We are now in a position to explain the end of the proof of Proposition 7.1.

**Proof of Proposition 7.1.** From the construction of the equivalence $Ras$ and the fact that $\omega_{Gr}$ is I-equivariant, one sees that $Ras(\omega_{Gr}) \cong \omega_{Gr}$. Now, since the forgetful functor $\mathrm{Shv}(I\backslash Gr) \to \mathrm{Shv}(Gr)$ is t-exact and conservative, Proposition 7.4 implies that $\omega_{Gr} \in \mathrm{Shv}(I\backslash Gr)$ is in degree $-\infty$ for the perverse t-structure. The desired claim follows, since by Proposition 7.1 the functor $Ras^{-1}$ is right t-exact up to a shift. $\square$

**7.2. Cohomological amplitude of Raskin's equivalences.** In this subsection we explain the proof of Proposition 7.2. We will only consider the case of sheaves on $\mathrm{Fl}$; the case of $Gr$ can be deduced using the fact that the functor $p^\dagger$ is t-exact and detects perversity (see Remark 6.7), or proved along similar lines.

We will use the following easy claim.

**Lemma 7.5.** Consider any t-structure on the $\infty$-category $\mathrm{Shv}(I\backslash \mathrm{Fl})$, and fix a simple reflection $s \in W_{\mathrm{ext}}$. If an object $\mathcal{F}$ belongs to the “$\le 0$” part of the t-structure, then the complex $\Delta_{s_\alpha} \star^1 \mathcal{F}$ belongs the “$\le 1$” part of the t-structure iff the complex $\nabla_{s_\alpha} \star^1 \mathcal{F}$ belongs to the “$\le 1$” part of the t-structure.

**Proof.** Recall that we have exact sequences of perverse sheaves

$$IC_1 \hookrightarrow \Delta_s \twoheadrightarrow IC_s, \quad IC_s \hookrightarrow \nabla_s \twoheadrightarrow IC_1.$$

The desired claim follows, after convolving of the right with $\mathcal{F}$. $\square$

First, the fact that $Ras$ is right t-exact immediately follows from Proposition 5.6, using the fact that this functor is continuous, that the Wakimoto sheaves are perverse (see §4.4), and the definition of the perverse t-structure on $\mathrm{Shv}(I^{\infty/2}\backslash \mathrm{Fl})$.

Then, in order to prove that $Ras[-\ell(w_\circ)]$ is left t-exact, we have to prove that for any $\mathcal{F}$ in $\mathrm{Shv}(I^{\infty/2}\backslash \mathrm{Fl})$, any $w \in W_{\mathrm{ext}}$ and any $n \in \mathbb{Z}_{>0}$ we have

$$\mathrm{Hom}_{\mathrm{Shv}(I\backslash \mathrm{Fl})}(\Delta_w[n], \mathrm{Ras}(\mathcal{F})[-\ell(w_\circ)]) = 0,$$

which is equivalent to proving that for any $w \in W_{\mathrm{ext}}$ the object $Ras^{-1}(\Delta_w)[\ell(w_\circ)]$ belongs to $\mathrm{Shv}(I^{\infty/2}\backslash \mathrm{Fl})^{\le 0}$, i.e. that $Ras^{-1}(\Delta_w)$ belongs to $\mathrm{Shv}(I^{\infty/2}\backslash \mathrm{Fl})^{\le \ell(w_\circ)}$.

The main remaining ingredient of the proof is the following statement.

**Lemma 7.6.** For any $z \in W_f$, the functor

$$\mathrm{Ras}^{-1}(\nabla_z \star^1 \mathrm{Ras}(-))[\ell(z)]$$

is right t-exact with respect to the perverse t-structure.

**Proof.** Using Lemma 4.1 and induction on $\ell(z)$, it suffices to prove the lemma when $z = s_\alpha$ is the reflection associated with a simple root $\alpha$. Moreover, in view of Proposition 5.6, the desired statement amounts to the property that for any $y \in W_{\mathrm{ext}}$ we have $Ras^{-1}(\nabla_{s_\alpha} \star^1 W_y) \in \mathrm{Shv}(I^{\infty/2}\backslash \mathrm{Fl})^{\le 1}$.

By standard arguments, one can assume that the quotient of $X$ by the root lattice is free. (This condition is equivalent to the property that the dual group $G_k^{\vee}$ has simply connected derived subgroup.) In this case, writing $y = t_\lambda x_f$ with $\lambda \in \mathbb{Y}$ and $x_f \in W_f$, we will proceed by induction on $|\langle \lambda, \alpha \rangle|$. We will use the following facts:

(1) if $\langle \lambda, \alpha \rangle = 0$, then $\nabla_{s_\alpha} \star^1 W_{t_\lambda} \cong W_{t_\lambda} \star^1 \nabla_{s_\alpha}$;

(2) if $\langle \lambda, \alpha \rangle = 1$, then we have $\Delta_{s_\alpha} \star^1 W_{t_{s_\alpha}(\lambda)} \star^1 \Delta_{s_\alpha} \cong W_{t_\lambda}$.
(To justify these claims, one observes that Lemma 4.1 implies that the assignment $w \mapsto \Delta_w$ extends to a morphism from the braid group associated with $W_{\text{ext}}$ to the group of isomorphism classes of invertible objects in $\text{Shv}(\mathbb{I}\backslash\text{Fl})$. By construction the Wakimoto objects ($W_{t_\lambda} : \lambda \in \mathbf{Y}$) are the images under this map of the Bernstein elements in this braid group, see e.g. [38, §2.6]. Then our claims follow from standard identities in this braid group, established in [38, Lemma 2.7].)

To start the induction we consider the case when $\langle\lambda, \alpha\rangle = 0$. Here, for any $x_f \in W_f$ we have

$$ \nabla_{s_\alpha} \star^I W_{t_\lambda x_f} \cong \nabla_{s_\alpha} \star^I W_{t_\lambda} \star^I \Delta_{x_f} \cong W_{t_\lambda} \star^I \nabla_{s_\alpha} \star^I \Delta_{x_f}, $$

where we have used (1) above. By Lemma 4.1(3) the complex $\nabla_{s_\alpha} \star^I \Delta_{x_f}$ is a perverse sheaf supported on $G/B \subset \text{Fl}$; in particular it belongs to the full subcategory of $\text{Shv}(\mathbb{I}\backslash\text{Fl})$ generated under extensions and colimits by the objects $\Delta_z$ with $z \in W_f$. Using once again Proposition 5.6 we deduce that $\text{Ras}^{-1}(\nabla_{s_\alpha} \star^I W_{t_\lambda x_f}) \in \text{Shv}(\mathbb{I}^{\infty/2}\backslash\text{Fl})^{\le 0}$ in this case.

Next we fix some $\lambda \in \mathbf{Y}$ such that $\langle\lambda, \alpha\rangle \ne 0$, and assume that the desired claim is known for any element $\mu \in \mathbf{Y}$ such that $|\langle\mu, \alpha\rangle| < |\langle\lambda, \alpha\rangle|$. Assume first that $\langle\lambda, \alpha\rangle > 0$. Our assumption above implies that one can write $\lambda = \lambda_1 + \lambda_2$ with $\langle\lambda_1, \alpha\rangle = 1$. Then if $x_f \in W_f$ we have

$$ \nabla_{s_\alpha} \star^I W_{t_\lambda x_f} \cong \nabla_{s_\alpha} \star^I W_{t_{\lambda_1}} \star^I W_{t_{\lambda_2} x_f} \cong W_{t_{s_\alpha(\lambda_1)}} \star^I \Delta_{s_\alpha} \star^I W_{t_{\lambda_2} x_f}, $$

where we have used (2) above. By induction we know that $\text{Ras}^{-1}(\nabla_{s_\alpha} \star^I W_{t_{\lambda_2} x_f})$ belongs to $\text{Shv}(\mathbb{I}^{\infty/2}\backslash\text{Fl})^{\le 1}$. Using Lemma 7.5 we deduce that $\text{Ras}^{-1}(\Delta_{s_\alpha} \star^I W_{t_{\lambda_2} x_f}) \in \text{Shv}(\mathbb{I}^{\infty/2}\backslash\text{Fl})^{\le 1}$, and then that $\text{Ras}^{-1}(\nabla_{s_\alpha} \star^I W_{t_\lambda x_f}) \in \text{Shv}(\mathbb{I}^{\infty/2}\backslash\text{Fl})^{\le 1}$.

Finally we consider the case when $\langle\lambda, \alpha\rangle < 0$. Here we write $\lambda = \lambda_1 + \lambda_2$ with $\langle\lambda_1, \alpha\rangle = -1$. Then we have

$$ \Delta_{s_\alpha} \star^I W_{t_\lambda x_f} \cong \Delta_{s_\alpha} \star^I W_{t_{\lambda_1}} \star^I W_{t_{\lambda_2} x_f} \cong W_{t_{s_\alpha(\lambda_1)}} \star^I \nabla_{s_\alpha} \star^I W_{t_{\lambda_2} x_f}, $$

so that using induction one sees that the image under $\text{Ras}^{-1}$ of this object belongs to $\text{Shv}(\mathbb{I}^{\infty/2}\backslash\text{Fl})^{\le 1}$. Using again Lemma 7.5 one deduces that $\text{Ras}^{-1}(\nabla_{s_\alpha} \star^I W_{t_\lambda x_f}) \in \text{Shv}(\mathbb{I}^{\infty/2}\backslash\text{Fl})^{\le 1}$, which concludes the proof. $\square$

One can now conclude the proof of Proposition 7.2 as follows. It is a standard fact that the maximal length representatives of the cosets in $W_f\backslash W_{\text{ext}}$ are of the form $t_\lambda x$ with $\lambda \in \mathbf{Y}_+$ and $x \in W_f$. (This can e.g. be deduced from the description of minimal coset representatives in $W_{\text{ext}}/W_f$ recalled in §3.1.) For such elements the Wakimoto sheaf coincides with the associated standard perverse sheaf, see Lemma 4.3(2). Using also Lemma 4.1, one deduces that for any $w \in W_{\text{ext}}$ there exist $z \in W_f$ and $y \in W_{\text{ext}}$ such that $\Delta_w \cong \nabla_z \star^I W_y$. Then we have

$$ \text{Ras}^{-1}(\Delta_w) \cong \text{Ras}^{-1}(\nabla_z \star^I W_y) \cong \text{Ras}^{-1}(\nabla_z \star^I \text{Ras}(\Delta_y^{\infty/2})) $$

by Proposition 5.6, so that the desired claim follows from Lemma 7.6.

## 8. THE GAITSGORY SHEAF

8.1. **Construction.** Recall the objects ($\mathcal{M}_\lambda : \lambda \in \mathbf{Y}^+$) considered in §6.7. Recall also from Remark 6.17(1) that we have an identification

$$ \mathcal{M}_\lambda \cong \text{Ras}^{-1}(\overline{W}_{-\lambda} \star^{\mathcal{L}+G} \mathcal{I}_*(\lambda)). $$
Consider the preorder $\leq$ on $\mathbf{Y}$ defined by

$$ \lambda \leq \mu \quad \text{iff} \quad \mu - \lambda \in \mathbf{Y}_{+}. $$

If $\mu, \lambda \in \mathbf{Y}_{+}$ satisfy $\mu \leq \lambda$, we will explain how to define a canonical map

$$ \theta_{\mu, \lambda} : \mathcal{M}_{\mu} \to \mathcal{M}_{\lambda} $$

in $\mathrm{Shv}(\mathbb{I}^{\infty/2} \backslash \mathrm{Gr})^{\heartsuit}$.

The construction will use the following ingredients. First, if $\nu \in \mathbf{Y}_{+}$ we have $\overline{\mathcal{W}}_{\nu} = \overline{\Delta}_{\nu}$. Since $\mathcal{I}_{*}(\nu)|_{\mathrm{Gr}_{\nu}} = \mathbb{K}_{\mathrm{Gr}_{\nu}}[\langle \nu, 2\rho \rangle]$, by adjunction there exists a canonical map

$$ (8.1) \qquad \overline{\mathcal{W}}_{\nu} \to \mathcal{I}_{*}(\nu). $$

Next, if $\nu, \nu' \in \mathbf{Y}_{+}$, it is a standard fact that

$$ (\mathcal{I}_{*}(\nu) \star^{\mathcal{L}^{+}G} \mathcal{I}_{*}(\nu'))|_{\mathrm{Gr}_{\nu+\nu'}} \cong \mathbb{K}_{\mathrm{Gr}_{\nu+\nu'}}[\langle \nu + \nu', 2\rho \rangle]. $$

By adjunction, we deduce that there exists a canonical map

$$ (8.2) \qquad \mathcal{I}_{*}(\nu) \star^{\mathcal{L}^{+}G} \mathcal{I}_{*}(\nu') \to \mathcal{I}_{*}(\nu + \nu'). $$

Now we come back to the setting above. We need to construct a map

$$ \overline{\mathcal{W}}_{-\mu} \star^{\mathcal{L}^{+}G} \mathcal{I}_{*}(\mu) \to \overline{\mathcal{W}}_{-\lambda} \star^{\mathcal{L}^{+}G} \mathcal{I}_{*}(\lambda). $$

Applying the equivalence $\mathcal{W}_{t_{\lambda}} \star^{\mathrm{I}} (-)$ and using (4.13), this is equivalent to constructing a map

$$ \overline{\mathcal{W}}_{\lambda-\mu} \star^{\mathcal{L}^{+}G} \mathcal{I}_{*}(\mu) \to \mathcal{I}_{*}(\lambda). $$

We take this map to be the composition

$$ \overline{\mathcal{W}}_{\lambda-\mu} \star^{\mathcal{L}^{+}G} \mathcal{I}_{*}(\mu) \to \mathcal{I}_{*}(\lambda-\mu) \star^{\mathcal{L}^{+}G} \mathcal{I}_{*}(\mu) \to \mathcal{I}_{*}(\lambda), $$

where the first map is obtained from (8.1) (for the dominant weight $\lambda - \mu$) by right convolution with $\mathcal{I}_{*}(\mu)$, and the second one is (8.2) applied to the dominant weights $\lambda - \mu$ and $\mu$.

The collection of maps $\theta_{\mu, \lambda} : \mathcal{M}_{\mu} \to \mathcal{M}_{\lambda}$ makes $(\mathcal{M}_{\lambda})_{\lambda \in \mathbf{Y}_{+}}$ a directed system in the abelian category $\mathrm{Shv}(\mathbb{I}^{\infty/2} \backslash \mathrm{Gr})^{\heartsuit}$ with respect to $(\mathbf{Y}_{+}, \leq)$. Since the mapping spaces between objects in the heart of a t-structure on a stable $\infty$-category are discrete, this system canonically lifts to a functor

$$ \mathbf{Y}_{+} \to \mathrm{Shv}(\mathbb{I}^{\infty/2} \backslash \mathrm{Gr}), $$

so that we can define the *Gaitsgory sheaf* to be the object

$$ \mathrm{Ga}^{\infty/2} := \operatorname{colim}_{\lambda \in \mathbf{Y}_{+}} \mathcal{M}_{\lambda} $$

(where the colimit is taken with respect to the preorder $\leq$). By Remark 6.1(3), this object belongs to the heart of the perverse t-structure.

Let us note the following property for later use.

**Lemma 8.1.** If $\mu \leq \lambda$, the transition map $\mathcal{M}_{\mu} \to \mathcal{M}_{\lambda}$ is injective.

*Proof.* It is clear that for any $\nu \in \mathbf{Y}_{+}$ the map $\theta_{0,\nu}$ is nonzero; in view of Corollary 6.22, this map therefore identifies with the embedding of the socle of $\mathcal{M}_{\nu}$. Since $\theta_{\mu,\lambda} \circ \theta_{0,\mu} = \theta_{0,\lambda}$, we deduce the claim. $\square$
8.2. Main properties. The following statement describes the stalks and costalks of $Ga^{\infty/2}$.

**Theorem 8.2.** (1) The object $Ga^{\infty/2}$ is supported on $\overline{S}_0$, and its stalks are given for $\nu \in \mathbf{Y}$ by

$$ \mathrm{rank} \, {}^p\mathcal{H}^n((\mathbf{i}_\nu)^* \mathrm{Ga}^{\infty/2}) = \begin{cases} \mathcal{P}(-\nu, 1) & \text{if } n=0, \\ 0 & \text{otherwise.} \end{cases} $$

(2) For $\nu \in \mathbf{Y}$, we have

$$ \mathrm{rank} \, {}^p\mathcal{H}^0((\mathbf{i}_\nu)^! \mathrm{Ga}^{\infty/2}) = \begin{cases} 1 & \text{if } \nu = 0, \\ 0 & \text{otherwise,} \end{cases} \quad \mathrm{rank} \, {}^p\mathcal{H}^1((\mathbf{i}_\nu)^! \mathrm{Ga}^{\infty/2}) = 0. $$

(3) Assume that $\mathrm{char}(\mathbb{k})$ is good for $G$. For $\nu \in \mathbf{Y}$, the corestriction of $Ga^{\infty/2}$ to $S_\nu$ is given by

$$ \sum_n \mathrm{rank} \, {}^p\mathcal{H}^n((\mathbf{i}_\nu)^! \mathrm{Ga}^{\infty/2}) \cdot q^n = \mathcal{P}(-\nu, q^2). $$

*Proof.* As explained in Lemma 6.18(1), each object $M_\lambda$ is supported on $\overline{S}_0$; the same is therefore true for $Ga^{\infty/2}$, so that all the statements are obvious unless $\nu \le 0$.

Fix now $\nu \in \mathbf{Y}$ such that $\nu \le 0$, and a finite union $Z$ of closures $S_\eta$ such that $Z \subset \overline{S}_0$ and $V := \overline{S}_0 \setminus Z$ is a finite union of $I^{\infty/2}$-orbits containing $S_\nu$. We can consider the objects $Ga^{\infty/2}$ and $M_\lambda$ as objects of $\mathrm{Shv}(I^{\infty/2} \setminus \overline{S}_0)^\heartsuit$, and the restriction and corestriction of $Ga^{\infty/2}$ to $S_\nu$ clearly only depend on the object

$$ (Ga^{\infty/2})|_V \cong \underset{\lambda}{\mathrm{colim}} M_{\lambda|V}. $$

We claim that there exists $\lambda \in \mathbf{Y}_+$ such that for any $\mu, \mu' \in \mathbf{Y}_+$ such that $\lambda \sqsubseteq \mu \sqsubseteq \mu'$ the transition morphism

$$ (\theta_{\mu, \mu'})|_V : M_{\mu|V} \to M_{\mu'|V} $$

is an isomorphism. In fact, by Lemma 8.1 and t-exactness of restriction to $V$ these morphisms are injective. We will prove that the objects have the same finite length, which will imply the claim.

By (the analogue for $\mathrm{Gr}$ of) Lemma 6.23 and the general theory of recollement, we know that the restriction functor $\mathrm{Shv}(I^{\infty/2} \setminus \overline{S}_0)^\heartsuit \to \mathrm{Shv}(I^{\infty/2} \setminus V)^\heartsuit$ sends simple objects either to 0 or to simple objects. Since each $M_\mu$ has finite length (see Corollary 6.22), it follows that the same is true for each $M_{\mu|V}$. Moreover, as in Remark 6.25 the simple objects in $\mathrm{Shv}(I^{\infty/2} \setminus V)^\heartsuit$ are the restrictions of the objects $\overline{\Delta}^{\infty/2}_\eta$ where $\eta$ runs over the (finitely many) elements $\eta \in \mathbf{Y}$ such that $S_\eta \subset V$. The multiplicity of the restriction of $\overline{\Delta}^{\infty/2}_\eta$ in an object of $\mathrm{Shv}(I^{\infty/2} \setminus V)^\heartsuit$ is equal to the rank of the image of the latter object under the functor $(\mathbf{i}_\eta)^*$. For the objects $M_{\mu|V}$, this rank has been computed in Lemma 6.18(2); to conclude it therefore suffices to prove that the integer given by this formula is independent of $\mu$ for all $\eta$ as above, provided $\mu$ is sufficiently large with respect to $\le$. However, for any fixed $\eta$, if $\mu$ is sufficiently large we have

$$ w(\mu + \rho^\vee) - (\mu + \eta + \rho^\vee) \not\ge 0 \quad \text{for all } w \in W_f \setminus \{0\}. $$

(Of course, this property is closely related to the facts stated in Remark 6.19.) Therefore, for such $\mu$, in the sum appearing in Lemma 6.18(2), only the term
corresponding to $w = e$ contributes. Since this contribution does not depend on $\mu$, this finishes the proof of our claim.

The statements now follow from this claim, its proof, and Lemma 6.18. $\square$

*Remark 8.3.* In view of Remark 6.19, for any $\nu \in \mathbf{Y}$ the image under the equivalence from (3.6) of the object $(i_\nu)^*\mathbf{Ga}^{\infty/2}$ identifies with $\mathcal{O}(U_k^{\vee,+})_\nu$. For characteristic-0 coefficients this statement is [25, Proposition 2.5.4]. In this setting Gaitsgory also provides a canonical description of the cofibers of $\mathbf{Ga}^{\infty/2}$ in [25, Proposition 2.5.2]. We do not establish this description for general coefficients here, although we expect that it still holds in good characteristic.

We establish additional properties of $\mathbf{Ga}^{\infty/2}$ in the following statement.

**Proposition 8.4.** (1) We have $\mathbf{Ga}^{\infty/2} \cong {}^p\mathcal{H}^0(\bar{\nabla}_0^{\infty/2})$.

(2) The object $\mathrm{oblv}^{1/2|I_u^{\infty/2}}(\mathbf{Ga}^{\infty/2}) \in \mathrm{Shv}(I_u^{\infty/2}\backslash\mathrm{Gr})^{\heartsuit}$ has a unique simple subobject, isomorphic to $\mathrm{oblv}^{1/2|I_u^{\infty/2}}(\bar{\Delta}_0^{\infty/2})$, and it is injective.

*Proof.* (1) By (the analogue for Gr of) Lemma 6.23 applied to $\bar{S}_0$ and $\bar{S}_0 \backslash S_0$, and the general theory of recollement (see in particular [11, comments following Corollaire 1.4.24]), ${}^p\mathcal{H}^0(\bar{\nabla}_0^{\infty/2})$ is uniquely characterized in $\mathrm{Shv}(I^{\infty/2}\backslash\bar{S}_0)^{\heartsuit}$ by the following properties:

* ${}^p\mathcal{H}^0(\bar{\nabla}_0^{\infty/2})|_{S_0} \cong \omega_{S_0}[-\mathrm{psdim}\,S_0]$;

* if $i : \bar{S}_0 \backslash S_0 \to \bar{S}_0$ is the embedding, $i^1\mathcal{H}^0(\bar{\nabla}_0^{\infty/2})$ belongs to $\mathrm{Shv}(I^{\infty/2}\backslash\mathrm{Gr})^{\ge 2}$.

The object $\mathbf{Ga}^{\infty/2}$ satisfies these properties, by definition and Theorem 8.2(2), so it must be isomorphic to ${}^p\mathcal{H}^0(\bar{\nabla}_0^{\infty/2})$.

(2) It follows from Theorem 8.2(2) and adjunction that for any $\nu \in \mathbf{Y}$ we have

$$ (8.3) \quad \operatorname{Hom}_{\operatorname{Shv}(I_u^{\infty/2} \backslash \mathrm{Gr})^{\heartsuit}} (\operatorname{oblv}^{1/2} |_{I_u^{\infty/2}} (\bar{\Delta}_{\nu}^{\infty/2}), \operatorname{oblv}^{1/2} |_{I_u^{\infty/2}} (\mathbf{Ga}^{\infty/2})) = \begin{cases} \mathbb{k} & \text{if } \nu = 0; \\ 0 & \text{otherwise} \end{cases} $$

and

$$ (8.4) \quad \operatorname{Ext}^1_{\operatorname{Shv}(I_u^{\infty/2} \backslash \mathrm{Gr})^{\heartsuit}} (\operatorname{oblv}^{1/2} |_{I_u^{\infty/2}} (\bar{\Delta}_{\nu}^{\infty/2}), \operatorname{oblv}^{1/2} |_{I_u^{\infty/2}} (\mathbf{Ga}^{\infty/2})) = 0. $$

Here (8.3) immediately implies the first claim, in view of Remark 6.21(1). We will show that (8.4) implies the second one. Consider an injective morphism $X \hookrightarrow Y$ in $\mathrm{Shv}(I_u^{\infty/2}\backslash\mathrm{Gr})^{\heartsuit}$. What we have to prove is that any morphism $g : X \to \mathrm{oblv}^{1/2|I_u^{\infty/2}}(\mathbf{Ga}^{\infty/2})$ factors through $Y$. For that we consider the set of pairs $(Z, h)$ where $Z$ is a subobject of $Y$ containing $X$ and $h : Z \to \mathrm{oblv}^{1/2|I_u^{\infty/2}}(\mathbf{Ga}^{\infty/2})$ is a morphism such that $h|_X = g$, which we endow with the order given by $(Z, h) \le (Z', h')$ if $Z \subset Z'$ and $h'_Z = h$. By Zorn's lemma this poset admits a maximal element $(Z, h)$. But if $Z \ne Y$, then by Lemma 6.4(4) there exists a nonzero morphism $\mathrm{oblv}^{1/2|I_u^{\infty/2}}(\bar{\Delta}_{\nu}^{\infty/2}) \to Y/Z$. By Remark 6.21(1) the domain of this map is simple, so the morphism must be injective. If $Z'$ is the preimage in $Y$ of the image of
oblv$^{I^{\infty}/2} |I_{u^2}^{\infty} (\bar{\Delta}_{\nu}^{\infty})$, then the exact sequence

$$
\begin{align*}
\operatorname{Hom}(Z', \operatorname{oblv}^{I^{\infty}/2} |I_{u^2}^{\infty} (\operatorname{Ga}^{\infty/2})) &\to \operatorname{Hom}(Z, \operatorname{oblv}^{I^{\infty}/2} |I_{u^2}^{\infty} (\operatorname{Ga}^{\infty/2})) \\
&\to \operatorname{Ext}^1(\operatorname{oblv}^{I^{\infty}/2} |I_{u^2}^{\infty} (\bar{\Delta}_{\nu}^{\infty}), \operatorname{oblv}^{I^{\infty}/2} |I_{u^2}^{\infty} (\operatorname{Ga}^{\infty/2}))
\end{align*}
$$

and (8.4) imply that $h$ extends to $Z'$, contradicting the maximality of $(Z, h)$. We conclude that $Z = Y$, which finishes the proof. □

Remark 8.5. Since the restriction of oblv$^{I^{\infty}/2} |I_{u^2}^{\infty}$ to Shv$(I^{\infty}/2 \backslash Gr)^{\heartsuit}$ is fully faithful (see Lemma 6.5), Proposition 8.4(2) also implies that the object $Ga^{\infty/2}$ is injective in Shv$(I^{\infty}/2 \backslash Gr)^{\heartsuit}$.

8.3. Another description of the Gaitsgory sheaf. The definition of $Ga^{\infty/2}$ in [25] (for characteristic-0 coefficients) is different from ours. In this subsection we show that the two definitions are equivalent (for general coefficients). Namely, for $\lambda \in \mathbf{Y}_+$ we consider the shifted perverse sheaf

$$
z^{-\lambda} \cdot \mathcal{I}_*(\lambda)[\langle \lambda, 2\rho \rangle] \in \mathrm{Shv}(T\backslash \mathrm{Gr}).
$$

(Here, as in the proof of Lemma 5.5, we denote by $z^{-\lambda} \cdot (-)$ the pushforward functor under the automorphism of Gr given by left multiplication by the element $z^{-\lambda} \in (\mathcal{L}T)(\mathbb{F})$.) For $\lambda, \mu \in \mathbf{Y}_+$ such that $\lambda \trianglelefteq \mu$, there exists a canonical morphism

$$
(8.5) \qquad z^{-\lambda} \cdot \mathcal{I}_*(\lambda)[\langle \lambda, 2\rho \rangle] \to z^{-\mu} \cdot \mathcal{I}_*(\mu)[\langle \mu, 2\rho \rangle]
$$

defined as follows.

First, for any $\nu \in \mathbf{Y}_+$, as explained in the construction of (8.1) there exists a canonical isomorphism

$$
(i_\nu)^! \mathcal{I}_*(\nu) \cong \mathbb{K}_{\mathrm{Gr},\nu} [\langle \nu, 2\rho \rangle].
$$

Denoting by $k_\nu$ the embedding of $z^\nu \mathcal{L}^+ G$ in Gr, we deduce a canonical isomorphism

$$
(k_{\nu})^{!} \mathcal{I}_{*}(\nu) \cong \mathbb{K}[-\langle \nu, 2\rho \rangle].
$$

Now we have

$$
(i_0)^!(z^{-\nu} \cdot \mathcal{I}_*(\nu)[\langle \nu, 2\rho \rangle]) \cong (k_\nu)^! \mathcal{I}_*(\nu)[\langle \nu, 2\rho \rangle],
$$

hence this identification provides by adjunction a canonical morphism

$$
(8.6) \qquad \delta_{\mathrm{Gr}} \to z^{-\nu} \cdot \mathcal{I}_*(\nu)[\langle \nu, 2\rho \rangle].
$$

Now we come back to the construction of the morphism (8.5). Namely, fix
$\lambda, \mu \in \mathbf{Y}_+$ such that $\lambda \le \mu$. Consider the morphism (8.6) for $\nu = \mu - \lambda$, and then
convolve on the right with $\mathcal{I}_*(\lambda)$, and apply $z^{-\lambda} \cdot (-)[\langle \lambda, 2\rho \rangle]$; we obtain a morphism

$$
z^{-\lambda} \cdot \mathcal{I}_*(\lambda)[\langle \lambda, 2\rho \rangle] \to z^{-\mu} \cdot (\mathcal{I}_*(\mu - \lambda) \star \mathcal{L}^+ G \mathcal{I}_*(\lambda))[\langle \mu, 2\rho \rangle].
$$

Composing this morphism with the morphism induced by (8.2) for the weights $\mu-\lambda$
and $\lambda$, we deduce the wished-for morphism (8.5).

Gaitsgory explains in [25, §2.7] that the datum of the objects $z^{-\lambda} \cdot \mathcal{I}_*(\lambda)[\langle \lambda, 2\rho \rangle]$ and the morphisms (8.6) upgrades to a functor from the $\infty$-category associated with the directed set $(\mathbf{Y}_+, \trianglelefteq)$ to $\mathrm{Shv}(T\backslash\mathrm{Gr})$. His arguments apply to general coefficients. Once this fact is established, one can consider the colimit

$$
\operatorname{colim}_{\lambda \in \mathbf{Y}_+} z^{-\lambda} \cdot \mathcal{I}_*(\lambda)[\langle \lambda, 2\rho \rangle] \in \mathrm{Shv}(T\backslash\mathrm{Gr}).
$$
**Proposition 8.6.** There exists a canonical isomorphism

$$ \mathrm{Ga}^{\infty} \cong \underset{\lambda \in \mathbf{Y}_+}{\mathrm{colim}} z^{-\lambda} \cdot \mathcal{I}_*(\lambda)[\langle \lambda, 2\rho \rangle]. $$

*Proof.* First, the same arguments as in [25, §2.4.1] show that the colimit in the right-hand side belongs to $\mathrm{Shv}(\mathrm{I}^{\infty}\backslash\mathrm{Gr})$. This object therefore coincides with its image under $\mathrm{Ras}^{-1} \circ \mathrm{Ras}$. From the definition of $\mathrm{Ras}$ (see §2.7) we obtain that

$$ \mathrm{Ras} \left( \underset{\lambda \in \mathbf{Y}_+}{\mathrm{colim}} z^{-\lambda} \cdot \mathcal{I}_*(\lambda)[\langle \lambda, 2\rho \rangle] \right) \cong \underset{\lambda \in \mathbf{Y}_+}{\mathrm{colim}} \mathrm{Av}_*^{\mathrm{I}|\mathcal{L}^{+T}}(z^{-\lambda} \cdot \mathcal{I}_*(\lambda))[\langle \lambda, 2\rho \rangle]. $$

Now that for any $\lambda \in \mathbf{Y}_+$ we have

$$ \mathrm{Av}_*^{\mathrm{I}|\mathcal{L}^{+T}}(z^{-\lambda} \cdot \mathcal{I}_*(\lambda)) \cong ((i_{-\lambda})_*\mathrm{K}_{\mathrm{Gr}_{-\lambda}}) \star^{\mathcal{L}^{+G}} \mathcal{I}_*(\lambda). $$

Since $\ell(t_{-\lambda}) = \langle \lambda, 2\rho \rangle$, we deduce that

$$ \begin{aligned} \mathrm{Av}_*^{\mathrm{I}|\mathcal{L}^{+T}}(z^{-\lambda} \cdot \mathcal{I}_*(\lambda))[\langle \lambda, 2\rho \rangle] &\cong \nabla_{t_{-\lambda}} \star^1 \mathrm{oblv}^{\mathcal{L}^{+G}|1}(\mathcal{I}_*(\lambda)) \\ &\cong \mathcal{W}_{t_{-\lambda}} \star^1 \mathrm{oblv}^{\mathcal{L}^{+G}|1}(\mathcal{I}_*(\lambda)) \cong \mathrm{Ras}(\mathcal{M}_{\lambda}). \end{aligned} $$

Since $\mathrm{Ras}^{-1}$ commutes with colimits we deduce an isomorphism

$$ \underset{\lambda \in \mathbf{Y}_+}{\mathrm{colim}} z^{-\lambda} \cdot \mathcal{I}_*(\lambda)[\langle \lambda, 2\rho \rangle] \cong \underset{\lambda \in \mathbf{Y}_+}{\mathrm{colim}} \mathcal{M}_{\lambda}. $$

One can check that if $\lambda, \mu \in \mathbf{Y}_+$ satisfy $\lambda \le \mu$ the image under the functor $\mathrm{Ras}^{-1} \circ \mathrm{Av}_*^{\mathrm{I}|\mathcal{L}^{+T}}$ of the morphism (8.5) coincides with the morphism $\theta_{\lambda, \mu}$ of §8.1, which finishes the proof. (This property can be checked at the level of homotopy categories, since these are morphisms between objects in the heart of a t-structure.) □

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DEPARTMENT OF MATHEMATICS, LOUISIANA STATE UNIVERSITY, BATON ROUGE, LA 70803, U.S.A.

Email address: pramod@math.lsu.edu

UCLA MATHEMATICS DEPARTMENT, LOS ANGELES, CA 90095-1555, USA.

Email address: gsd@math.ucla.edu

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Email address: simon.riche@uca.fr