbf{Z}, \mathbf{Z}''} \langle \mathbf{Z} \rangle $$

is the multiplicity of $\mathcal{Y}(\mathbf{Z}')$ in the intersection product $\mathcal{X}(\mathbf{Z}'', \mathbf{n}) \cdot (\mathcal{G}r_n^\lambda)|_{\Delta}$ computed in the ambient space $\mathcal{G}r_n^\lambda|_{\Delta_n}$.

The proof does not require any new ingredient and is left to the reader.

## 5.7. Triangularity

In this section, we show that the transition matrix described in Theorem 5.8 is unitriangular with respect to a suitable order on $\mathcal{Z}(\lambda)_{\mu}$.

**Proposition 5.11.** Let $(\mu_1, \dots, \mu_n)$ and $(\nu_1, \dots, \nu_n)$ in $\Lambda^n$ and let $\mathcal{S}$ be a stratum for the ind-structure of $\mathcal{G}r_n$. If $\Psi(T_{\nu_1} \propto \dots \propto T_{\nu_n})$ meets the closure of $\mathcal{S} \cap \Psi(T_{\mu_1} \propto \dots \propto T_{\mu_n})$, then

$$ \nu_1 \ge \mu_1, \quad \nu_1 + \nu_2 \ge \mu_1 + \mu_2, \quad \dots, \quad \nu_1 + \dots + \nu_n \ge \mu_1 + \dots + \mu_n. $$

*Proof.* Given a tuple $\xi = (\xi_1, \dots, \xi_n)$ in $(\Lambda/\mathbb{Z}\Phi)^n$, we set

$$ \mathcal{G}r_{n,\xi} = \bigsqcup_{\substack{\lambda \in (\Lambda^+)^n \\ \lambda_1 \in \xi_1, \dots, \lambda_n \in \xi_n}} \mathcal{G}r_n^\lambda. $$

From (25), we deduce that each $\mathcal{G}r_{n,\xi}$ is closed and connected in the ind-topology. As these subsets form a finite partition of the space $\mathcal{G}r_n$, they are its connected components. We easily verify that a subset of the form $\Psi(T_{\mu_1} \propto \dots \propto T_{\mu_n})$ is contained in $\mathcal{G}r_{n,\xi}$ if each $\xi_j$ is the coset of $\mu_j$ modulo $\mathbb{Z}\Phi$. Therefore, a necessary condition for the set $\Psi(T_{\nu_1} \propto \dots \propto T_{\nu_n})$ to meet the closure of $\mathcal{S} \cap \Psi(T_{\mu_1} \propto \dots \propto T_{\mu_n})$ is that $\mu_j - \nu_j \in \mathbb{Z}\Phi$ for each $j \in \{1, \dots, n\}$.

Let $\lambda^\vee \in \mathrm{Hom}_{\mathbb{Z}}(\Lambda, \mathbb{Z})$ be a dominant integral weight for the group $G^\vee$ and let $V$ be the finite-dimensional irreducible representation of $G^\vee$ of highest weight $\lambda^\vee$. Then $G^\vee(\mathbb{C}(z))$ acts on $V \otimes \mathbb{C}(z)$. The standard lattice $L_0 = V \otimes \mathbb{C}[z]$ is left stable by $G^\vee(\mathbb{C}[z])$.

We choose a nonzero linear form $p: V \to \mathbb{C}$ that vanishes on all weight subspaces of $V$ but the highest weight subspace. Extending the scalars, we regard $p$ as a linear form $V \otimes \mathbb{C}(z) \to \mathbb{C}(z)$.

For $\mathbf{x} = (x_1, \dots, x_n) \in \mathbb{C}^n$, we set $f_{\mathbf{x}} = (z - x_1) \cdots (z - x_n)$. Let $\mathcal{S}$ be a stratum for the ind-structure of $\mathcal{G}r_n$. There exists a positive integer $k$ such that $f_{\mathbf{x}}^k L_0 \subseteq \beta_1 \cdots \beta_n (L_0) \subseteq f_{\mathbf{x}}^{-k} L_0$ for each $(x_1, \dots, x_n; [\beta_1, \dots, \beta_n]) \in \mathcal{S}$.
where the bar means closure in $(\dot{T}_{\mu})|_{\Delta_n}$. The $\mathcal{X}(\mathbf{Z}, \mathbf{n})$ generalize the set $\mathcal{X}(\mathbf{Z})$ defined in Sect. 5.4, as the latter corresponds to the composition $(1, \dots, 1)$.

Theorem 5.8 can then be extended to this context in a straightforward fashion, as demonstrated by the following statement.

**Proposition 5.10.** Let $(\mathbf{Z}', \mathbf{Z}'') \in (\mathcal{Z}(\lambda))_{\mu}^2$. The coefficient $b_{\mathbf{Z}',\mathbf{Z}''}$ in the expansion

$$ \langle \mathbf{Z}''_{(1)} \rangle \otimes \dots \otimes \langle \mathbf{Z}''_{(r)} \rangle = \sum_{\mathbf{Z} \in \mathcal{Z}(\lambda)_{\mu}} b_{\mathbf{Z}, \mathbf{Z}''} \langle \mathbf{Z} \rangle $$

is the multiplicity of $\mathcal{Y}(\mathbf{Z}')$ in the intersection product $\mathcal{X}(\mathbf{Z}'', \mathbf{n}) \cdot (\mathcal{G}r_n^{\lambda})|_{\Delta}$ computed in the ambient space $\mathcal{G}r_n^{\lambda}|_{\Delta_n}$.

The proof does not require any new ingredient and is left to the reader.

## 5.7. Triangularity

In this section, we show that the transition matrix described in Theorem 5.8 is unitriangular with respect to a suitable order on $\mathcal{Z}(\lambda)_{\mu}$.

**Proposition 5.11.** Let $(\mu_1, \dots, \mu_n)$ and $(\nu_1, \dots, \nu_n)$ in $\Lambda^n$ and let $\mathcal{S}$ be a stratum for the ind-structure of $\mathcal{G}r_n$. If $\Psi(T_{\nu_1} \propto \dots \propto T_{\nu_n})$ meets the closure of $\mathcal{S} \cap \Psi(T_{\mu_1} \propto \dots \propto T_{\mu_n})$, then

$$ \nu_1 \ge \mu_1, \quad \nu_1 + \nu_2 \ge \mu_1 + \mu_2, \quad \dots, \quad \nu_1 + \dots + \nu_n \ge \mu_1 + \dots + \mu_n. $$

*Proof.* Given a tuple $\xi = (\xi_1, \dots, \xi_n)$ in $(\Lambda/\mathbb{Z}\Phi)^n$, we set

$$ \mathcal{G}r_{n,\xi} = \bigsqcup_{\substack{\lambda \in (\Lambda^+)^n \\ \lambda_1 \in \xi_1, \dots, \lambda_n \in \xi_n}} \mathcal{G}r_n^{\lambda}. $$

From (25), we deduce that each $\mathcal{G}r_{n,\xi}$ is closed and connected in the ind-topology. As these subsets form a finite partition of the space $\mathcal{G}r_n$, they are its connected components. We easily verify that a subset of the form $\Psi(T_{\mu_1} \propto \dots \propto T_{\mu_n})$ is contained in $\mathcal{G}r_{n,\xi}$ if each $\xi_j$ is the coset of $\mu_j$ modulo $\mathbb{Z}\Phi$. Therefore, a necessary condition for the set $\Psi(T_{\nu_1} \propto \dots \propto T_{\nu_n})$ to meet the closure of $\mathcal{S} \cap \Psi(T_{\mu_1} \propto \dots \propto T_{\mu_n})$ is that $\mu_j - \nu_j \in \mathbb{Z}\Phi$ for each $j \in \{1, \dots, n\}$.

Let $\lambda^\vee \in \mathrm{Hom}_{\mathbb{Z}}(\Lambda, \mathbb{Z})$ be a dominant integral weight for the group $G^\vee$ and let $V$ be the finite-dimensional irreducible representation of $G^\vee$ of highest weight $\lambda^\vee$. Then $G^\vee(\mathbb{C}(z))$ acts on $V \otimes \mathbb{C}(z)$. The standard lattice $L_0 = V \otimes \mathbb{C}[z]$ is left stable by $G^\vee(\mathbb{C}[z])$.

We choose a nonzero linear form $p: V \to \mathbb{C}$ that vanishes on all weight subspaces of $V$ but the highest weight subspace. Extending the scalars, we regard $p$ as a linear form $V \otimes \mathbb{C}(z) \to \mathbb{C}(z)$.

For $\mathbf{x} = (x_1, \dots, x_n) \in \mathbb{C}^n$, we set $f_{\mathbf{x}} = (z - x_1) \cdots (z - x_n)$. Let $\mathcal{S}$ be a stratum for the ind-structure of $\mathcal{G}r_n$. There exists a positive integer $k$ such that $f_{\mathbf{x}}^k L_0 \subseteq \beta_1 \cdots \beta_n(L_0) \subseteq f_{\mathbf{x}}^{-k} L_0$ for each $(x_1, \dots, x_n; [\beta_1, \dots, \beta_n]) \in \mathcal{S}$.
Now we take $(\mu_1, \dots, \mu_n) \in \Lambda^n$ and $(x_1, \dots, x_n; [\beta_1, \dots, \beta_n])$ in the intersection $S \cap \Psi(T_{\mu_1} \propto \dots \propto T_{\mu_n})$. Then $p(\beta_1 \cdots \beta_n(L_0))$ is the fractional ideal

$$ (z - x_1)^{(\lambda^{\vee}, \mu_1)} \cdots (z - x_n)^{(\lambda^{\vee}, \mu_n)} \mathbb{C}[z], $$

and therefore

$$ \dim(p(\beta_1 \cdots \beta_n(L_0))/f_x^k \mathbb{C}[z]) = kn - \langle \lambda^{\vee}, \mu_1 + \cdots + \mu_n \rangle. $$

If the point $(x_1, \dots, x_n; [\beta_1, \dots, \beta_n])$ degenerates to

$$ (y_1, \dots, y_n; [\gamma_1, \dots, \gamma_n]) \in \Psi(T_{v_1} \propto \dots \propto T_{v_n}), $$

then

$$ \dim(p(\gamma_1 \cdots \gamma_n(L_0))/f_y^k \mathbb{C}[z]) \le \dim(p(\beta_1 \cdots \beta_n(L_0))/f_x^k \mathbb{C}[z]) $$

which translates to

$$ \langle \lambda^{\vee}, v_1 + \cdots + v_n \rangle \ge \langle \lambda^{\vee}, \mu_1 + \cdots + \mu_n \rangle. $$

This inequality holds for any dominant coweight $\lambda^{\vee}$, hence $v_1 + \cdots + v_n \ge \mu_1 + \cdots + \mu_n$.

This proves the last of the stated inequalities. The other ones can be obtained in a similar way, by taking the image under the obvious truncation map $Gr_n \to Gr_j$ for each $j \in \{1, \dots, n\}$. ■

**Corollary 5.12.** In the setup of Theorem 5.8, let $(\mu_1, \dots, \mu_n)$ and $(v_1, \dots, v_n)$ in $\Lambda^n$ be such that $Z' \in \mathcal{Z}(\lambda_1)_{v_1} \times \cdots \times \mathcal{Z}(\lambda_n)_{v_n}$ and $Z'' \in \mathcal{Z}(\lambda_1)_{\mu_1} \times \cdots \times \mathcal{Z}(\lambda_n)_{\mu_n}$. A necessary condition for $a_{Z',Z''} \ne 0$ is that

$$ v_1 \ge \mu_1, \quad v_1 + v_2 \ge \mu_1 + \mu_2, \quad \dots, \quad v_1 + \cdots + v_{n-1} \ge \mu_1 + \cdots + \mu_{n-1}. $$

We can obtain more stringent conditions regarding the transition matrix by looking at the associativity properties from Sect. 5.6. The sharpest result is obtained with a composition $(n_1, n_2)$ of $n$ with two parts. Accordingly, we write $\lambda$ as a concatenation $(\lambda_{(1)}, \lambda_{(2)})$ and similarly write each $Z \in \mathcal{Z}(\lambda)$ as $(Z_{(1)}, Z_{(2)})$. Here $Z_{(1)}$ is an element in $\mathcal{Z}(\lambda_1) \times \cdots \times \mathcal{Z}(\lambda_{n_1})$, but owing to the bijection (2) it can also be regarded as a cycle in $\overline{\mathrm{Gr}}_{n_1}^{\lambda_{(1)}}$.

**Theorem 5.13.** Let $(Z', Z'') \in (\mathcal{Z}(\lambda))_{\mu}^2$. Consider the expansion

$$ \langle Z''_{(1)} \rangle \otimes \langle Z''_{(2)} \rangle = \sum_{Z \in \mathcal{Z}(\lambda)_{\mu}} b_{Z,Z''} \langle Z \rangle. $$

If $b_{Z',Z''} \ne 0$, then either $Z' = Z''$ or $Z'_{(1)} \subsetneq \overline{Z''_{(1)}}$ as cycles in $\overline{\mathrm{Gr}}_{n_1}^{\lambda_{(1)}}$. In addition, $b_{Z'',Z''} = 1$.

*Proof.* Let $Z' = (Z'_1, \dots, Z'_n)$ and $Z'' = (Z''_1, \dots, Z''_n)$ in $\mathcal{Z}(\lambda)_{\mu}$.
Now we take $(\mu_1, \dots, \mu_n) \in \Lambda^n$ and $(x_1, \dots, x_n; [\beta_1, \dots, \beta_n])$ in the intersection $S \cap \Psi(T_{\mu_1} \propto \dots \propto T_{\mu_n})$. Then $p(\beta_1 \cdots \beta_n(L_0))$ is the fractional ideal

$$ (z - x_1)^{(\lambda^{\vee}, \mu_1)} \cdots (z - x_n)^{(\lambda^{\vee}, \mu_n)} \mathbb{C}[z], $$

and therefore

$$ \dim(p(\beta_1 \cdots \beta_n(L_0))/f_x^k \mathbb{C}[z]) = kn - \langle \lambda^{\vee}, \mu_1 + \cdots + \mu_n \rangle. $$

If the point $(x_1, \dots, x_n; [\beta_1, \dots, \beta_n])$ degenerates to

$$ (y_1, \dots, y_n; [\gamma_1, \dots, \gamma_n]) \in \Psi(T_{v_1} \propto \dots \propto T_{v_n}), $$

then

$$ \dim(p(\gamma_1 \cdots \gamma_n(L_0))/f_y^k \mathbb{C}[z]) \le \dim(p(\beta_1 \cdots \beta_n(L_0))/f_x^k \mathbb{C}[z]) $$

which translates to

$$ \langle \lambda^{\vee}, v_1 + \cdots + v_n \rangle \ge \langle \lambda^{\vee}, \mu_1 + \cdots + \mu_n \rangle. $$

This inequality holds for any dominant coweight $\lambda^{\vee}$, hence $v_1 + \cdots + v_n \ge \mu_1 + \cdots + \mu_n$.

This proves the last of the stated inequalities. The other ones can be obtained in a similar way, by taking the image under the obvious truncation map $Gr_n \to Gr_j$ for each $j \in \{1, \dots, n\}$. ■

**Corollary 5.12.** In the setup of Theorem 5.8, let $(\mu_1, \dots, \mu_n)$ and $(v_1, \dots, v_n)$ in $\Lambda^n$ be such that $Z' \in \mathcal{Z}(\lambda_1)_{v_1} \times \cdots \times \mathcal{Z}(\lambda_n)_{v_n}$ and $Z'' \in \mathcal{Z}(\lambda_1)_{\mu_1} \times \cdots \times \mathcal{Z}(\lambda_n)_{\mu_n}$. A necessary condition for $a_{Z',Z''} \ne 0$ is that

$$ v_1 \ge \mu_1, \quad v_1 + v_2 \ge \mu_1 + \mu_2, \quad \dots, \quad v_1 + \cdots + v_{n-1} \ge \mu_1 + \cdots + \mu_{n-1}. $$

We can obtain more stringent conditions regarding the transition matrix by looking at the associativity properties from Sect. 5.6. The sharpest result is obtained with a composition $(n_1, n_2)$ of $n$ with two parts. Accordingly, we write $\lambda$ as a concatenation $(\lambda_{(1)}, \lambda_{(2)})$ and similarly write each $Z \in \mathcal{Z}(\lambda)$ as $(Z_{(1)}, Z_{(2)})$. Here $Z_{(1)}$ is an element in $\mathcal{Z}(\lambda_1) \times \cdots \times \mathcal{Z}(\lambda_{n_1})$, but owing to the bijection (2) it can also be regarded as a cycle in $\overline{\mathrm{Gr}}_{n_1}^{\lambda_{(1)}}$.

**Theorem 5.13.** Let $(Z', Z'') \in (\mathcal{Z}(\lambda))_{\mu}^2$. Consider the expansion

$$ \langle Z''_{(1)} \rangle \otimes \langle Z''_{(2)} \rangle = \sum_{Z \in \mathcal{Z}(\lambda)_{\mu}} b_{Z,Z''} \langle Z \rangle. $$

If $b_{Z',Z''} \ne 0$, then either $Z' = Z''$ or $Z'_{(1)} \subsetneq \overline{Z''_{(1)}}$ as cycles in $\overline{\mathrm{Gr}}_{n_1}^{\lambda_{(1)}}$. In addition, $b_{Z'',Z''} = 1$.

*Proof.* Let $Z' = (Z'_1, \dots, Z'_n)$ and $Z'' = (Z''_1, \dots, Z''_n)$ in $\mathcal{Z}(\lambda)_{\mu}$.
For $j \in \{1, \dots, n\}$, let $\mu_j$ be the weight such that $Z_j'' \in \mathcal{P}(\lambda_j)_{\mu_j}$. Using the gallery models from [17] (or Theorem 4.6 and Proposition 4.8), we find a nonnegative integer $d_j$ and construct a map $\phi_j : \mathbb{C}^{d_j} \to \mathbb{N}^{\cdot, \vee}(\mathbb{C}[z, z^{-1}])$ such that $\{[\phi_j(\mathbf{a})z^{\mu_j}] \mid \mathbf{a} \in \mathbb{C}^{d_j}\}$ is a dense subset of $Z_j''$. Then

$$ \phi : (\mathbf{x}; \mathbf{a}_1, \dots, \mathbf{a}_n) \mapsto (\mathbf{x}; [\phi_1(\mathbf{a}_1)|_{x_1}(z-x_1)^{\mu_1}, \dots, \phi_n(\mathbf{a}_n)|_{x_n}(z-x_n)^{\mu_n}]) $$

maps $\mathbb{C}^n \times \mathbb{C}^{d_1} \times \dots \times \mathbb{C}^{d_n}$ onto a dense subset of $\Psi(Z_1'' \alpha \dots \alpha Z_n'')$, where the notation $(\dots)|_x$ means the result of substituting $z-x$ for $z$ in $(\dots)$.

Assume that $b_{Z',Z''} \neq 0$. By Proposition 5.10, $\mathcal{Y}(Z')$ is contained in $\mathcal{X}(Z'', (n_1, n_2))$, hence in the closure of $\Psi(Z_1'' \alpha \dots \alpha Z_n'')|_{\Delta(n_1, n_2)}$.

Take a point in $Z'_{(1)} \cap \mathrm{Gr}^{\lambda_{n_1}}$, written as $[g_1, \dots, g_{n_1}]$ where each $g_j$ is in $G^{\vee}(\mathbb{C}[z, z^{-1}])$. We can complete this datum to get an element

$$ \Gamma = (0, \dots, 0; [g_1, \dots, g_n]) $$

of $\mathcal{Y}(Z')$. Working in the analytic topology for expositional simplicity, we see that $\Gamma$ is the limit of a sequence $(\phi(\mathbf{x}_p; \mathbf{a}_{1,p}, \dots, \mathbf{a}_{n,p}))_{p \in \mathbb{N}}$ with $\mathbf{x}_p \in \Delta_{(n_1,n_2)}$ and $(\mathbf{a}_{1,p}, \dots, \mathbf{a}_{n,p}) \in \mathbb{C}^{d_1} \times \dots \times \mathbb{C}^{d_n}$. We write

$$ \mathbf{x}_p = \underbrace{(x_{1,p}, \dots, x_{1,p}, x_{2,p}, \dots, x_{2,p})}_{n_1 \text{ times}} \quad \text{with of course} \quad \lim_{p \to \infty} x_{1,p} = \lim_{p \to \infty} x_{2,p} = 0. \qquad (30) $$

Then

$$ \begin{aligned} [g_1, \dots, g_{n_1}] &= \lim_{p \to \infty} [\phi_1(\mathbf{a}_{1,p})z^{\mu_1}, \dots, \phi_{n_1}(\mathbf{a}_{1,n_1})z^{\mu_{n_1}}]|_{x_{1,p}} \\ &= \lim_{p \to \infty} [\phi_1(\mathbf{a}_{1,p})z^{\mu_1}, \dots, \phi_{n_1}(\mathbf{a}_{1,n_1})z^{\mu_{n_1}}] \end{aligned} $$

is the limit of a sequence of points in $Z''_{(1)}$. Therefore $Z'_{(1)} \cap \mathrm{Gr}^{\lambda_{n_1}} \subseteq \overline{Z''_{(1)}}$, whence the inclusion $Z'_{(1)} \subseteq \overline{Z''_{(1)}}$.

In addition to $b_{Z',Z''} \neq 0$, assume that the last inclusion is an equality. Then we have $Z'_{(1)} = Z''_{(1)}$ because these two MV cycles are irreducible components of the same $\overline{\mathrm{Gr}}^{\lambda_{n_1}} \cap (m_{n_1})^{-1}(T_{\mu_{(1)}})$, with indeed $\mu_{(1)} = \mu_1 + \dots + \mu_{n_1}$. We regard $Z'_{(2)}$ and $Z''_{(2)}$ as cycles in $\overline{\mathrm{Gr}}^{\lambda_{n_2}}$. Take a point in $Z'_{(2)} \cap \mathrm{Gr}^{\lambda_{n_2}}$, written as $[g_{n_1+1}, \dots, g_n]$ where each $g_j$ is in $G^{\vee}(\mathbb{C}[z, z^{-1}])$. We can then look at the element

$$ \Gamma = (0, \dots, 0; [z^{\mu_1}, \dots, z^{\mu_{n_1}}, g_{n_1+1}, \dots, g_n]) $$

of $\mathcal{Y}(Z')$. Again $\Gamma$ is the limit of a sequence $(\phi(\mathbf{x}_p; \mathbf{a}_{1,p}, \dots, \mathbf{a}_{n,p}))_{p \in \mathbb{N}}$ with $\mathbf{x}_p \in \Delta_{(n_1,n_2)}$ and $(\mathbf{a}_{1,p}, \dots, \mathbf{a}_{n,p}) \in \mathbb{C}^{d_1} \times \dots \times \mathbb{C}^{d_n}$. We set

$$ B_p = z^{-\mu_{(1)}} \phi_1(\mathbf{a}_{1,p}) z^{\mu_1} \cdots \phi_{n_1}(\mathbf{a}_{n_1,p}) z^{\mu_{n_1}}. $$
For $j \in \{1, \dots, n\}$, let $\mu_j$ be the weight such that $Z_j'' \in \mathcal{P}(\lambda_j)_{\mu_j}$. Using the gallery models from [17] (or Theorem 4.6 and Proposition 4.8), we find a nonnegative integer $d_j$ and construct a map $\phi_j : \mathbb{C}^{d_j} \to \mathbb{N}^{\cdot, \vee}(\mathbb{C}[z, z^{-1}])$ such that $\{[\phi_j(\mathbf{a})z^{\mu_j}] \mid \mathbf{a} \in \mathbb{C}^{d_j}\}$ is a dense subset of $Z_j''$. Then

$$ \phi : (\mathbf{x}; \mathbf{a}_1, \dots, \mathbf{a}_n) \mapsto (\mathbf{x}; [\phi_1(\mathbf{a}_1)|_{x_1}(z-x_1)^{\mu_1}, \dots, \phi_n(\mathbf{a}_n)|_{x_n}(z-x_n)^{\mu_n}]) $$

maps $\mathbb{C}^n \times \mathbb{C}^{d_1} \times \dots \times \mathbb{C}^{d_n}$ onto a dense subset of $\Psi(Z_1'' \alpha \dots \alpha Z_n'')$, where the notation $(\dots)|_x$ means the result of substituting $z-x$ for $z$ in $(\dots)$.

Assume that $b_{Z',Z''} \neq 0$. By Proposition 5.10, $\mathcal{Y}(Z')$ is contained in $\mathcal{X}(Z'', (n_1, n_2))$, hence in the closure of $\Psi(Z_1'' \alpha \dots \alpha Z_n'')|_{\Delta(n_1, n_2)}$.

Take a point in $Z'_{(1)} \cap \mathrm{Gr}^{\lambda_{n_1}}$, written as $[g_1, \dots, g_{n_1}]$ where each $g_j$ is in $G^{\vee}(\mathbb{C}[z, z^{-1}])$. We can complete this datum to get an element

$$ \Gamma = (0, \dots, 0; [g_1, \dots, g_n]) $$

of $\mathcal{Y}(Z')$. Working in the analytic topology for expositional simplicity, we see that $\Gamma$ is the limit of a sequence $(\phi(\mathbf{x}_p; \mathbf{a}_{1,p}, \dots, \mathbf{a}_{n,p}))_{p \in \mathbb{N}}$ with $\mathbf{x}_p \in \Delta(n_1, n_2)$ and $(\mathbf{a}_{1,p}, \dots, \mathbf{a}_{n,p}) \in \mathbb{C}^{d_1} \times \dots \times \mathbb{C}^{d_n}$. We write

$$ \mathbf{x}_p = \underbrace{(x_{1,p}, \dots, x_{1,p}, x_{2,p}, \dots, x_{2,p})}_{n_1 \text{ times}} \quad \text{with of course} \quad \lim_{p \to \infty} x_{1,p} = \lim_{p \to \infty} x_{2,p} = 0. \qquad (30) $$

Then

$$ \begin{aligned} [g_1, \dots, g_{n_1}] &= \lim_{p \to \infty} [\phi_1(\mathbf{a}_{1,p})z^{\mu_1}, \dots, \phi_{n_1}(\mathbf{a}_{1,n_1})z^{\mu_{n_1}}]|_{x_{1,p}} \\ &= \lim_{p \to \infty} [\phi_1(\mathbf{a}_{1,p})z^{\mu_1}, \dots, \phi_{n_1}(\mathbf{a}_{1,n_1})z^{\mu_{n_1}}] \end{aligned} $$

is the limit of a sequence of points in $Z''_{(1)}$. Therefore $Z'_{(1)} \cap \mathrm{Gr}^{\lambda_{n_1}} \subseteq \overline{Z''_{(1)}}$, whence the inclusion $Z'_{(1)} \subseteq \overline{Z''_{(1)}}$.

In addition to $b_{Z',Z''} \neq 0$, assume that the last inclusion is an equality. Then we have $Z'_{(1)} = Z''_{(1)}$ because these two MV cycles are irreducible components of the same $\overline{\mathrm{Gr}}^{\lambda_{n_1}}(m_{n_1})^{-1}(T_{\mu_{(1)}})$, with indeed $\mu_{(1)} = \mu_1 + \dots + \mu_{n_1}$. We regard $Z'_{(2)}$ and $Z''_{(2)}$ as cycles in $\overline{\mathrm{Gr}}^{\lambda_{n_2}}$. Take a point in $Z'_{(2)} \cap \mathrm{Gr}^{\lambda_{n_2}}$, written as $[g_{n_1+1}, \dots, g_n]$ where each $g_j$ is in $G^{\vee}(\mathbb{C}[z, z^{-1}])$. We can then look at the element

$$ \Gamma = (0, \dots, 0; [z^{\mu_1}, \dots, z^{\mu_{n_1}}, g_{n_1+1}, \dots, g_n]) $$

of $\mathcal{Y}(Z')$. Again $\Gamma$ is the limit of a sequence $(\phi(\mathbf{x}_p; \mathbf{a}_{1,p}, \dots, \mathbf{a}_{n,p}))_{p \in \mathbb{N}}$ with $\mathbf{x}_p \in \Delta(n_1, n_2)$ and $(\mathbf{a}_{1,p}, \dots, \mathbf{a}_{n,p}) \in \mathbb{C}^{d_1} \times \dots \times \mathbb{C}^{d_n}$. We set

$$ B_p = z^{-\mu_{(1)}} \phi_1(\mathbf{a}_{1,p}) z^{\mu_1} \cdots \phi_{n_1}(\mathbf{a}_{n_1,p}) z^{\mu_{n_1}}. $$
Writing again (30), we have

$$L_{\mu_{(1)}} = \lim_{p \to \infty} [z^{\mu_{(1)}} B_p]|_{x_{1,p}} = \lim_{p \to \infty} [z^{\mu_{(1)}} B_p] \quad (31)$$

and

$$z^{\mu(1)}[g_{n_1+1}, \dots, g_n] = \lim_{p \to \infty} (z^{\mu(1)} B_p)|_{x_{1,p}} [\phi_{n_1+1}(\mathbf{a}_{n_1+1,p})z^{\mu n_1+1}, \dots, \phi_n(\mathbf{a}_{n,p})z^{\mu n}]|_{x_{2,p}}. \quad (32)$$

Let $K$ be the kernel of the evaluation map $N^{-,\vee}(\mathbb{C}[z^{-1}]) \to N^{-,\vee}(\mathbb{C})$ at $z = \infty$. The multiplication induces a bijection

$$K \times N^{-,\vee}(\mathbb{C}[z]) \xrightarrow{\approx} N^{-,\vee}(\mathbb{C}[z, z^{-1}]).$$

We decompose $B_p$ as a product $B_{-,p}B_{+,p}$ according to this bijection. Using (31) and identifying the ind-variety $T_0$ with $K$, we find that $B_{-,p} \to 1$ as $p \to \infty$. Inserting this information in (32), we obtain

$$[g_{n_1+1}, \dots, g_n] = \lim_{p \to \infty} B_{+,p}[\phi_{n_1+1}(\mathbf{a}_{n_1+1,p})z^{\mu n_1+1}, \dots, \phi_n(\mathbf{a}_{n,p})z^{\mu n}],$$

so $[g_{n_1+1}, \dots, g_n]$ is the limit of a sequence of points in $Z'_{(2)}$. We conclude that $Z'_{(2)} \subseteq \overline{Z''_{(2)}}$, and since these two cycles have the same dimension, actually $Z'_{(2)} = Z''_{(2)}$.

To sum up: if $b_{Z',Z''} \neq 0$, then $Z'_{(1)} \subseteq \overline{Z''_{(1)}}$, and in case $Z'_{(1)} = Z''_{(1)}$, we additionally have $Z'_{(2)} = Z''_{(2)}$. This proves the first statement in the theorem. The second one is proved in the same manner as Proposition 5.9. ■

**Remark 5.14.** Using Theorem 5.13, one easily sharpens Corollary 5.12: with the notation of the latter, if $a_{Z',Z''} \neq 0$, then either $Z' = Z''$ or one of the displayed inequalities is strict. The proof is left to the reader.

*Application to standard monomial theory.* Let $\lambda \in \Lambda^+$ and let $\ell \subseteq V(\lambda)^*$ be the line spanned by a highest weight vector. The group $G$ acts on the projective space $\mathbb{P}(V(\lambda)^*)$; let $Q$ be the stabilizer of $\ell$, a parabolic subgroup of $G$. The map $g \mapsto g\ell$ induces an embedding of the partial flag variety $X = G/Q$ in $\mathbb{P}(V(\lambda)^*)$. We denote by $\mathcal{L}$ the pull-back of the line bundle $\mathcal{O}(1)$ by this embedding. Then the homogeneous coordinate ring of $X$ is

$$R_{\lambda} = \bigoplus_{m \ge 0} H^0(X, \mathcal{L}^{\otimes m});$$

here $H^0(X, \mathcal{L}^{\otimes m})$ is isomorphic to $V(m\lambda)$ and the multiplication in $R_{\lambda}$ is given by the projection onto the Cartan component

$$V(m\lambda) \otimes V(n\lambda) \to V((m+n)\lambda).$$

The algebra $R_{\lambda}$ is endowed with an MV basis, obtained by gathering the MV bases of the summands $V(m\lambda)$.
Writing again (30), we have

$$L_{\mu_{(1)}} = \lim_{p \to \infty} [z^{\mu_{(1)}} B_p]|_{x_{1,p}} = \lim_{p \to \infty} [z^{\mu_{(1)}} B_p] \quad (31)$$

and

$$z^{\mu(1)}[g_{n_1+1}, \dots, g_n] = \lim_{p \to \infty} (z^{\mu(1)} B_p)|_{x_{1,p}} [\phi_{n_1+1}(\mathbf{a}_{n_1+1,p})z^{\mu_{n_1+1}}, \dots, \phi_n(\mathbf{a}_{n,p})z^{\mu_n}]|_{x_{2,p}}. \quad (32)$$

Let $K$ be the kernel of the evaluation map $N^{-,\vee}(\mathbb{C}[z^{-1}]) \to N^{-,\vee}(\mathbb{C})$ at $z = \infty$. The multiplication induces a bijection

$$K \times N^{-,\vee}(\mathbb{C}[z]) \xrightarrow{\approx} N^{-,\vee}(\mathbb{C}[z, z^{-1}]).$$

We decompose $B_p$ as a product $B_{-,p}B_{+,p}$ according to this bijection. Using (31) and identifying the ind-variety $T_0$ with $K$, we find that $B_{-,p} \to 1$ as $p \to \infty$. Inserting this information in (32), we obtain

$$[g_{n_1+1}, \dots, g_n] = \lim_{p \to \infty} B_{+,p}[\phi_{n_1+1}(\mathbf{a}_{n_1+1,p})z^{\mu_{n_1+1}}, \dots, \phi_n(\mathbf{a}_{n,p})z^{\mu_n}],$$

so $[g_{n_1+1}, \dots, g_n]$ is the limit of a sequence of points in $Z'_{(2)}$. We conclude that $Z'_{(2)} \subseteq \overline{Z''_{(2)}}$, and since these two cycles have the same dimension, actually $Z'_{(2)} = Z''_{(2)}$.

To sum up: if $bZ',Z'' \neq 0$, then $Z'_{(1)} \subseteq \overline{Z''_{(1)}}$, and in case $Z'_{(1)} = Z''_{(1)}$, we additionally have $Z'_{(2)} = Z''_{(2)}$. This proves the first statement in the theorem. The second one is proved in the same manner as Proposition 5.9. ■

**Remark 5.14.** Using Theorem 5.13, one easily sharpens Corollary 5.12: with the notation of the latter, if $aZ',Z'' \neq 0$, then either $Z' = Z''$ or one of the displayed inequalities is strict. The proof is left to the reader.

*Application to standard monomial theory.* Let $\lambda \in \Lambda^+$ and let $\ell \subseteq V(\lambda)^*$ be the line spanned by a highest weight vector. The group $G$ acts on the projective space $\mathbb{P}(V(\lambda)^*)$; let $Q$ be the stabilizer of $\ell$, a parabolic subgroup of $G$. The map $g \mapsto g\ell$ induces an embedding of the partial flag variety $X = G/Q$ in $\mathbb{P}(V(\lambda)^*)$. We denote by $\mathcal{L}$ the pull-back of the line bundle $\mathcal{O}(1)$ by this embedding. Then the homogeneous coordinate ring of $X$ is

$$R_{\lambda} = \bigoplus_{m \ge 0} H^0(X, \mathcal{L}^{\otimes m});$$

here $H^0(X, \mathcal{L}^{\otimes m})$ is isomorphic to $V(m\lambda)$ and the multiplication in $R_{\lambda}$ is given by the projection onto the Cartan component

$$V(m\lambda) \otimes V(n\lambda) \to V((m+n)\lambda).$$

The algebra $R_{\lambda}$ is endowed with an MV basis, obtained by gathering the MV bases of the summands $V(m\lambda)$.
Each MV cycle $Z \in \mathcal{Z}(\lambda)$ defines a basis element $\langle Z \rangle \in V(\lambda)$. Given an $m$-tuple $Z = (Z_1, \dots, Z_m)$ of elements of $\mathcal{Z}(\lambda)$, the product $\langle Z_1 \rangle \cdots \langle Z_m \rangle$ in the algebra $R_\lambda$ is the image of $\langle \mathbb{Z} \rangle = \langle Z_1 \rangle \otimes \cdots \otimes \langle Z_m \rangle$ under the projection $V(\lambda)^{\otimes m} \to V(m\lambda)$. This product is called *standard* if $Z$ lies in the Cartan component of the crystal $\mathcal{Z}(\lambda)^{\otimes m}$.

Remark 3.5 implies that the MV basis element $\langle Z \rangle \in V(\lambda)^{\otimes m}$ goes, under the projection $V(\lambda)^{\otimes m} \to V(m\lambda)$, either to an element in the MV basis of $V(m\lambda)$ or to 0, depending on whether $Z$ lies or not in the Cartan component of $\mathcal{Z}(\lambda)^{\otimes m}$.

Using Corollary 5.12 and Remark 5.14, we can then endow, for each degree $m$, the Cartan component of $\mathcal{Z}(\lambda)^{\otimes m}$ with an order, so that the transition matrix expressing the standard monomials in the MV basis of $R_\lambda$ is unitriangular. In particular, the standard monomials form a basis for the algebra $R_\lambda$ too, and straightening laws can be obtained from Theorem 5.8.

The dual of the MV basis is compatible with the Demazure modules contained in $V(m\lambda)^*$; this property is recorded in [3, Remark 2.6(ii)] but the crux of the argument is due to Kashiwara [26]. This implies that for any Schubert variety $Y \subseteq X$, the kernel of the restriction map

$$ \bigoplus_{m \ge 0} H^0(X, \mathcal{L}^{\otimes m}) \to \bigoplus_{m \ge 0} H^0(Y, \mathcal{L}^{\otimes m}) $$

is spanned by a subset of the MV basis of $R_\lambda$. Therefore the homogeneous coordinate ring of $Y$ is also endowed with an MV basis.

These observations suggest that the MV basis could be a relevant tool for the study of the standard monomial theory.

## 5.8. A conjectural symmetry

Recall the notation set up in Sects. 3.1–3.2. Given $\lambda \in \Lambda^+$, we set $\lambda^* = -w_0\lambda$, where as usual $w_0$ denotes the longest element in the Weyl group $W$. As is well-known, there exists a unique bijection $\sigma : B(\lambda) \to B(\lambda^*)$ which for each $i \in I$ exchanges the actions of $\tilde{e}_i$ and $\tilde{f}_i$. In our context, we will regard $\sigma$ as a bijection $\mathcal{Z}(\lambda) \to \mathcal{Z}(\lambda^*)$ and may define it by means of [33, Lemma 2.1 (e)] and Theorem 4.9.

Now let $n \ge 1$ and let $\lambda = (\lambda_1, \dots, \lambda_n)$ in $(\Lambda^+)^n$. We set $\lambda^* = (\lambda_n^*, \dots, \lambda_1^*)$ and define a bijection

$$ \sigma : \mathcal{Z}(\lambda_1) \times \cdots \times \mathcal{Z}(\lambda_n) \to \mathcal{Z}(\lambda_n^*) \times \cdots \times \mathcal{Z}(\lambda_1^*) $$

by $\sigma(Z_1, \dots, Z_n) = (\sigma(Z_n), \dots, \sigma(Z_1))$. (Using the same symbol $\sigma$ to denote different bijections is certainly abusive, but adding extra indices to disambiguate would overload the notation without clear benefit.) The Cartesian products above are in fact tensor product of crystals, and here again $\sigma$ exchanges the actions of $\tilde{e}_i$ and $\tilde{f}_i$ for each $i \in I$ [23, Theorem 2].

Let $\mu \in \Lambda$ and choose $(Z', Z'') \in (\mathcal{Z}(\lambda))_{\mu}^2$; we then obtain $\sigma(Z')$ and $\sigma(Z'')$ in $\mathcal{Z}(\lambda^*)_{-\mu}$. Recall the notation introduced in Theorem 5.8 to denote the entries of the transition matrix between the two bases of $V(\lambda)$ and adopt a similar notation for $V(\lambda^*)$.
Each MV cycle $Z \in \mathcal{Z}(\lambda)$ defines a basis element $\langle Z \rangle \in V(\lambda)$. Given an $m$-tuple $\mathbf{Z} = (Z_1, \dots, Z_m)$ of elements of $\mathcal{Z}(\lambda)$, the product $\langle Z_1 \rangle \cdots \langle Z_m \rangle$ in the algebra $R_\lambda$ is the image of $\langle \mathbf{Z} \rangle = \langle Z_1 \rangle \otimes \cdots \otimes \langle Z_m \rangle$ under the projection $V(\lambda)^{\otimes m} \to V(m\lambda)$. This product is called *standard* if $\mathbf{Z}$ lies in the Cartan component of the crystal $\mathcal{Z}(\lambda)^{\otimes m}$.

Remark 3.5 implies that the MV basis element $\langle \mathbf{Z} \rangle \in V(\lambda)^{\otimes m}$ goes, under the projection $V(\lambda)^{\otimes m} \to V(m\lambda)$, either to an element in the MV basis of $V(m\lambda)$ or to 0, depending on whether $\mathbf{Z}$ lies or not in the Cartan component of $\mathcal{Z}(\lambda)^{\otimes m}$.

Using Corollary 5.12 and Remark 5.14, we can then endow, for each degree $m$, the Cartan component of $\mathcal{Z}(\lambda)^{\otimes m}$ with an order, so that the transition matrix expressing the standard monomials in the MV basis of $R_\lambda$ is unitriangular. In particular, the standard monomials form a basis for the algebra $R_\lambda$ too, and straightening laws can be obtained from Theorem 5.8.

The dual of the MV basis is compatible with the Demazure modules contained in $V(m\lambda)^*$; this property is recorded in [3, Remark 2.6(ii)] but the crux of the argument is due to Kashiwara [26]. This implies that for any Schubert variety $Y \subseteq X$, the kernel of the restriction map

$$ \bigoplus_{m \ge 0} H^0(X, \mathcal{L}^{\otimes m}) \to \bigoplus_{m \ge 0} H^0(Y, \mathcal{L}^{\otimes m}) $$

is spanned by a subset of the MV basis of $R_\lambda$. Therefore the homogeneous coordinate ring of $Y$ is also endowed with an MV basis.

These observations suggest that the MV basis could be a relevant tool for the study of the standard monomial theory.

## 5.8. A conjectural symmetry

Recall the notation set up in Sects. 3.1–3.2. Given $\lambda \in \Lambda^+$, we set $\lambda^* = -w_0\lambda$, where as usual $w_0$ denotes the longest element in the Weyl group $W$. As is well-known, there exists a unique bijection $\sigma : B(\lambda) \to B(\lambda^*)$ which for each $i \in I$ exchanges the actions of $\tilde{e}_i$ and $\tilde{f}_i$. In our context, we will regard $\sigma$ as a bijection $\mathcal{Z}(\lambda) \to \mathcal{Z}(\lambda^*)$ and may define it by means of [33, Lemma 2.1 (e)] and Theorem 4.9.

Now let $n \ge 1$ and let $\lambda = (\lambda_1, \dots, \lambda_n)$ in $(\Lambda^+)^n$. We set $\lambda^* = (\lambda_n^*, \dots, \lambda_1^*)$ and define a bijection

$$ \sigma : \mathcal{Z}(\lambda_1) \times \cdots \times \mathcal{Z}(\lambda_n) \to \mathcal{Z}(\lambda_n^*) \times \cdots \times \mathcal{Z}(\lambda_1^*) $$

by $\sigma(Z_1, \dots, Z_n) = (\sigma(Z_n), \dots, \sigma(Z_1))$. (Using the same symbol $\sigma$ to denote different bijections is certainly abusive, but adding extra indices to disambiguate would overload the notation without clear benefit.) The Cartesian products above are in fact tensor product of crystals, and here again $\sigma$ exchanges the actions of $\tilde{e}_i$ and $\tilde{f}_i$ for each $i \in I$ [23, Theorem 2].

Let $\mu \in \Lambda$ and choose $(\mathbf{Z}', \mathbf{Z}'') \in (\mathcal{Z}(\lambda))_{\mu}^2$; we then obtain $\sigma(\mathbf{Z}')$ and $\sigma(\mathbf{Z}'')$ in $\mathcal{Z}(\lambda^*)_{-\mu}$. Recall the notation introduced in Theorem 5.8 to denote the entries of the transition matrix between the two bases of $V(\lambda)$ and adopt a similar notation for $V(\lambda^*)$.
**Conjecture 5.15.** The equality $a_{Z',Z''} = a_{\sigma(Z'),\sigma(Z'')}$ holds.

In type $A_1$, this conjecture follows from the results in [1]. Its general validity would have two interesting consequences.

Firstly, one could then strengthen Theorem 5.13. Indeed, $b_{Z',Z''} \neq 0$ would imply not only $Z'_{(1)} \subseteq \bar{Z}''_{(1)}$, but also $\sigma(Z'_{(2)}) \subseteq \bar{\sigma}(\bar{Z}''_{(2)})$, restoring the symmetry between the two tensor factors.

Secondly, the MV basis of an irreducible representation $V(\lambda)$ would then satisfy the analogue of [37, Proposition 21.1.2]. In fact, one easily verifies that the MV basis enjoys this property if $\lambda$ is minuscule or quasi-minuscule. Our conjecture would allow one to deduce the general case by taking suitable tensor products, mimicking the strategy of proof from [40].

# 6. The basis of the invariant subspace

Let $n \ge 1$ and let $\lambda \in (\Lambda^+)^n$. The MV basis of $V(\lambda)$ is compatible with the isotypic filtration, hence provides a basis of the fixed-point subspace $V(\lambda)^G$, called the *Satake basis* in [13]. In this section we study two properties of this basis.

## 6.1. Cyclic permutations

Let us write $\lambda = (\lambda_1, \dots, \lambda_n)$ and consider the rotated sequence $\lambda^{[1]} = (\lambda_2, \dots, \lambda_n, \lambda_1)$. Thus,

$$V(\lambda) = V(\lambda_1) \otimes \cdots \otimes V(\lambda_n) \quad \text{and} \quad V(\lambda^{[1]}) = V(\lambda_2) \otimes \cdots \otimes V(\lambda_n) \otimes V(\lambda_1).$$

The signed cyclic permutation

$$x_1 \otimes \cdots \otimes x_n \mapsto (-1)^{2\rho(\lambda_1)} x_2 \otimes \cdots \otimes x_n \otimes x_1,$$

defines an isomorphism of $G$-modules $R: V(\lambda) \to V(\lambda^{[1]})$. In particular, $R$ induces a linear bijection between the invariant subspaces.

**Theorem 6.1.** *The signed cyclic permutation $R$ maps the Satake basis of $V(\lambda)^G$ to the Satake basis of $V(\lambda^{[1]})^G$.*

Theorem 6.1 replicates a similar result for the dual canonical basis due to Lusztig [37, 28.2.9], and our proof below mirrors Lusztig's argument. In the case where all the weights $\lambda_j$ are minuscule, Theorem 6.1 has been proved by Fontaine, Kamnitzer and Kuperberg [13, Theorem 4.5]. The bijection induced by $R$ between the two Satake bases has a nice interpretation, both in terms of crystals (see [12]) and in terms of cluster combinatorics (see [22, Sect. 2.1.6]).

The rest of this section is devoted to the proof of Theorem 6.1.

As in Sect. 3, we denote by $\{\alpha_i | i \in I\}$ the set of simple roots and choose simple root vectors $e_i$ and $f_i$ in the Lie algebra of $G$ of weights $\pm \alpha_i$ such that $[e_i, f_i] = -\alpha_i^\vee$. The Weyl group $W$ is generated by the simple reflections $s_i$ and contains a longest element $w_0$.
**Conjecture 5.15.** The equality $a_{Z',Z''} = a_{\sigma(Z'),\sigma(Z'')}$ holds.

In type $A_1$, this conjecture follows from the results in [1]. Its general validity would have two interesting consequences.

Firstly, one could then strengthen Theorem 5.13. Indeed, $b_{Z',Z''} \neq 0$ would imply not only $Z'_{(1)} \subseteq \bar{Z}''_{(1)}$, but also $\sigma(Z'_{(2)}) \subseteq \bar{\sigma}(\bar{Z}''_{(2)})$, restoring the symmetry between the two tensor factors.

Secondly, the MV basis of an irreducible representation $V(\lambda)$ would then satisfy the analogue of [37, Proposition 21.1.2]. In fact, one easily verifies that the MV basis enjoys this property if $\lambda$ is minuscule or quasi-minuscule. Our conjecture would allow one to deduce the general case by taking suitable tensor products, mimicking the strategy of proof from [40].

# 6. The basis of the invariant subspace

Let $n \ge 1$ and let $\lambda \in (\Lambda^+)^n$. The MV basis of $V(\lambda)$ is compatible with the isotypic filtration, hence provides a basis of the fixed-point subspace $V(\lambda)^G$, called the *Satake basis* in [13]. In this section we study two properties of this basis.

## 6.1. Cyclic permutations

Let us write $\lambda = (\lambda_1, \dots, \lambda_n)$ and consider the rotated sequence $\lambda^{[1]} = (\lambda_2, \dots, \lambda_n, \lambda_1)$. Thus,

$$V(\lambda) = V(\lambda_1) \otimes \cdots \otimes V(\lambda_n) \quad \text{and} \quad V(\lambda^{[1]}) = V(\lambda_2) \otimes \cdots \otimes V(\lambda_n) \otimes V(\lambda_1).$$

The signed cyclic permutation

$$x_1 \otimes \cdots \otimes x_n \mapsto (-1)^{2\rho(\lambda_1)} x_2 \otimes \cdots \otimes x_n \otimes x_1,$$

defines an isomorphism of $G$-modules $R: V(\lambda) \to V(\lambda^{[1]})$. In particular, $R$ induces a linear bijection between the invariant subspaces.

**Theorem 6.1.** *The signed cyclic permutation $R$ maps the Satake basis of $V(\lambda)^G$ to the Satake basis of $V(\lambda^{[1]})^G$.*

Theorem 6.1 replicates a similar result for the dual canonical basis due to Lusztig [37, 28.2.9], and our proof below mirrors Lusztig's argument. In the case where all the weights $\lambda_j$ are minuscule, Theorem 6.1 has been proved by Fontaine, Kamnitzer and Kuperberg [13, Theorem 4.5]. The bijection induced by $R$ between the two Satake bases has a nice interpretation, both in terms of crystals (see [12]) and in terms of cluster combinatorics (see [22, Sect. 2.1.6]).

The rest of this section is devoted to the proof of Theorem 6.1.

As in Sect. 3, we denote by $\{\alpha_i | i \in I\}$ the set of simple roots and choose simple root vectors $e_i$ and $f_i$ in the Lie algebra of $G$ of weights $\pm \alpha_i$ such that $[e_i, f_i] = -\alpha_i^\vee$. The Weyl group $W$ is generated by the simple reflections $s_i$ and contains a longest element $w_0$.
Given $\lambda \in \Lambda^+$ and $w \in W$, we can pick a reduced word $(i_1, \dots, i_\ell)$ of $w$ and form the product of divided powers

$$ \theta(w, \lambda) = f_{i_1}^{(n_1)} \cdots f_{i_\ell}^{(n_\ell)}, \quad \text{where} \quad n_j = \langle \alpha_{ij}^\vee, s_{i_j+1} \cdots s_{i_\ell} \lambda \rangle. $$

This element does not depend on the choice of $(i_1, \dots, i_\ell)$ (see [37, Proposition 28.1.2]), which legitimizes the notation. We note that $\theta(w_0, \lambda)$ acts on $V(\lambda)$ by mapping highest weight vectors to lowest weight vectors.

We set $\lambda = \lambda_1$, the first element in the sequence $\lambda$. With the notation of Sect. 5.4, the highest and lowest weight elements in the MV basis of $V(\lambda)$ are

$$ v_{\lambda} = \langle \{L_{\lambda}\} \rangle \quad \text{and} \quad v_{w_{0}\lambda} = \langle \overline{\mathrm{Gr}}^{\lambda} \rangle. $$

Under suitable normalizations in the geometric Satake equivalence, these two elements are related by $v_{w_0\lambda} = \theta(w_0, \lambda) \cdot v_{\lambda}$ (see [3, Theorem 5.2 and Remark 2.10]). We define $v_{\lambda}^*$ to be the linear form on $V(\lambda)$ such that $\langle v_{\lambda}^*, v_{\lambda} \rangle = 1$ and $v_{\lambda}^*$ vanishes on all weight subspaces of weight different from $\lambda$. Similarly, we define $v_{w_0\lambda}^*$ to be the linear form on $V(\lambda)$ such that $\langle v_{w_0\lambda}^*, v_{w_0\lambda} \rangle = 1$ and $v_{w_0\lambda}^*$ vanishes on all weight subspaces of weight different from $w_0\lambda$.

Let $M$ be a representation of $G$. The assignment $v \otimes m \mapsto (-1)^{2\rho(\lambda)}m \otimes v$ defines an isomorphism $P : V(\lambda) \otimes M \to M \otimes V(\lambda)$.

We set $\lambda^* = -w_0\lambda$. Let $M^\circ$ be the isotypic component of $M$ corresponding to the highest weight $\lambda^*$, namely, the sum of all subrepresentations isomorphic to $V(\lambda^*)$. Given a weight $\mu \in \Lambda$, we denote by $M_\mu$ the corresponding weight subspace of $M$ and set $M_\mu^\circ = M^\circ \cap M_\mu$. Then $M_{\lambda^*}^\circ$ is the set of all vectors in $M_{\lambda^*}$ that are annihilated by all the root vectors $e_i$, and $M_{w_0\lambda^*}^\circ$ is the set of all vectors in $M_{w_0\lambda^*}$ that are annihilated by all the root vectors $f_i$.

**Lemma 6.2.** *The following diagram commutes and consists of isomorphisms of vector spaces:*

$$ \begin{tikzcd}[column sep=2.8em, row sep=2.8em]
(V(\lambda) \otimes M)^G \arrow[r, "P"] & (M \otimes V(\lambda))^G \\
v_{w_0\lambda}^* \otimes \mathrm{id}_M \arrow[u, "M_\mu^\circ"'] \arrow[u, "\theta(w_0, \lambda^*)"'] & \mathrm{id}_M \otimes v_{\lambda^*} \\
M_{\lambda^*}^\circ \arrow[u, "M_{w_0\lambda^*}^\circ"'] & M_{w_0\lambda^*}^\circ
\end{tikzcd} \quad (33) $$

*Proof.* By additivity, we can reduce to the case where $M$ is a simple representation. If $M$ is not isomorphic to the dual of $V(\lambda)$, then all four spaces are zero and the statement is banal. We therefore assume that $M \cong V(\lambda^*)$; in this case, all four spaces are one-dimensional.

Let $m_{\lambda^*}$ be a highest weight vector in $M$ and set $m_{w_0\lambda^*} = \theta(w_0, \lambda^*) \cdot m_{\lambda^*}$. There exists a unique $G$-invariant bilinear form $\Phi : V(\lambda) \times M \to \mathbb{C}$ such that $\Phi(v_{w_0\lambda}, m_{\lambda^*}) = 1$. This form $\Phi$ is nondegenerate and a standard computation gives $\Phi(v_{\lambda}, m_{w_0\lambda^*}) = (-1)^{2\rho(\lambda)}$.
Given $\lambda \in \Lambda^+$ and $w \in W$, we can pick a reduced word $(i_1, \dots, i_\ell)$ of $w$ and form the product of divided powers

$$ \theta(w, \lambda) = f_{i_1}^{(n_1)} \cdots f_{i_\ell}^{(n_\ell)}, \quad \text{where} \quad n_j = \langle \alpha_{ij}^\vee, s_{i_j+1} \cdots s_{i_\ell} \lambda \rangle. $$

This element does not depend on the choice of $(i_1, \dots, i_\ell)$ (see [37, Proposition 28.1.2]), which legitimizes the notation. We note that $\theta(w_0, \lambda)$ acts on $V(\lambda)$ by mapping highest weight vectors to lowest weight vectors.

We set $\lambda = \lambda_1$, the first element in the sequence $\lambda$. With the notation of Sect. 5.4, the highest and lowest weight elements in the MV basis of $V(\lambda)$ are

$$ v_{\lambda} = \langle \{L_{\lambda}\} \rangle \quad \text{and} \quad v_{w_{0}\lambda} = \langle \overline{\mathrm{Gr}}^{\lambda} \rangle. $$

Under suitable normalizations in the geometric Satake equivalence, these two elements are related by $v_{w_0\lambda} = \theta(w_0, \lambda) \cdot v_{\lambda}$ (see [3, Theorem 5.2 and Remark 2.10]). We define $v_{\lambda}^*$ to be the linear form on $V(\lambda)$ such that $\langle v_{\lambda}^*, v_{\lambda} \rangle = 1$ and $v_{\lambda}^*$ vanishes on all weight subspaces of weight different from $\lambda$. Similarly, we define $v_{w_0\lambda}^*$ to be the linear form on $V(\lambda)$ such that $\langle v_{w_0\lambda}^*, v_{w_0\lambda} \rangle = 1$ and $v_{w_0\lambda}^*$ vanishes on all weight subspaces of weight different from $w_0\lambda$.

Let $M$ be a representation of $G$. The assignment $v \otimes m \mapsto (-1)^{2\rho(\lambda)}m \otimes v$ defines an isomorphism $P : V(\lambda) \otimes M \to M \otimes V(\lambda)$.

We set $\lambda^* = -w_0\lambda$. Let $M^\circ$ be the isotypic component of $M$ corresponding to the highest weight $\lambda^*$, namely, the sum of all subrepresentations isomorphic to $V(\lambda^*)$. Given a weight $\mu \in \Lambda$, we denote by $M_\mu$ the corresponding weight subspace of $M$ and set $M_\mu^\circ = M^\circ \cap M_\mu$. Then $M_{\lambda^*}^\circ$ is the set of all vectors in $M_{\lambda^*}$ that are annihilated by all the root vectors $e_i$, and $M_{w_0\lambda^*}^\circ$ is the set of all vectors in $M_{w_0\lambda^*}$ that are annihilated by all the root vectors $f_i$.

**Lemma 6.2.** *The following diagram commutes and consists of isomorphisms of vector spaces:*

$$ \begin{tikzcd}[column sep=2.8em, row sep=2.8em]
(V(\lambda) \otimes M)^G \arrow[r, "P"] & (M \otimes V(\lambda))^G \\
v_{w_0\lambda}^* \otimes \mathrm{id}_M \arrow[u, "M_\mu^\circ"'] \arrow[u, "\theta(w_0, \lambda^*)"'] & \mathrm{id}_M \otimes v_{\lambda^*}^* \\
M_{\lambda^*}^\circ \arrow[u, "M_{w_0\lambda^*}^\circ"'] & M_{w_0\lambda^*}^\circ
\end{tikzcd} \quad (33) $$

*Proof.* By additivity, we can reduce to the case where $M$ is a simple representation. If $M$ is not isomorphic to the dual of $V(\lambda)$, then all four spaces are zero and the statement is banal. We therefore assume that $M \cong V(\lambda^*)$; in this case, all four spaces are one-dimensional.

Let $m_{\lambda^*}$ be a highest weight vector in $M$ and set $m_{w_0\lambda^*} = \theta(w_0, \lambda^*) \cdot m_{\lambda^*}$. There exists a unique $G$-invariant bilinear form $\Phi : V(\lambda) \times M \to \mathbb{C}$ such that $\Phi(v_{w_0\lambda}, m_{\lambda^*}) = 1$. This form $\Phi$ is nondegenerate and a standard computation gives $\Phi(v_{\lambda}, m_{w_0\lambda^*}) = (-1)^{2\rho(\lambda)}$.
The assignment $v \otimes m \mapsto \Phi(v, ?)m$ defines a $G$-equivariant isomorphism $V(\lambda) \otimes M \to \text{End}(M)$. The preimage $x$ of $\text{id}_M$ by this bijection spans the vector space $(V(\lambda) \otimes M)^G$. By construction, $(v_{w_0\lambda}^* \otimes \text{id}_M)(x) = m_{\lambda^*}$ and $(v_\lambda^* \otimes \text{id}_M)(x) = (-1)^{2\rho(\lambda)}m_{w_0\lambda^*}$. Thus, both paths around the diagram map $x$ to $m_{w_0\lambda^*}$. ■

We take $M = V(\lambda_2) \otimes \cdots \otimes V(\lambda_n)$. We define $M^\bullet$ to be the step in the isotypic filtration of $M$ where the component $M^\circ$ is appended to smaller ones. There is a natural quotient map $p: M^\bullet \to M^\circ$.

We set $\mathcal{M} = \mathcal{L}(\lambda_2) \times \cdots \times \mathcal{L}(\lambda_n)$. Using the notation introduced in Sect. 5.4, the MV basis of $M$ consists of elements $\langle \mathbf{Z} \rangle$ for $\mathbf{Z} \in \mathcal{M}$. Let $\mathcal{M}^\bullet$ be the set of all $Z \in \mathcal{M}$ such that $\langle \mathbf{Z} \rangle \in \mathcal{M}^\bullet$; since MV bases are $L$-perfect, $\{\langle \mathbf{Z} \rangle \mid \mathbf{Z} \in \mathcal{M}^\bullet\}$ is a basis of $\mathcal{M}^\bullet$. Let $\mathcal{M}^\circ$ be the set of all $Z \in \mathcal{M}^\bullet$ such that $\langle \mathbf{Z} \rangle \notin \ker p$; then $\{p(\langle \mathbf{Z} \rangle) \mid \mathbf{Z} \in \mathcal{M}^\circ\}$ is a basis of $\mathcal{M}^\circ$. In consequence, each weight subspace of $\mathcal{M}^\circ$ is endowed with a basis.

As a crystal, $\mathcal{M}$ decomposes as the disjoint union (direct sum) of its connected components, and $\mathcal{M}^\circ$ is the union of the connected components of $\mathcal{M}$ that are isomorphic to $\mathcal{L}(\lambda^*)$. For each connected component $\mathcal{C} \subseteq \mathcal{M}^\circ$, the subspace of $\mathcal{M}^\circ$ spanned by $B_{\mathcal{C}} = \{p(\langle \mathbf{Z} \rangle) \mid \mathbf{Z} \in \mathcal{C}\}$ is a subrepresentation isomorphic to $V(\lambda^*)$, and by Remark 3.5, $B_{\mathcal{C}}$ identifies with the MV basis of $V(\lambda^*)$. The action of $\theta(w_0, \lambda^*)$ therefore maps the highest weight element in $B_{\mathcal{C}}$ to the lowest element in $B_{\mathcal{C}}$. We conclude that the bottom horizontal arrow in (33) maps the basis of $M_{\lambda^*}^\circ$ to the basis of $M_{w_0\lambda^*}^\circ$.

Each element in the MV basis of $V(\lambda^{[1]}) = M \otimes V(\lambda)$ is of the form $\langle \mathbf{Z} \rangle$ with $\mathbf{Z}$ in $\mathcal{L}(\lambda^{[1]}) = \mathcal{M} \times \mathcal{L}(\lambda)$. Let $V(\lambda)_{\neq \lambda}$ be the sum of all the weight subspaces of $V(\lambda)$ other than the higher weight subspace. Theorem 5.13 implies that for each $\mathbf{Z}_{(1)} \in \mathcal{M}$,

$$ \langle \mathbf{Z}_{(1)} \rangle \otimes \langle \{L_{\lambda}\} \rangle \equiv \langle (\mathbf{Z}_{(1)}, \{L_{\lambda}\}) \rangle (\text{mod } M \otimes V(\lambda)_{\neq \lambda}). $$

Thus, for $\mathbf{Z}_{(1)} \in \mathcal{M}$ and $\mathbf{Z} = (\mathbf{Z}_{(1)}, \{L_{\lambda}\})$, we have $(\text{id}_M \otimes v_{\lambda}^*)(\langle \mathbf{Z} \rangle) = \langle \mathbf{Z}_{(1)} \rangle$.

As evidenced by the crystal structure on $\mathcal{M} \otimes \mathcal{L}(\lambda)$, the Satake basis of the space $(M \otimes V(\lambda))^G$ consists of the vectors $\langle \mathbf{Z} \rangle$ for the pairs $\mathbf{Z} = (\mathbf{Z}_{(1)}, \{L_{\lambda}\})$ such that $\mathbf{Z}_{(1)} \in \mathcal{M}_{w_0\lambda^*}^\circ$. Noting that $\langle \mathbf{Z}_{(1)} \rangle \in M_{w_0\lambda^*}^\circ$ for those $\mathbf{Z}_{(1)}$, we conclude that the right vertical arrow in (33) maps basis elements to basis elements.

Similarly, the left vertical arrow in (33) maps the Satake basis of $(V(\lambda) \otimes M)^G$ to the basis of $M_{\lambda^*}^\circ$. Lemma 6.2 then concludes the proof of Theorem 6.1.

## 6.2. Tensor product with an invariant element

Let $(n', n'')$ be a composition of $n$ with two parts. Correspondingly, we write $\lambda \in (\Lambda^+)^n$ as a concatenation $(\lambda', \lambda'')$ and view each element in $\mathcal{L}(\lambda)$ as a pair $(\mathbf{Z}', \mathbf{Z}'') \in \mathcal{L}(\lambda') \times \mathcal{L}(\lambda'')$.

The following proposition implies that the Satake basis of the invariant subspace of $V(\lambda)$ satisfies the second item in Khovanov and Kuperberg's list of properties for the dual canonical basis (see the introduction of [30]).

**Proposition 6.3.** Let $(\mathbf{Z}', \mathbf{Z}'') \in \mathcal{L}(\lambda)$. If $\langle \mathbf{Z}' \rangle \in V(\lambda')^G$, then $\langle \mathbf{Z}' \rangle \otimes \langle \mathbf{Z}'' \rangle = \langle (\mathbf{Z}', \mathbf{Z}'') \rangle$.
The assignment $v \otimes m \mapsto \Phi(v, ?)m$ defines a $G$-equivariant isomorphism $V(\lambda) \otimes M \to \text{End}(M)$. The preimage $x$ of $\text{id}_M$ by this bijection spans the vector space $(V(\lambda) \otimes M)^G$. By construction, $(v_{w_0\lambda}^* \otimes \text{id}_M)(x) = m_{\lambda^*}$ and $(v_\lambda^* \otimes \text{id}_M)(x) = (-1)^{2\rho(\lambda)}m_{w_0\lambda^*}$. Thus, both paths around the diagram map $x$ to $m_{w_0\lambda^*}$. ■

We take $M = V(\lambda_2) \otimes \cdots \otimes V(\lambda_n)$. We define $M^\bullet$ to be the step in the isotypic filtration of $M$ where the component $M^\circ$ is appended to smaller ones. There is a natural quotient map $p: M^\bullet \to M^\circ$.

We set $\mathcal{M} = \mathcal{L}(\lambda_2) \times \cdots \times \mathcal{L}(\lambda_n)$. Using the notation introduced in Sect. 5.4, the MV basis of $M$ consists of elements $\langle \mathbf{Z} \rangle$ for $\mathbf{Z} \in \mathcal{M}$. Let $\mathcal{M}^\bullet$ be the set of all $Z \in \mathcal{M}$ such that $\langle \mathbf{Z} \rangle \in \mathcal{M}^\bullet$; since MV bases are $L$-perfect, $\{\langle \mathbf{Z} \rangle \mid \mathbf{Z} \in \mathcal{M}^\bullet\}$ is a basis of $\mathcal{M}^\bullet$. Let $\mathcal{M}^\circ$ be the set of all $Z \in \mathcal{M}^\bullet$ such that $\langle \mathbf{Z} \rangle \notin \ker p$; then $\{p(\langle \mathbf{Z} \rangle) \mid \mathbf{Z} \in \mathcal{M}^\circ\}$ is a basis of $\mathcal{M}^\circ$. In consequence, each weight subspace of $\mathcal{M}^\circ$ is endowed with a basis.

As a crystal, $\mathcal{M}$ decomposes as the disjoint union (direct sum) of its connected components, and $\mathcal{M}^\circ$ is the union of the connected components of $\mathcal{M}$ that are isomorphic to $\mathcal{L}(\lambda^*)$. For each connected component $\mathcal{C} \subseteq \mathcal{M}^\circ$, the subspace of $\mathcal{M}^\circ$ spanned by $B_{\mathcal{C}} = \{p(\langle \mathbf{Z} \rangle) \mid \mathbf{Z} \in \mathcal{C}\}$ is a subrepresentation isomorphic to $V(\lambda^*)$, and by Remark 3.5, $B_{\mathcal{C}}$ identifies with the MV basis of $V(\lambda^*)$. The action of $\theta(w_0, \lambda^*)$ therefore maps the highest weight element in $B_{\mathcal{C}}$ to the lowest element in $B_{\mathcal{C}}$. We conclude that the bottom horizontal arrow in (33) maps the basis of $M_{\lambda^*}^\circ$ to the basis of $M_{w_0\lambda^*}^\circ$.

Each element in the MV basis of $V(\lambda^{[1]}) = M \otimes V(\lambda)$ is of the form $\langle \mathbf{Z} \rangle$ with $\mathbf{Z}$ in $\mathcal{L}(\lambda^{[1]}) = \mathcal{M} \times \mathcal{L}(\lambda)$. Let $V(\lambda)_{\neq \lambda}$ be the sum of all the weight subspaces of $V(\lambda)$ other than the higher weight subspace. Theorem 5.13 implies that for each $\mathbf{Z}_{(1)} \in \mathcal{M}$,

$$ \langle \mathbf{Z}_{(1)} \rangle \otimes \langle \{L_\lambda\} \rangle \equiv \langle (\mathbf{Z}_{(1)}, \{L_\lambda\}) \rangle (\text{mod } M \otimes V(\lambda)_{\neq \lambda}). $$

Thus, for $\mathbf{Z}_{(1)} \in \mathcal{M}$ and $\mathbf{Z} = (\mathbf{Z}_{(1)}, \{L_\lambda\})$, we have $(\text{id}_M \otimes v_\lambda^*)(\langle \mathbf{Z} \rangle) = \langle \mathbf{Z}_{(1)} \rangle$.

As evidenced by the crystal structure on $\mathcal{M} \otimes \mathcal{L}(\lambda)$, the Satake basis of the space $(M \otimes V(\lambda))^G$ consists of the vectors $\langle \mathbf{Z} \rangle$ for the pairs $\mathbf{Z} = (\mathbf{Z}_{(1)}, \{L_\lambda\})$ such that $\mathbf{Z}_{(1)} \in \mathcal{M}_{w_0\lambda^*}^\circ$. Noting that $\langle \mathbf{Z}_{(1)} \rangle \in M_{w_0\lambda^*}^\circ$ for those $\mathbf{Z}_{(1)}$, we conclude that the right vertical arrow in (33) maps basis elements to basis elements.

Similarly, the left vertical arrow in (33) maps the Satake basis of $(V(\lambda) \otimes M)^G$ to the basis of $M_{\lambda^*}^\circ$. Lemma 6.2 then concludes the proof of Theorem 6.1.

## 6.2. Tensor product with an invariant element

Let $(n', n'')$ be a composition of $n$ with two parts. Correspondingly, we write $\lambda \in (\Lambda^+)^n$ as a concatenation $(\lambda', \lambda'')$ and view each element in $\mathcal{L}(\lambda)$ as a pair $(\mathbf{Z}', \mathbf{Z}'') \in \mathcal{L}(\lambda') \times \mathcal{L}(\lambda'')$.

The following proposition implies that the Satake basis of the invariant subspace of $V(\lambda)$ satisfies the second item in Khovanov and Kuperberg's list of properties for the dual canonical basis (see the introduction of [30]).

**Proposition 6.3.** Let $(\mathbf{Z}', \mathbf{Z}'') \in \mathcal{L}(\lambda)$. If $\langle \mathbf{Z}' \rangle \in V(\lambda')^G$, then $\langle \mathbf{Z}' \rangle \otimes \langle \mathbf{Z}'' \rangle = \langle (\mathbf{Z}', \mathbf{Z}'') \rangle$.
Proof. Let $\mathbf{Z}' \in \mathcal{P}(\lambda')$. Recall the map $m_{n'} : \mathrm{Gr}_{n'} \to \mathrm{Gr}$ defined in Sect. 2.2 and the notation $\mu_I$ from Sect. 3.4 and set $\mu = \mu_I(\mathbf{Z}')$. Then $m_{n'}(\mathbf{Z}') \subseteq \overline{\mathrm{Gr}}^{\mu}$ and $\langle \mathbf{Z}' \rangle$ appear in the isotypic filtration of $V(\lambda')$ at the step where the component of type $V(\mu)$ is appended.

If $\langle \mathbf{Z}' \rangle \in V(\lambda')^G$, then $\mu = 0$, accordingly $\overline{\mathrm{Gr}}^{\mu} = \{L_0\}$, and as a result

$$ \overline{\mathbf{Z}}' \subseteq (m_{n'})^{-1}(\{L_0\}) \subseteq (m_{n'})^{-1}(T_0). $$

This implies that no MV cycle in $\mathcal{P}(\lambda')$ can be strictly contained in $\overline{\mathbf{Z}}'$. (Such a cycle would be contained in $(m_{n'})^{-1}(T_0)$, so would be an irreducible component of $\overline{\mathrm{Gr}}_{n'}^{\lambda'} \cap (m_{n'})^{-1}(T_0)$, and would end up having dimension $\rho(|\lambda'|)$, the same as $\mathbf{Z}'$.) The desired result now directly follows from Theorem 5.13. ■

# 7. Applications to the MV basis of $\mathbb{C}[N]$

We adopt the notation set up in the preamble of Sect. 3. Let $N$ be the unipotent radical of the Borel subgroup $B$ and let $\mathbb{C}[N]$ be the algebra of regular functions on $N$. At the expense of an isogeny, which does not alter $N$, we can assume that $G$ is simply-connected.

For each dominant weight $\lambda \in \Lambda^+$, we can choose a highest weight vector $v_\lambda$ in the representation $V(\lambda)$ and define the linear form $v_\lambda^* : V(\lambda) \to \mathbb{C}$ such that $\langle v_\lambda^*, v_\lambda \rangle = 1$ and $\langle v_\lambda^*, v \rangle = 0$ for all weight vectors $v$ of weight other than $\lambda$. This yields an embedding $\Psi_\lambda : v \mapsto \langle v_\lambda^*, ?v \rangle$ of $V(\lambda)$ into $\mathbb{C}[N]$, where $\langle v_\lambda^*, ?v \rangle$ stands for the function $n \mapsto \langle v_\lambda^*, nv \rangle$. The MV bases of the representations $V(\lambda)$ can be transported to $\mathbb{C}[N]$ through these maps $\Psi_\lambda$, and they glue together to form a basis of $\mathbb{C}[N]$, which we call the MV basis of $\mathbb{C}[N]$ (see [3]).

The algebra $\mathbb{C}[N]$ comes with several remarkable bases: the MV basis, subject of our current investigation, but also the dual canonical basis of Lusztig also known as the upper global basis of Kashiwara, and (in simply laced type) the dual semicanonical basis. The theory of cluster algebras was developed in order to compute effectively these bases (or at least, the dual canonical basis). Concretely, the cluster structure of $\mathbb{C}[N]$ allows one to define specific elements, called cluster monomials, which are linearly independent and easily amenable to calculations. It is known that both the dual canonical and the dual semicanonical bases contain all the cluster monomials [20, 25], but also that these bases differ (except when cluster monomials span $\mathbb{C}[N]$).

The methods developed in Sect. 5 allow us to effectively compute products of elements of the MV basis of $\mathbb{C}[N]$. This allows us to prove that this basis contains quite a few cluster monomials (Proposition 7.2) and that it generally differs from both the dual canonical and the dual semicanonical bases (Proposition 7.3).

## 7.1. Cluster monomials

As explained in [21, Sect. 4.3], each reduced word $(i_1, \dots, i_\ell)$ of the longest element $w_0$ in the Weyl group $W$ yields a seed of the cluster structure of $\mathbb{C}[N]$. The main result
Proof. Let $\mathbb{Z}' \in \mathcal{P}(\lambda')$. Recall the map $m_{n'} : \mathrm{Gr}_{n'} \to \mathrm{Gr}$ defined in Sect. 2.2 and the notation $\mu_I$ from Sect. 3.4 and set $\mu = \mu_I(\mathbb{Z}')$. Then $m_{n'}(\mathbb{Z}') \subseteq \overline{\mathrm{Gr}}^{\mu}$ and $\langle \mathbb{Z}' \rangle$ appear in the isotypic filtration of $V(\lambda')$ at the step where the component of type $V(\mu)$ is appended.

If $\langle \mathbb{Z}' \rangle \in V(\lambda')^G$, then $\mu = 0$, accordingly $\overline{\mathrm{Gr}}^{\mu} = \{L_0\}$, and as a result

$$ \overline{\mathbb{Z}}' \subseteq (m_{n'})^{-1}(\{L_0\}) \subseteq (m_{n'})^{-1}(T_0). $$

This implies that no MV cycle in $\mathcal{P}(\lambda')$ can be strictly contained in $\overline{\mathbb{Z}}'$. (Such a cycle would be contained in $(m_{n'})^{-1}(T_0)$, so would be an irreducible component of $\overline{\mathrm{Gr}}_{n'}^{\lambda'} \cap (m_{n'})^{-1}(T_0)$, and would end up having dimension $\rho(|\lambda'|)$, the same as $\mathbb{Z}'$.) The desired result now directly follows from Theorem 5.13. ■

# 7. Applications to the MV basis of $\mathbb{C}[N]$

We adopt the notation set up in the preamble of Sect. 3. Let $N$ be the unipotent radical of the Borel subgroup $B$ and let $\mathbb{C}[N]$ be the algebra of regular functions on $N$. At the expense of an isogeny, which does not alter $N$, we can assume that $G$ is simply-connected.

For each dominant weight $\lambda \in \Lambda^+$, we can choose a highest weight vector $v_\lambda$ in the representation $V(\lambda)$ and define the linear form $v_\lambda^* : V(\lambda) \to \mathbb{C}$ such that $\langle v_\lambda^*, v_\lambda \rangle = 1$ and $\langle v_\lambda^*, v \rangle = 0$ for all weight vectors $v$ of weight other than $\lambda$. This yields an embedding $\Psi_\lambda : v \mapsto \langle v_\lambda^*, ?v \rangle$ of $V(\lambda)$ into $\mathbb{C}[N]$, where $\langle v_\lambda^*, ?v \rangle$ stands for the function $n \mapsto \langle v_\lambda^*, nv \rangle$. The MV bases of the representations $V(\lambda)$ can be transported to $\mathbb{C}[N]$ through these maps $\Psi_\lambda$, and they glue together to form a basis of $\mathbb{C}[N]$, which we call the MV basis of $\mathbb{C}[N]$ (see [3]).

The algebra $\mathbb{C}[N]$ comes with several remarkable bases: the MV basis, subject of our current investigation, but also the dual canonical basis of Lusztig also known as the upper global basis of Kashiwara, and (in simply laced type) the dual semicanonical basis. The theory of cluster algebras was developed in order to compute effectively these bases (or at least, the dual canonical basis). Concretely, the cluster structure of $\mathbb{C}[N]$ allows one to define specific elements, called cluster monomials, which are linearly independent and easily amenable to calculations. It is known that both the dual canonical and the dual semicanonical bases contain all the cluster monomials [20, 25], but also that these bases differ (except when cluster monomials span $\mathbb{C}[N]$).

The methods developed in Sect. 5 allow us to effectively compute products of elements of the MV basis of $\mathbb{C}[N]$. This allows us to prove that this basis contains quite a few cluster monomials (Proposition 7.2) and that it generally differs from both the dual canonical and the dual semicanonical bases (Proposition 7.3).

## 7.1. Cluster monomials

As explained in [21, Sect. 4.3], each reduced word $(i_1, \dots, i_\ell)$ of the longest element $w_0$ in the Weyl group $W$ yields a seed of the cluster structure of $\mathbb{C}[N]$. The main result
of this section, Proposition 7.2, presents a sufficient condition for the cluster monomials built from one of these seeds to belong to the MV basis of $\mathbb{C}[N]$.

Set $t_{\mathbb{R}}^{\nabla} = \text{Hom}_{\mathbb{Z}}(\Lambda, \mathbb{R})$ and let $C = \{x \in t_{\mathbb{R}}^{\nabla} | \forall i \in I, \langle x, \alpha_i \rangle > 0\}$ be the Weyl chamber in $t_{\mathbb{R}}^{\nabla}$. We consider the following condition on a reduced word $(i_1, \dots, i_\ell)$:

(A) There exist $x_1 \in s_{i_1}(C)$, $x_2 \in (s_{i_1}s_{i_2})(C)$, ..., $x_\ell \in (s_{i_1}\cdots s_{i_\ell})(C)$ such that $x_k - x_{k+1} \in C$ for each $k \in \{1, \dots, \ell-1\}$.

For instance, choose $(x, y) \in C^2$ in such a way that the straight line joining $x$ to $-y$ avoids all the two-codimensional faces of the Weyl fan in $t_{\mathbb{R}}^{\nabla}$. List in order the chambers successively crossed by this line: $C, s_{i_1}C, (s_{i_1}s_{i_2})(C), \dots$. The word $(i_1, i_2, \dots)$ produced in this manner is then reduced and obviously satisfies condition (A).

Let $Q \subseteq \Lambda$ be the root lattice. We denote by $Q_+$ the positive cone in $Q$ with respect to the dominance order, that is, the set of all linear combinations of the simple roots $\alpha_i$ with nonnegative integral coefficients. We set $Q_- = -Q_+$.

**Lemma 7.1.** Let $(i_1, \dots, i_\ell)$ be a reduced word, set $w_k = s_{i_1} \cdots s_{i_k}$ for $k \in \{1, \dots, \ell\}$, and let $(v_1, \dots, v_\ell) \in w_1(Q_-) \times \cdots \times w_\ell(Q_-)$. Assume that $v_1 + \cdots + v_k \in Q_+$ for all $k \in \{1, \dots, \ell-1\}$, and $v_1 + \cdots + v_\ell = 0$, and $(i_1, \dots, i_\ell)$ satisfies condition (A). Then $v_1 = \cdots = v_\ell = 0$.

*Proof.* We set $\mu_0 = 0$ and $\mu_k = v_1 + \cdots + v_k$ for $k \in \{1, \dots, \ell\}$. We pick $x_1, \dots, x_\ell$ as stated in condition (A). Then

$$ \sum_{k=1}^{\ell} \langle x_k, v_k \rangle = \sum_{k=1}^{\ell} \langle x_k, \mu_k - \mu_{k-1} \rangle = \sum_{k=1}^{\ell-1} \langle x_k - x_{k+1}, \mu_k \rangle. $$

From $x_k \in w_k(C)$ and $v_k \in w_k(Q_-)$, we deduce that $\langle x_k, v_k \rangle \le 0$ for each $k \in \{1, \dots, \ell\}$. On the other hand, from $x_k - x_{k+1} \in C$ and $\mu_k \in Q_+$, we deduce that $\langle x_k - x_{k+1}, \mu_k \rangle \ge 0$ for each $k \in \{1, \dots, \ell-1\}$. We conclude that each $\langle x_k, v_k \rangle$ is indeed zero, which implies $v_k = 0$. ■

In Sect. 6.1, we defined, for each $(\lambda, w) \in \Lambda^+ \times W$, a product $\theta(\lambda, w)$ of divided powers of the root vectors $f_i$. We can then set $v_{w\lambda} = \theta(\lambda, w) \cdot v_\lambda$; this is a vector of weight $w\lambda$ in $V(\lambda)$. We define $\Delta_{\lambda, w\lambda} = \Psi_\lambda(v_{w\lambda})$, usually called a *flag minor* if $\lambda$ is minuscule. We denote by $\{\omega_i | i \in I\}$ the set of fundamental weights.

**Proposition 7.2.** Let $(i_1, \dots, i_\ell)$ be a reduced word and define $x_k = \Delta_{\omega_{i_k}, s_{i_1}\cdots s_{i_k}} \omega_{i_k}$ for each $k \in \{1, \dots, \ell\}$. If $(i_1, \dots, i_\ell)$ satisfies condition (A), then any monomial in $x_1, \dots, x_\ell$ belongs to the MV basis of $\mathbb{C}[N]$.

*Proof.* We choose $\lambda = (\lambda_1, \dots, \lambda_\ell)$ in $(\Lambda^+)^\ell$. For $k \in \{1, \dots, \ell\}$, we set $w_k = s_{i_1} \cdots s_{i_k}$. The extremal weight vector $v_{w_k\lambda_k} \in V(\lambda_k)$ belongs to the MV basis [3, Remark 2.10 and Theorem 5.2], so $v_{w_k\lambda_k} = \langle Z_k \rangle$ where $Z_k$ is the cycle $\text{Gr}^{\lambda_k} \cap T_{w_k\lambda_k}$. We set $\mu = w_1\lambda_1 + \cdots + w_\ell\lambda_\ell$ and $Z = (Z_1, \dots, Z_\ell)$. We adopt the convention of Sect. 5.4 and regard $Z$ as an element of $\mathcal{Z}(\lambda)_\mu$; then $\langle\langle Z \rangle\rangle = v_{w_1\lambda_1} \otimes \cdots \otimes v_{w_\ell\lambda_\ell}$.
of this section, Proposition 7.2, presents a sufficient condition for the cluster monomials built from one of these seeds to belong to the MV basis of $\mathbb{C}[N]$.

Set $t_{\mathbb{R}}^{\nabla} = \text{Hom}_{\mathbb{Z}}(\Lambda, \mathbb{R})$ and let $C = \{x \in t_{\mathbb{R}}^{\nabla} | \forall i \in I, \langle x, \alpha_i \rangle > 0\}$ be the Weyl chamber in $t_{\mathbb{R}}^{\nabla}$. We consider the following condition on a reduced word $(i_1, \dots, i_\ell)$:

(A) There exist $x_1 \in s_{i_1}(C)$, $x_2 \in (s_{i_1}s_{i_2})(C)$, ..., $x_\ell \in (s_{i_1}\cdots s_{i_\ell})(C)$ such that $x_k - x_{k+1} \in C$ for each $k \in \{1, \dots, \ell-1\}$.

For instance, choose $(x, y) \in C^2$ in such a way that the straight line joining $x$ to $-y$ avoids all the two-codimensional faces of the Weyl fan in $t_{\mathbb{R}}^{\nabla}$. List in order the chambers successively crossed by this line: $C, s_{i_1}C, (s_{i_1}s_{i_2})(C), \dots$. The word $(i_1, i_2, \dots)$ produced in this manner is then reduced and obviously satisfies condition (A).

Let $Q \subseteq \Lambda$ be the root lattice. We denote by $Q_+$ the positive cone in $Q$ with respect to the dominance order, that is, the set of all linear combinations of the simple roots $\alpha_i$ with nonnegative integral coefficients. We set $Q_- = -Q_+$.

**Lemma 7.1.** Let $(i_1, \dots, i_\ell)$ be a reduced word, set $w_k = s_{i_1} \cdots s_{i_k}$ for $k \in \{1, \dots, \ell\}$, and let $(v_1, \dots, v_\ell) \in w_1(Q_-) \times \cdots \times w_\ell(Q_-)$. Assume that $v_1 + \cdots + v_k \in Q_+$ for all $k \in \{1, \dots, \ell-1\}$, and $v_1 + \cdots + v_\ell = 0$, and $(i_1, \dots, i_\ell)$ satisfies condition (A). Then $v_1 = \cdots = v_\ell = 0$.

*Proof.* We set $\mu_0 = 0$ and $\mu_k = v_1 + \cdots + v_k$ for $k \in \{1, \dots, \ell\}$. We pick $x_1, \dots, x_\ell$ as stated in condition (A). Then

$$ \sum_{k=1}^{\ell} \langle x_k, v_k \rangle = \sum_{k=1}^{\ell} \langle x_k, \mu_k - \mu_{k-1} \rangle = \sum_{k=1}^{\ell-1} \langle x_k - x_{k+1}, \mu_k \rangle. $$

From $x_k \in w_k(C)$ and $v_k \in w_k(Q_-)$, we deduce that $\langle x_k, v_k \rangle \le 0$ for each $k \in \{1, \dots, \ell\}$. On the other hand, from $x_k - x_{k+1} \in C$ and $\mu_k \in Q_+$, we deduce that $\langle x_k - x_{k+1}, \mu_k \rangle \ge 0$ for each $k \in \{1, \dots, \ell-1\}$. We conclude that each $\langle x_k, v_k \rangle$ is indeed zero, which implies $v_k = 0$. ■

In Sect. 6.1, we defined, for each $(\lambda, w) \in \Lambda^+ \times W$, a product $\theta(\lambda, w)$ of divided powers of the root vectors $f_i$. We can then set $v_{w\lambda} = \theta(\lambda, w) \cdot v_\lambda$; this is a vector of weight $w\lambda$ in $V(\lambda)$. We define $\Delta_{\lambda, w\lambda} = \Psi_\lambda(v_{w\lambda})$, usually called a *flag minor* if $\lambda$ is minuscule. We denote by $\{\omega_i | i \in I\}$ the set of fundamental weights.

**Proposition 7.2.** Let $(i_1, \dots, i_\ell)$ be a reduced word and define $x_k = \Delta_{\omega_{i_k}, s_{i_1}\cdots s_{i_k}} \omega_{i_k}$ for each $k \in \{1, \dots, \ell\}$. If $(i_1, \dots, i_\ell)$ satisfies condition (A), then any monomial in $x_1, \dots, x_\ell$ belongs to the MV basis of $\mathbb{C}[N]$.

*Proof.* We choose $\lambda = (\lambda_1, \dots, \lambda_\ell)$ in $(\Lambda^+)^\ell$. For $k \in \{1, \dots, \ell\}$, we set $w_k = s_{i_1} \cdots s_{i_k}$. The extremal weight vector $v_{w_k\lambda_k} \in V(\lambda_k)$ belongs to the MV basis [3, Remark 2.10 and Theorem 5.2], so $v_{w_k\lambda_k} = \langle Z_k \rangle$ where $Z_k$ is the cycle $\text{Gr}^{\lambda_k} \cap T_{w_k\lambda_k}$. We set $\mu = w_1\lambda_1 + \cdots + w_\ell\lambda_\ell$ and $\mathbf{Z} = (Z_1, \dots, Z_\ell)$. We adopt the convention of Sect. 5.4 and regard $\mathbf{Z}$ as an element of $\mathcal{Z}(\lambda)_\mu$; then $\langle\langle \mathbf{Z} \rangle\rangle = v_{w_1\lambda_1} \otimes \cdots \otimes v_{w_\ell\lambda_\ell}$.
Let us expand this element relative to the MV basis of $V(\lambda)$. As in Theorem 5.8, we write

$$ \langle\langle \mathbf{Z} \rangle\rangle = \sum_{Z' \in \mathcal{Z}(\lambda)_{\mu}} a_{Z',Z} \langle\langle Z' \rangle\rangle. \quad (34) $$

Suppose $\mathbf{Z}' \in \mathcal{Z}(\lambda)_{\mu}$ satisfies $a_{\mathbf{Z}',\mathbf{Z}} \neq 0$. Let $(v_1, \dots, v_{\ell}) \in \Lambda^{\ell}$ be the tuple of weights such that $\mathbf{Z}' \in \mathcal{Z}(\lambda_1)_{v_1} \times \dots \times \mathcal{Z}(\lambda_{\ell})_{v_{\ell}}$. For each $k \in \{1, \dots, \ell\}$, we have $\mathcal{Z}(\lambda_k)_{v_k} \neq \emptyset$, so $w_k^{-1}v_k$ is a weight of $V(\lambda_k)$, whence $(v_k - w_k\lambda_k) \in w_k(Q_-)$. From $v_1 + \dots + v_{\ell} = \mu$, we deduce that $(v_1 - w_1\lambda_1) + \dots + (v_{\ell} - w_{\ell}\lambda_{\ell}) = 0$. And by Corollary 5.12, we get

$$ (v_1 - w_1 \lambda_1) + \dots + (v_k - w_k \lambda_k) \in Q_+ $$

for each $k \in \{1, \dots, \ell - 1\}$. Then, assuming that $(i_1, \dots, i_{\ell})$ satisfies condition (A) and applying Lemma 7.1, we find $v_k = w_k\lambda_k$ for each $k \in \{1, \dots, \ell - 1\}$. In other words, none of the inequalities given in Corollary 5.12 is strict. By Remark 5.14, this forces $\mathbf{Z}' = \mathbf{Z}$. Thus, the expansion (34) contains a single term, namely $\langle\langle \mathbf{Z} \rangle\rangle$.

Set $\lambda = \lambda_1 + \dots + \lambda_{\ell}$ and let $p: V(\lambda) \to V(\lambda)$ be the unique morphism that maps $v_{\lambda_1} \otimes \dots \otimes v_{\lambda_{\ell}}$ to $v_{\lambda}$. Noting that $p$ is the quotient map to the top factor in the isotypic filtration of $V(\lambda)$ and applying Remark 3.5, we find that $p(\langle\langle \mathbf{Z} \rangle\rangle)$ belongs to the MV basis of $V(\lambda)$. From the equality $v_{w_1\lambda_1} \otimes \dots \otimes v_{w_{\ell}\lambda_{\ell}} = \langle\langle \mathbf{Z} \rangle\rangle$, we deduce that

$$ \Delta_{\lambda_1, w_1 \lambda_1} \cdots \Delta_{\lambda_{\ell}, w_{\ell} \lambda_{\ell}} = \langle v_{\lambda}^*, p(?\langle\langle \mathbf{Z} \rangle\rangle) \rangle = \Psi_{\lambda}(p(\langle\langle \mathbf{Z} \rangle\rangle)) $$

belongs to the MV basis of $\mathbb{C}[N]$. The claim in the proposition is the particular case where each $\lambda_k$ is a multiple of $\omega_{i_k}$.

## 7.2. A computation in type $D_4$

In [29], Kashiwara and Saito found an example in type $A_5$ where the singular support of a simple perverse sheaf related to the canonical basis is not irreducible. Looking again at this situation, Geiß, Leclerc and Schröer [19] computed the dual canonical and dual semicanonical elements and found that they were different. They also observed that a similar phenomenon occurs in type $D_4$. In [3, Sect. 2.7], this setting in type $D_4$ was examined anew: the MV basis is a third basis, different from the other ones. In an appendix to [3], Dranowski, Kamnitzer and Morton-Ferguson extended this observation to the spot in type $A_5$ not covered by Kashiwara and Saito.

Let us have a closer look at the $D_4$ case. As usual, we label the vertices of the Dynkin diagram from 1 to 4, with 2 for the central node. Our three bases are indexed by the crystal $B(\infty)$: given $b \in B(\infty)$, we denote the corresponding dual semicanonical basis element by $C(b)$, the dual canonical basis element by $C'(b)$, and the MV basis element by $C''(b)$. Letting $b_0$ be the highest weight element in $B(\infty)$, we set

$$ b_1 = (\tilde{f}_2(\tilde{f}_1 \tilde{f}_3 \tilde{f}_4) \tilde{f}_2)^2 b_0 \quad \text{and} \quad b_{12} = (\tilde{f}_2)^2 (\tilde{f}_1 \tilde{f}_3 \tilde{f}_4)^2 (\tilde{f}_2)^2 b_0. $$

**Proposition 7.3.** *The basis elements are related by the equations*

$$ C(b_{12}) = C''(b_{12}) + 2C(b_1) \quad \text{and} \quad C'(b_{12}) = C''(b_{12}) + C(b_1). $$
Let us expand this element relative to the MV basis of $V(\lambda)$. As in Theorem 5.8, we write

$$ \langle\langle \mathbf{Z} \rangle\rangle = \sum_{\mathbf{Z}' \in \mathcal{Z}(\lambda)_{\mu}} a_{\mathbf{Z}', \mathbf{Z}} \langle\langle \mathbf{Z}' \rangle\rangle. \quad (34) $$

Suppose $\mathbf{Z}' \in \mathcal{Z}(\lambda)_\mu$ satisfies $a_{\mathbf{Z}', \mathbf{Z}} \neq 0$. Let $(v_1, \dots, v_\ell) \in \Lambda^\ell$ be the tuple of weights such that $\mathbf{Z}' \in \mathcal{Z}(\lambda_1)_{v_1} \times \dots \times \mathcal{Z}(\lambda_\ell)_{v_\ell}$. For each $k \in \{1, \dots, \ell\}$, we have $\mathcal{Z}(\lambda_k)_{v_k} \neq \emptyset$, so $w_k^{-1}v_k$ is a weight of $V(\lambda_k)$, whence $(v_k - w_k\lambda_k) \in w_k(Q_-)$. From $v_1 + \dots + v_\ell = \mu$, we deduce that $(v_1 - w_1\lambda_1) + \dots + (v_\ell - w_\ell\lambda_\ell) = 0$. And by Corollary 5.12, we get

$$ (v_1 - w_1 \lambda_1) + \dots + (v_k - w_k \lambda_k) \in Q_+ $$

for each $k \in \{1, \dots, \ell - 1\}$. Then, assuming that $(i_1, \dots, i_\ell)$ satisfies condition (A) and applying Lemma 7.1, we find $v_k = w_k\lambda_k$ for each $k \in \{1, \dots, \ell - 1\}$. In other words, none of the inequalities given in Corollary 5.12 is strict. By Remark 5.14, this forces $\mathbf{Z}' = \mathbf{Z}$. Thus, the expansion (34) contains a single term, namely $\langle\langle \mathbf{Z} \rangle\rangle$.

Set $\lambda = \lambda_1 + \dots + \lambda_\ell$ and let $p : V(\lambda) \to V(\lambda)$ be the unique morphism that maps $v_{\lambda_1} \otimes \dots \otimes v_{\lambda_\ell}$ to $v_\lambda$. Noting that $p$ is the quotient map to the top factor in the isotypic filtration of $V(\lambda)$ and applying Remark 3.5, we find that $p(\langle\langle \mathbf{Z} \rangle\rangle)$ belongs to the MV basis of $V(\lambda)$. From the equality $v_{w_1\lambda_1} \otimes \dots \otimes v_{w_\ell\lambda_\ell} = \langle\langle \mathbf{Z} \rangle\rangle$, we deduce that

$$ \Delta_{\lambda_1, w_1 \lambda_1} \cdots \Delta_{\lambda_\ell, w_\ell \lambda_\ell} = \langle v_{\lambda}^*, p(?\langle\langle \mathbf{Z} \rangle\rangle) \rangle = \Psi_{\lambda}(p(\langle\langle \mathbf{Z} \rangle\rangle)) $$

belongs to the MV basis of $\mathbb{C}[N]$. The claim in the proposition is the particular case where each $\lambda_k$ is a multiple of $\omega_{i_k}$.

## 7.2. A computation in type $D_4$

In [29], Kashiwara and Saito found an example in type $A_5$ where the singular support of a simple perverse sheaf related to the canonical basis is not irreducible. Looking again at this situation, Geiß, Leclerc and Schröer [19] computed the dual canonical and dual semicanonical elements and found that they were different. They also observed that a similar phenomenon occurs in type $D_4$. In [3, Sect. 2.7], this setting in type $D_4$ was examined anew: the MV basis is a third basis, different from the other ones. In an appendix to [3], Dranowski, Kamnitzer and Morton-Ferguson extended this observation to the spot in type $A_5$ not covered by Kashiwara and Saito.

Let us have a closer look at the $D_4$ case. As usual, we label the vertices of the Dynkin diagram from 1 to 4, with 2 for the central node. Our three bases are indexed by the crystal $B(\infty)$: given $b \in B(\infty)$, we denote the corresponding dual semicanonical basis element by $C(b)$, the dual canonical basis element by $C'(b)$, and the MV basis element by $C''(b)$. Letting $b_0$ be the highest weight element in $B(\infty)$, we set

$$ b_1 = (\tilde{f}_2(\tilde{f}_1 \tilde{f}_3 \tilde{f}_4) \tilde{f}_2)^2 b_0 \quad \text{and} \quad b_{12} = (\tilde{f}_2)^2 (\tilde{f}_1 \tilde{f}_3 \tilde{f}_4)^2 (\tilde{f}_2)^2 b_0. $$

**Proposition 7.3.** *The basis elements are related by the equations*

$$ C(b_{12}) = C''(b_{12}) + 2C(b_1) \quad \text{and} \quad C'(b_{12}) = C''(b_{12}) + C(b_1). $$
The proof is given in [3], except for one justification left to the present paper. We here fill the gap.

The fundamental weight $\omega_2$ is the highest root $\alpha_1 + 2\alpha_2 + \alpha_3 + \alpha_4$. The crystal of the representation $V(\omega_2)$ (the adjoint representation) is pictured below. Highest weights are towards the left, vertices are represented as keys $p_q$ with $p, q$ in $\{1, 2, 3, 4, \bar{1}, \bar{2}, \bar{3}, \bar{4}\}$, and operators $\tilde{f}_1, \tilde{f}_2, \tilde{f}_3$ and $\tilde{f}_4$ are indicated by dashed, solid, dotted and dash-dotted arrows, respectively.

<figure><img src="image_19.png" /></figure>

If we endow the weight lattice $\Lambda$ with its usual basis $(\varepsilon_1, \varepsilon_2, \varepsilon_3, \varepsilon_4)$, then the weight of the element $p_q$ is simply $\varepsilon_p + \varepsilon_q$, with the convention that $\varepsilon_{\bar{i}} = -\varepsilon_i$ for $i \in \{1, 2, 3, 4\}$. The crystal contains four elements of weight zero, namely $\frac{2}{2}, \frac{3}{3}, \frac{4}{4}$ and $\frac{4}{4}$.

We set $\lambda = (\omega_2, \omega_2)$ and look at the tensor square $V(\lambda) = V(\omega_2)^{\otimes 2}$. As in Sect. 5.4, its MV basis consists of symbols $\langle Z \rangle$, where $Z = (Z_1, Z_2)$ is a pair in $\mathcal{L}(\omega_2) \times \mathcal{L}(\omega_2)$. In addition, $V(\lambda)$ is endowed with the tensor product basis. To keep the notation straightforward, we indicate MV cycles by the keys $p_q$, making use of the isomorphism between $\mathcal{L}(\omega_2)$ and the crystal pictured above.

We claim that

$$ \langle \frac{2}{3} \rangle \otimes \langle \frac{3}{2} \rangle = 2 \left( \langle \begin{pmatrix} 1 & \frac{2}{1} \\ 2 & \frac{3}{1} \end{pmatrix} \rangle + \langle \begin{pmatrix} 1 & \frac{3}{1} \\ 3 & \frac{2}{1} \end{pmatrix} \rangle + \langle \begin{pmatrix} 2 & \frac{3}{2} \\ 3 & \frac{4}{1} \end{pmatrix} \rangle + \langle \begin{pmatrix} 1 & \frac{4}{1} \\ 4 & \frac{3}{2} \end{pmatrix} \rangle \right. \\ \left. + \langle \begin{pmatrix} 1 & \frac{4}{1} \\ 4 & \frac{2}{3} \end{pmatrix} \rangle + \langle \begin{pmatrix} 2 & \frac{4}{2} \\ 4 & \frac{4}{3} \end{pmatrix} \rangle + \langle \begin{pmatrix} 2 & \frac{4}{2} \\ 4 & \frac{3}{4} \end{pmatrix} \rangle + \langle \begin{pmatrix} 1 & \frac{3}{1} \\ 3 & \frac{2}{3} \end{pmatrix} \rangle + \langle \begin{pmatrix} 2 & \frac{3}{2} \\ 3 & \frac{3}{4} \end{pmatrix} \rangle. \right) \quad (35) $$

Let $p: V(\omega_2)^{\otimes 2} \to V(2\omega_2)$ be the unique morphism that maps $v_{\omega_2} \otimes v_{\omega_2}$ to $v_{2\omega_2}$. Applying $\Psi_{2\omega_2} \circ p$ to (35), we obtain the equality

$$ C''(b_{13})C''(b_{14}) = 2C''(b_1) + \sum_{i=2}^{8} C''(b_i) + C''(b_{12}) $$

asserted without proof in [3]. Establishing (35) will therefore complete the proof of Proposition 7.3. Actually, an inspection of the proof in loc. cit. reveals that it is enough to justify that the coefficient of $\langle (\frac{1}{2}, \frac{2}{1}) \rangle$ is strictly larger than 1.

We will use Theorem 5.8 to prove this fact. Here the group $G^\vee$ is $SO_8$. For $(i, j) \in \{1, \dots, 8\}$, we denote by $E_{i,j}$ the $8 \times 8$ matrix with zeros everywhere except for a 1 at position $(i, j)$. For each coroot $\alpha^\vee \in \Phi^\vee$, we define a subgroup $x_{\alpha^\vee}: \mathbb{C} \to G^\vee$ by
The proof is given in [3], except for one justification left to the present paper. We here fill the gap.

The fundamental weight $\omega_2$ is the highest root $\alpha_1 + 2\alpha_2 + \alpha_3 + \alpha_4$. The crystal of the representation $V(\omega_2)$ (the adjoint representation) is pictured below. Highest weights are towards the left, vertices are represented as keys $p_q$ with $p, q$ in $\{1, 2, 3, 4, \bar{1}, \bar{2}, \bar{3}, \bar{4}\}$, and operators $\tilde{f}_1, \tilde{f}_2, \tilde{f}_3$ and $\tilde{f}_4$ are indicated by dashed, solid, dotted and dash-dotted arrows, respectively.

<figure><img src="image_17.png" /></figure>

If we endow the weight lattice $\Lambda$ with its usual basis $(\varepsilon_1, \varepsilon_2, \varepsilon_3, \varepsilon_4)$, then the weight of the element $p_q$ is simply $\varepsilon_p + \varepsilon_q$, with the convention that $\varepsilon_{\bar{i}} = -\varepsilon_i$ for $i \in \{1, 2, 3, 4\}$. The crystal contains four elements of weight zero, namely $\frac{2}{2}, \frac{3}{3}, \frac{4}{4}$ and $\frac{4}{4}$.

We set $\lambda = (\omega_2, \omega_2)$ and look at the tensor square $V(\lambda) = V(\omega_2)^{\otimes 2}$. As in Sect. 5.4, its MV basis consists of symbols $\langle Z \rangle$, where $Z = (Z_1, Z_2)$ is a pair in $\mathcal{L}(\omega_2) \times \mathcal{L}(\omega_2)$. In addition, $V(\lambda)$ is endowed with the tensor product basis. To keep the notation straightforward, we indicate MV cycles by the keys $p_q$, making use of the isomorphism between $\mathcal{L}(\omega_2)$ and the crystal pictured above.

We claim that

$$ \langle \frac{2}{3} \rangle \otimes \langle \frac{3}{2} \rangle = 2 \left( \langle \begin{pmatrix} 1 & \frac{2}{1} \\ 2 & \frac{3}{1} \end{pmatrix} \rangle + \langle \begin{pmatrix} 1 & \frac{3}{1} \\ 3 & \frac{2}{1} \end{pmatrix} \rangle + \langle \begin{pmatrix} 2 & \frac{3}{2} \\ 3 & \frac{4}{1} \end{pmatrix} \rangle + \langle \begin{pmatrix} 1 & \frac{4}{1} \\ 4 & \frac{3}{2} \end{pmatrix} \rangle \right. \\ \left. + \langle \begin{pmatrix} 1 & \frac{4}{1} \\ 4 & \frac{2}{3} \end{pmatrix} \rangle + \langle \begin{pmatrix} 2 & \frac{4}{2} \\ 4 & \frac{4}{3} \end{pmatrix} \rangle + \langle \begin{pmatrix} 2 & \frac{4}{2} \\ 4 & \frac{3}{4} \end{pmatrix} \rangle + \langle \begin{pmatrix} 1 & \frac{3}{1} \\ 3 & \frac{2}{3} \end{pmatrix} \rangle + \langle \begin{pmatrix} 2 & \frac{3}{2} \\ 3 & \frac{3}{4} \end{pmatrix} \rangle. \right) \quad (35) $$

Let $p: V(\omega_2)^{\otimes 2} \to V(2\omega_2)$ be the unique morphism that maps $v_{\omega_2} \otimes v_{\omega_2}$ to $v_{2\omega_2}$. Applying $\Psi_{2\omega_2} \circ p$ to (35), we obtain the equality

$$ C''(b_{13})C''(b_{14}) = 2C''(b_1) + \sum_{i=2}^{8} C''(b_i) + C''(b_{12}) $$

asserted without proof in [3]. Establishing (35) will therefore complete the proof of Proposition 7.3. Actually, an inspection of the proof in loc. cit. reveals that it is enough to justify that the coefficient of $\langle \begin{pmatrix} 1 & \frac{2}{1} \\ 2 & \frac{3}{1} \end{pmatrix} \rangle$ is strictly larger than 1.

We will use Theorem 5.8 to prove this fact. Here the group $G^\vee$ is $SO_8$. For $(i, j) \in \{1, \dots, 8\}$, we denote by $E_{i,j}$ the $8 \times 8$ matrix with zeros everywhere except for a 1 at position $(i, j)$. For each coroot $\alpha^\vee \in \Phi^\vee$, we define a subgroup $x_{\alpha^\vee}: \mathbb{C} \to G^\vee$ by