To obtain the character formula for $\text{tr} T_{\tilde{\lambda}, \tilde{a}^*, \tau_+}^G$, it suffices to verify it on the open set formed by the $X \in \mathcal{V}_e$ such that $e \exp X \in G_{ss \text{ reg}}$, included in $\mathfrak{g}(e)_{ss \text{ reg}}$ and of negligible complement (see [Bou 87, before Lem. 1.4.1 p. 6]), on which the two sides of the equality are analytic functions.

Fix $j_e \in \text{Car} \mathfrak{g}(e)$ and $X \in j_e \cap \mathcal{V}_e$ such that $e \exp X \in G_{ss \text{ reg}}$. We have:

$$k_e(X) = |\det(1 - \text{Ad } e)_{\mathfrak{g}/\mathfrak{g}(e)}|^{-1/2} \times |\mathcal{D}_{\mathfrak{g}(e)}(X)|^{-1/2} \times |D_G(e \exp X)|^{1/2}.$$

We paraphrase Bouaziz's calculations as follows. We fix representatives $g_1, \dots, g_n$ of double classes in $M' \cap G_0 \setminus \{g_0 \in G_0 \mid g_0 e g_0^{-1} \in M'\} / G(e)_0$, and then set $e_1 = g_1 e g_1^{-1}, \dots, e_n = g_n e g_n^{-1}$. The disjoint union of the sets

$M' \cap G(e_j)_0 \setminus \{x_0 \in G(e_j)_0 \mid x_0 g_j \in \mathfrak{m}'(e_j)\}$ where $1 \le j \le n$ (we can write "$\subseteq \mathfrak{m}'$" instead of "$\subseteq \mathfrak{m}'(e_j)$") is in bijection with the set

$M' \cap G_0 \setminus \{g_0 \in G_0 \mid g_0 e \exp j_e g_0^{-1} \subseteq M'\}$ via the map that sends $(j, \dot{x})$ to $x \dot{g}_j$. Write $(\text{tr} T_{\tilde{\lambda}, \tilde{a}^*, \tau_+}^G)_e(X)$ using Proposition 11.1 (a) by taking $\pi = T_{\tilde{\lambda}, \tau_{M'}}^{M'}$ and considering this bijection. We then use the expression for $(\text{tr} T_{\tilde{\lambda}, \tau_{M'}}^{M'})_{e_j}$ deduced from formula $(\mathbf{F}_{+,0})$ page 176 with $L = M' \cap G(e_j)_0$ instead of $M'(e_j)_0$, as explained there. This provides three summations. We invert the last two and arrive at:

$$\sum_{1 \le j \le n} \sum_{g \dot{\lambda}_+ \in M' \cap G(e_j)_0 \setminus M' \cdot \lambda_+ \cap \mathfrak{m}'(e_j)^*} \sum_{\dot{x} \in M' \cap G(e_j)_0 \setminus \{x_0 \in G(e_j)_0 \mid x_0 g_j \in \mathfrak{m}'\}} \dots \\ \text{such that } (g \tilde{\lambda})[e_j] \in \mathfrak{g}(e_j)^*_{\text{reg}}$$

The last sum is calculated by applying Proposition 11.1 (b) after replacing $G$ by $G(e_j)$, $\nu'$ by $\nu_+$, $\mathfrak{h}'$ by $(g\mathfrak{h})(e_j)$, $M'_1$ by $M' \cap G(e_j)_0$, $\lambda'$ by $g\lambda_+$ with $t \in ]0, 1]$ (see 1.4), $j$ by $g_j \in \mathfrak{g}$, and $X$ by $g_j X$, then by going to the limit $t \to 0^+$ (see 2.6 (b)). Furthermore, the disjoint union of the sets $M' \cap G(e_j)_0 \setminus M' \cdot \lambda_+ \cap \mathfrak{m}'(e_j)^*$ where $1 \le j \le n$ is in bijection with the set $G(e)_0 \setminus G \cdot \lambda_+ \cap \mathfrak{g}(e)^*$ by the map that sends $(j, \dot{l})$ to $\dot{g}_j^{-1} \dot{l}$. This makes it possible to combine the first two summations into a single one.

Let $\widehat{e}'_+ \in G(\lambda_+)^{\mathfrak{g}/\mathfrak{h}}$ and $\widehat{e}'_{\mathfrak{m}'} \in M'(\tilde{\lambda})^{\mathfrak{m}'/\mathfrak{h}}$ lie over an elliptic element $e'$ of $G(\lambda_+)$. The eigenvalues other than $-1$ and $1$ of the restriction of $\text{Ad } e'^{\mathbb{C}}$ to $\mathfrak{u}_{\mathbb{C}}$ are nonreal pairwise conjugates of absolute value $1$. Since $\mathcal{L}_{\lambda_t}/\mathfrak{h}_{\mathbb{C}}$ (see 5.2) is a direct sum of the $\text{Ad } e'^{\mathbb{C}}$-invariant subspaces $\mathfrak{b}_{M'}/\mathfrak{h}_{\mathbb{C}}$ and $\mathfrak{u}_{\mathbb{C}}$, we have $q \mathcal{L}_{\lambda_t}/\mathfrak{h}_{\mathbb{C}}((\text{Ad } e')_{\mathfrak{g}/\mathfrak{h}}) = q_{\mathfrak{b}_{M'}/\mathfrak{h}_{\mathbb{C}}}((\text{Ad } e')_{\mathfrak{m}'/\mathfrak{h}})$ relative to $B_{\lambda_+}$, then from 4.2 (d):

$$\frac{\mathcal{O}(e'_+)^{(1-\text{Ad } e')(\mathfrak{g}/\mathfrak{h})}}{\mathcal{O}(B_{\lambda_+})^{(1-\text{Ad } e')(\mathfrak{g}/\mathfrak{h})}} \rho_{\lambda_+}^{\mathfrak{g}/\mathfrak{h}}(e'_+) = \frac{\mathcal{O}(\widehat{e}'_{\mathfrak{m}'})^{(1-\text{Ad } e')(\mathfrak{m}'/\mathfrak{h})}}{\mathcal{O}(B_{\lambda_+}^{\mathfrak{m}'}, \mathfrak{m}')^{(1-\text{Ad } e')(\mathfrak{m}'/\mathfrak{h})}} \rho_{\lambda_+^{\mathfrak{m}'}, \mathfrak{m}'}^{\mathfrak{m}'/\mathfrak{h}}(\widehat{e}'_{\mathfrak{m}'}) \times i^{-1/2 \dim(1-\text{Ad } e')(\mathfrak{g}/\mathfrak{m}')}.$$

Using the equality written in the proof of Lemma 8.2 (a), we deduce the character formula for $\text{tr} T_{\tilde{\lambda}, \tilde{a}^*, \tau_+}^G$ (assuming this formula for $\text{tr} T_{\tilde{\lambda}, \tau_{M'}}^{M'}$). $\dots$

## 12. Translation in the sense of Zuckerman

In this section, the aim is to create a tool for the only tricky step: calculating the character of the representation defined after a passage to homology.

I begin by recalling a few classical results.

**Lemma 12.1.** Let $\pi$ be a continuous linear representation of $G_0$ in a complex Hilbert space $V$ and let $K_0$ be a maximal compact subgroup of $G_0$. Under these
tinuous endomorphisms $\pi(\varphi_k d_{G_0} * \prod_{\chi \in Q} (u-\chi(u))^n)$ of $V$ with $k \in \mathbb{N}$ vanish every-
where on $\sum_{\chi \in Q} (V^\infty)^\chi$ and then on $\sum_{\chi \in Q} V^\chi$, and, on the other hand, converge point-
wise on $(\sum_{\chi \in Q} V^\chi)^\infty$ when $k \to +\infty$ to the operator associated with $\prod_{\chi \in Q} (u-\chi(u))^n$.

For each $\chi_0 \in P$, we therefore have $(V^{\chi_0} \cap \overline{\sum_{\chi \in P \setminus \{\chi_0\}} V^\chi})^\infty = \{0\}$, which
shows that the sum of the vector spaces $V^\chi$ with $\chi \in P$ is a direct sum. $\square$

**Definition 12.2.** By “admissible $G_0$-module of finite length” (or “Harish-Chandra module of $G_0$”), we mean any complex Hilbert space $V$ equipped with a continuous linear representation $\pi$ of $G_0$ that satisfies the equivalent properties of the previous lemma.

The aim of the following proposition is to adapt Zuckerman’s translation
functor (see [Zuc 77]) to the case of a representation of a disconnected group.

**Proposition 12.3.** *Fix a real Lie group *G* whose neutral component is equal to *G*<sub>0</sub> and an element *a* of *G* which operates on *G*<sub>0</sub> by the inner automorphism of some element *a* of *G*. For simplicity, we assume that the group *G* is generated by *a*G<sub>0</sub>.*

Let $\pi$ be a continuous linear representation of $\mathcal{G}$ in a complex Hilbert
space $V$ whose restriction to $G_0$ is an admissible $G_0$-module of finite length.

(a) Define a generalized function $(\operatorname{tr} \pi)_{aG_0}$ on $aG_0$, invariant under int $G_0$, locally integrable on $aG_0$ and analytic on $aG_0 \cap G_{ss \text{ reg}}$, by setting

$(\operatorname{tr} \pi)_{aG_0}(\varphi \, d_G) = \operatorname{tr} \pi(\varphi \, d_G)$ for all $\varphi \in C_c^\infty(aG_0)$,
where $\varphi \in C_c^\infty(\underline{a}G_0)$ is defined by $\varphi(\underline{a}x_0) = \varphi(ax_0)$ for $x_0 \in G_0$ and, $d_G$ and $d_G$ are Haar measures on $G$ and $\underline{G}$ whose restrictions on $G_0$ coincide.

We have: $$(\operatorname{tr} \pi)_{aG_0} = \sum_{\chi \in P \text{ such that } \underline{a}\chi=\chi} (\operatorname{tr} \pi_{\tilde{\chi}})_{aG_0},$$

where $P$ is the set of weights of $Z(U\mathfrak{g}_c)$ in $V^\infty$ and $\pi_\chi$ denotes for each $\chi \in P$ such that $\underline{a}\chi = \chi$ the representation of $\underline{G}$ in $V^\chi$ resulting from $\pi$.

(b) It is assumed there exists a character $\chi$ of $Z(U\mathfrak{g}_c)$ such that $V = V^\chi$.

Let $x \in aG_0 \cap G_{ss \text{ reg}}$. Therefore $j_x := \mathfrak{g}(x)$ is abelian (see 10.1 (b)) and $j := C_{\mathfrak{g}}(j_x)$ is a Cartan subalgebra of $\mathfrak{g}$ (see [Bou 87, Lem. 1.4.1 p. 6]). Let $j(x)_1$ denote the connected component of 0 in $\{Y \in j(x) \mid x \exp Y \in G_{ss \text{ reg}}\}$.

There exist a family $(p_l)_l$ of elements of $S(j(x)^*_c)$, indexed by those $l \in j(x)^*_c$
that satisfy $\chi=\chi_l^{U\mathfrak{g}_c}$ (notation of 8.4 (a)), such that for every $Y \in j(x)_1$ we have

$$
|D_G(x \exp Y)|^{1/2} (\mathrm{tr} \, \pi)_{aG_0}(x \exp Y) = \sum_{l \in j(x)^*_c \text{ such that } \chi = \chi_l^{U\mathfrak{g}_c}} p_l(Y) e^{l(Y)}.
$$

Moreover, when $\chi$ is “regular” (i.e. associated with a regular semisimple
orbit of int $\mathfrak{g}_c$ in $\mathfrak{g}_c^*$) every $p_l$ is a scalar.

(c) Assume that $\pi$ has a regular infinitesimal character $\chi$. Choose $\Lambda \in \mathfrak{g}_c^*$
semisimple regular such that $\chi = \chi_\Lambda^{U\mathfrak{g}_c}$. Denote by $C$ the unique chamber of
$R(\mathfrak{g}_c, \mathfrak{g}_c(\Lambda))$ in $\mathfrak{g}_c(\Lambda)^* \cap \mathrm{D}(\mathfrak{g}_c)^*$ such that $\Lambda \in C + Z(\mathfrak{g}_c)^*$, and $\bar{C}$ its closure.

Let $\pi_F$ be a linear representation of $\underline{G}$ in a finite-dimensional complex Hilbert space $F$, whose restriction to $G_0$ is irreducible, and such that the lowest weight of $\mathfrak{g}_F(\Lambda)$ in $F$ relative to $C$, denoted $\Lambda_F$, satisfies $\Lambda + \Lambda_F \in \bar{C} + Z(\mathfrak{g}_F)^*$.
The restriction to $G_0$ of the representation $\pi \otimes \pi_F$ of $\mathcal{G}$ yields an admissible $G_0$-module of finite length and $\mathcal{G}$ operates in its $Z(\mathfrak{U}\mathcal{G})$-primary component $(V \otimes_{\mathcal{G}} F)^{\chi_{\Lambda+\Lambda_F}^{\mathcal{U}\mathcal{G}}}$ by means of a representation $\pi_{Zuc}$ still having this property.

Let $x \in aG_0 \cap G_{ss \text{ reg}}$. Associate to it, as in (b), a Cartan subalgebra $\mathfrak{j}$ of $\mathfrak{g}$ and complex numbers $p_l$ (see end of (b)). For all $Y \in \mathfrak{j}(x)_1$, we have

$$|D_G(x \exp Y)|^{1/2} (\text{tr } \pi_{Zuc})_{aG_0}(x \exp Y) = \sum_{l \in \mathfrak{j}(x)_* \mathcal{C} \text{ such that } \chi = \chi_l^{\mathcal{U}\mathcal{G}}} \underline{x}_{l_F} p_l e^{(l+l_F)(Y)},$$

where $l_F \in \mathfrak{j}_* \mathcal{C}$ is defined from $l$ as $\Lambda_F$ from $\Lambda$ and $\underline{x}_{l_F}$ is the scalar by which the element $\underline{x} := \underline{a}(a^{-1}x)$ of $\mathcal{G}$ (which fixes $l_F$) acts on the one-dimensional eigenspace of weight $l_F$ for the action of $\mathfrak{j}(x)_\mathcal{C}$ on $F$.

**Proof of the proposition.** We will use the canonical action by left-invariant differential operators of the enveloping algebra of a real Lie group on the vector space of generalized functions on this Lie group.

(a) The representation $\pi$ is trace class, by Lemma 12.1. The generalized function $(\text{tr } \pi)_{aG_0}$, equal to $\delta_a * (\delta_{\underline{a}^{-1}} * \text{tr } \pi|_{aG_0})$, is invariant under $\text{int } G_0$.

Let $u_0 \in Z(\mathfrak{U}\mathcal{G})$ and $\varphi \in C_c^\infty(\underline{a}G_0)$. Set $u = \prod_{\chi \in P} (u_0 - \chi(u_0))$. The proof of [KV 95, Cor. 7.207 p. 530] provides the existence of a finite sequence $W_0 = V \supset W_1 \supset \cdots \supset W_N = \{0\}$ of closed $\mathcal{G}$-submodules of $V$ such that the quotients $W_{k-1}/W_k$ with $1 \le k \le N'$ are topologically irreducible under $\mathcal{G}$.

Let $k \in \{1, \dots, N'\}$. Denote by $\pi_k$ the representation of $\mathcal{G}$ in $W_{k-1}/W_k$. Any subquotient of an admissible $G_0$-module of finite length is an admissible $G_0$-module of finite length. Therefore $W_{k-1}/W_k$ has a topologically irreducible closed $G_0$-submodule $E$ which has infinitesimal character. The action of $u$ on $\sum_{n \in \mathbb{Z}} \underline{a}^n E^\infty$ is zero. Therefore $\pi_k(\varphi d_{\mathcal{G}} * u) = 0$.

Thus: $\partial_u(\text{tr } \pi)(\varphi d_{\mathcal{G}}) = \text{tr } \pi_1(\varphi d_{\mathcal{G}} * u) + \cdots + \text{tr } \pi_{N'}(\varphi d_{\mathcal{G}} * u) = 0$ (see [Wal 88, Lem. 8.1.3 p. 293]). Furthermore, we have $\underline{a}u = u$. It follows that:

$$\partial_u((\text{tr } \pi)_{aG_0}) = \delta_a * ((\delta_{\underline{a}^{-1}} * \text{tr } \pi|_{aG_0}) * (\delta_{\underline{a}} * \tilde{u} * \delta_{\underline{a}^{-1}})) = 0.$$

By [Bou 87, Th. 2.1.1 p. 10] and [KV 95, Th. 4.95 p. 286 and Th. 7.30 (a) p. 450], $(\text{tr } \pi)_{aG_0}$ is locally integrable on $aG_0$ and analytic on $aG_0 \cap G_{ss \text{ reg}}$.

Let $\chi \in P$ such that $\underline{a}\chi \neq \chi$ and (again) $\varphi \in C_c^\infty(\underline{a}G_0)$. Denote by $m$ the cardinality of the orbit $\tilde{\chi}$ of $\chi$ under the action of the subgroup $\langle \underline{a} \rangle$ of $\mathcal{G}$ generated by $\underline{a}$, and denote by $\pi_{\tilde{\chi}}$ the representation of $\mathcal{G}$ in $W := \sum_{n \in \mathbb{Z}} \underline{a}^n V^\chi$.

The restriction of $\pi_{\tilde{\chi}}(\varphi d_{\mathcal{G}})$ to the direct sum $V^\chi \oplus \underline{a}V^\chi \oplus \cdots \oplus \underline{a}^{m-1}V^\chi$ decomposes into blocks with zero diagonal blocks. I exploit this property using a suggestion by G. Skandalis to return to the finite-dimensional case.

The subalgebra of $\mathcal{L}(W)$ formed by endomorphisms that simultaneously stabilize $V^\chi$, $\underline{a}V^\chi, \dots, \underline{a}^{m-1}V^\chi$ is closed. According to [B 67, Rem. p.55 and Prop. 7 p.47], the primary subspaces $W^z$ of the restriction of $\pi_{\tilde{\chi}}(\varphi d_{\mathcal{G}})^m$ to $W$ associated with $z \in \mathbb{C} \setminus \{0\}$, which are finite-dimensional, are all sums of their intersections with $V^\chi$ and \dots and $\underline{a}^{m-1}V^\chi$. Furthermore, each $W^z$ with $z \in \mathbb{C} \setminus \{0\}$ is equal to the sum of the primary subspaces of the restriction of $\pi_{\tilde{\chi}}(\varphi d_{\mathcal{G}})$ to $W$ associated with the $m^{\text{th}}$ roots of $z$. Lidskij's formula ("the trace is the sum of the eigenvalues") yields the equality $\text{tr } \pi_{\tilde{\chi}}(\varphi d_{\mathcal{G}}) = \sum_{z \in \mathbb{C} \setminus \{0\}} \text{tr } (\pi_{\tilde{\chi}}(\varphi d_{\mathcal{G}})|_{W^z}) = 0$.

Hence, now using [B 67] relative to $\mathcal{L}(V)$ (see proof of 12.1):
$$ (\mathrm{tr} \, \pi)_{aG_0} = \sum_{\dot{\chi} \in \langle a \rangle \setminus P} (\mathrm{tr} \, \pi_{\dot{\chi}})_{aG_0} = \sum_{\chi \in P \text{ such that } \underline{a}\chi = \chi} (\mathrm{tr} \, \pi_{\dot{\chi}})_{aG_0}. $$

(b) I will now explain the calculations of [Bou 87, p. 33 and p. 34].

Set $\theta(Y) = |D_G(x \exp Y)|^{1/2} (\mathrm{tr} \, \pi)_{aG_0}(x \exp Y)$ for all $Y \in j(x)_1$.

Let $W$ denote the group $W(\mathfrak{g}_C, j_C)$ and $\mathrm{pr}$ the canonical projection from $S j_C$ onto $S j(x)_C$ obtained as in Conventions 2 (b). The proof of (a) shows that $(\mathrm{tr} \, \pi)_{aG_0}$ is an eigenvector of $Z(U\mathfrak{g}_C)$ associated with $\chi$. Fix an element $l_0$ of $j_C^*$ such that $\chi = \chi_{l_0}^{U\mathfrak{g}_C}$. According to [Bou 87, line 4 of 2.5 p. 20], we have:

$$ q(l_0) \theta = \partial_{\mathrm{pr}(q)} \theta \quad \text{for all } q \in (S j_C)^W. $$

In what follows, we repeat the arguments of [Kna 86, p. 369 to p. 371].

Let $X \in j_C$. Denote by $q_0, \dots, q_{|W|-1}$ the elements of $(S j_C)^W$ for which the following equality of polynomials in the indeterminate $T$ holds:

$$ \prod_{w \in W} (T - wX) = T^{|W|} + q_{|W|-1} T^{|W|-1} + \dots + q_0 \quad \text{in } (S j_C)[T]. $$

From this, first replace $T$ by $X$, then take the value of each member at $l_0$ and replace $T$ by $X$. This yields, in $S j_C$:

$$ X^{|W|} + X^{|W|-1}q_{|W|-1} + \dots + q_0 = 0 $$

$$ \text{and } X^{|W|} + X^{|W|-1}q_{|W|-1}(l_0) + \dots + q_0(l_0) = \prod_{w \in W} (X - l_0(wX)). $$

Use the partial differential equation satisfied above by $\theta$ with $q$ successively equal to $q_{|W|-1}, \dots, q_0$. This yields the following equality, which remains to be exploited:

$$ \prod_{w \in W} (\partial_{\mathrm{pr}(X)} - l_0(wX)) \cdot \theta = 0. $$

Let $X$ range over a base of $j(x)_C$. By [Var 77, Prop. 3 p. 58], there exists a unique family $(p_l)_{l \in j(x)_C^*}$ of elements of $S(j(x)_C^*)$ with finite support such that

$$ \theta(Y) = \sum_{l \in j(x)_C^*} p_l(Y) e^{l(Y)} \quad \text{for all } Y \in j(x)_1. $$

More precise information will be obtained by substituting this expression of $\theta$ into the preceding equality (and extending analytically to $j(x)$).

Let $l \in j(x)_C^*$ such that $p_l \neq 0$. Denote by $d$ the degree of $p_l$ in $S(j(x)_C^*)$ and by $p_l^{[d]}$ the homogeneous component of $p_l$ of degree $d$. Let $X \in j_C$. We have $(\partial_{\mathrm{pr}(X)} - l_0(wX)) \cdot p_l e^l = (l(X) - l_0(wX)) p_l e^l + (\partial_{\mathrm{pr}(X)} p_l) e^l$ for all $w \in W$. The null polynomial $e^{-l} \prod_{w \in W} (\partial_{\mathrm{pr}(X)} - l_0(wX)) \cdot p_l e^l$ admits $\prod_{w \in W} (l - w^{-1}l_0)(X) \cdot p_l^{[d]}$ as its homogeneous component of degree $d$. Since the ring $S j_C^*$ is an integral domain, this shows that $l \in W \cdot l_0$, hence $\chi = \chi_l^{U\mathfrak{g}_C}$.

Now assume $p_l$ to be nonscalar. Choose $X \in j_C$ such that, for all $w \in W$ outside $W(l)$ we have $(w l)(X) \neq l(X)$, and $p_l^{[d]}(\mathrm{pr}(X)) \neq 0$. Let $(\mathrm{pr}(X)^*)$ denote the basis of $(\mathbb{C} \mathrm{pr}(X))^*$ dual of $(\mathrm{pr}(X))$. The restriction of the zero polynomial

$$ e^{-l} \prod_{w \in W(l)} (\partial_{\mathrm{pr}(X)} - l(wX)) \cdot \left( \prod_{w \in W \text{ and } w \notin W(l)} (\partial_{\mathrm{pr}(X)} - l(wX)) \cdot p_l e^l \right) $$

to $\mathbb{C} \mathrm{pr}(X)$ is the sum of a polynomial of the form $\zeta \times (\partial_{\mathrm{pr}(X)})^{|W(l)|} \cdot (\mathrm{pr}(X)^*)^d$ with $\zeta \in \mathbb{C} \setminus \{0\}$ and monomials of (strictly) smaller degrees. Hence $|W(l)| > 1$.

(c) I follow [Zuc 77, p. 301].

By [KV 95, Cor. 7.207 p. 530], $\pi \otimes \pi_F$ and the representations of $G_0$ on the $Z(U\mathfrak{g}_C)$-primary components of $V \otimes_C F$ provide admissible $G_0$-modules of finite length.
By hypothesis, we have $\underline{a}\chi = \chi$. Hence, there exists $\sigma \in (\mathrm{Ad} \underline{a})^{\mathbb{C}}$ int $\mathfrak{g}_{\mathbb{C}}$ such that $\sigma(\Lambda) = \Lambda$. We have: $\sigma(\mathfrak{g}_{\mathbb{C}}(\Lambda)) = \mathfrak{g}_{\mathbb{C}}(\Lambda)$ and $\sigma(C) = C$, then $\sigma(\Lambda_F) = \Lambda_F$. Therefore $\underline{a}\chi_{\Lambda+\Lambda_F}^{\mathfrak{g}_{\mathbb{C}}} = \sigma\chi_{\Lambda+\Lambda_F}^{\mathfrak{g}_{\mathbb{C}}} = \chi_{\Lambda+\Lambda_F}^{\mathfrak{g}_{\mathbb{C}}}$ and the group $\underline{G}$ stabilizes $(V \otimes_{\mathbb{C}} F)^{\chi_{\Lambda+\Lambda_F}^{\mathfrak{g}_{\mathbb{C}}}}$.

Denote by $P'$ the set of weights of $Z(\mathfrak{U}\mathfrak{g}_{\mathbb{C}})$ in $(V \otimes_{\mathbb{C}} F)^{\infty}$, by $P(F, j(x)_{\mathbb{C}})$ the set of weights of $j(x)_{\mathbb{C}}$ in $F$ and by $F^{\gamma}$ the eigenspace of $j(x)_{\mathbb{C}}$ associated with an element $\gamma$ of $P(F, j(x)_{\mathbb{C}})$. For all $Y \in j(x)_1$, we obtain thanks to (a):

$$ (\mathrm{tr} \, \pi)_{aG_0}(x \exp Y) \times \mathrm{tr} \, \pi_F(\underline{x} \exp Y) = \sum_{\chi' \in P' \text{ such that } \underline{a}\chi' = \chi'} (\mathrm{tr}(\pi \otimes \pi_F)_{\chi'})_{aG_0}(x \exp Y) $$ 
$$ \text{and } \mathrm{tr} \, \pi_F(\underline{x} \exp Y) = \sum_{\gamma \in P(F, j(x)_{\mathbb{C}})} \mathrm{tr}(\pi_F(\underline{x})|_{F^{\gamma}}) e^{\gamma(Y)}. $$

Apply (b) to $\pi$, and to $(\pi \otimes \pi_F)_{\chi'}$ with $\chi' \in P'$ such that $\underline{a}\chi' = \chi'$, among which is $\pi_{Zuc}$. By identifying the polynomial factors written in front of the functions $e^l$ with $l \in j(x)_{\mathbb{C}}^*$, we find that for all $Y \in j(x)_1$ we have

$$ |D_G(x \exp Y)|^{1/2} (\mathrm{tr} \, \pi_{Zuc})_{aG_0}(x \exp Y) = \sum_{(l, \gamma) \in \mathcal{A}} \mathrm{tr}(\pi_F(\underline{x})|_{F^{\gamma}}) p_l e^{(l+\gamma)(Y)}, $$ 
where we put $\mathcal{A} = \{(l, \gamma) \in j(x)_{\mathbb{C}}^* \times P(F, j(x)_{\mathbb{C}}) \mid \chi = \chi_l^{\mathfrak{g}_{\mathbb{C}}} \text{ and } \chi_{\Lambda+\Lambda_F}^{\mathfrak{g}_{\mathbb{C}}} = \chi_{l+\gamma}^{\mathfrak{g}_{\mathbb{C}}}\}.$

Finally, we verify that $\mathcal{A} = \{(l, \gamma) \in j(x)_{\mathbb{C}}^* \times j_{\mathbb{C}}^* \mid \chi = \chi_l^{\mathfrak{g}_{\mathbb{C}}} \text{ and } \gamma = l_F\}$, and $\dim F^{\gamma} = 1$ for $(l, \gamma) \in \mathcal{A}$. The inclusion $\supseteq$ is clear. Consider $(l, \gamma) \in \mathcal{A}$. We modify the choice of $\Lambda$ by taking $\Lambda = l$. We will show that $\gamma = \Lambda_F$, and calculate $\dim F^{\gamma}$. Fix $w \in W(\mathfrak{g}_{\mathbb{C}}, j_{\mathbb{C}})$ such that $l + \gamma = w(\Lambda + \Lambda_F)$. We have $\gamma - \Lambda_F = w(\Lambda + \Lambda_F) - (\Lambda + \Lambda_F)$ and $w(\Lambda + \Lambda_F) - (\Lambda + \Lambda_F) \le 0$ for the order associated with $C$, because $\Lambda + \Lambda_F \in \bar{C} + Z(\mathfrak{g}_{\mathbb{C}})^*$. On the other hand, there is a weight $\zeta$ of $j_{\mathbb{C}}$ in $F$ such that $\gamma = \zeta|_{j(x)_{\mathbb{C}}}$. The inclusion $j(x)_{\mathbb{C}}^* \subseteq j_{\mathbb{C}}^*$ identifies $\gamma$ with the isobarycenter of the finite subset $\{\mathrm{Ad}^* x^n \cdot \zeta\}_{n \in \mathbb{Z}}$ of the set of weights of $j_{\mathbb{C}}$ in $F$. Therefore $\gamma - \Lambda_F \ge 0$ for the order associated with $C$. Consequently, $\gamma = \Lambda_F$. Since $\gamma$ is an extremal weight of $j_{\mathbb{C}}$ in $F$ which is isobarycenter of a finite set of weights of $j_{\mathbb{C}}$ in $F$, it is equal in particular to $\zeta$. Thus the eigenspace of $F$ of weight $\gamma$ under the action of $j(x)_{\mathbb{C}}$ is of dimension 1. $\square$

### 13. Transition from $T_{\lambda_r, \sigma_r}^{M'}$ to $T_{\lambda, \tau_M}^{M'}$

To simplify the notation, in this section set: $M' = \mathbb{G}(\nu_+)$, $\rho = \rho_{m',\mathfrak{h}}$, $\sigma = \tau_{M'}$, and $\lambda_r = \lambda_{+,m'}$ (see page 163). This yields $\lambda_r = (\mu - 2i\rho) + \nu \in \mathfrak{m}_{ss \mathrm{reg}}^*$. We'll see that, roughly speaking, the character $\mathrm{tr} T_{\tilde{\lambda}, \sigma}^{M'}$ can be deduced from a character of the form $\mathrm{tr} T_{\lambda_r, \sigma_r}^{M'}$ by translation in the Zuckerman sense.

The following proposition is obtained by proving a variant of Bouaziz's results in [Bou 84, (i) and (ii) p. 550] (see also [KV 95, p. 547 and p. 548]). Its origin is Lemma 3.1 on p. 406 of [KZ 82], which yields equality of restrictions to $M'_0$ of the characters considered in (b) below.

**Proposition 13.1.** Fix $\hat{a} \in M'(\tilde{\lambda})^{m'/\mathfrak{h}}$ lying over an elliptical element $a$ of $M'(\tilde{\lambda})$. Denote by $\mathcal{M}'$ the semidirect product of $\mathbb{Z}$ by $M'_0$ such that $\underline{a} := 1 \in \mathbb{Z}$ acts on $M'_0$ by int $a$.

(a) There exists a continuous linear representation $\pi_F$ of $\mathcal{M}'$ in a finite-dimensional complex Hilbert space $F$, unique to within isomorphism, whose re-
striction to $M'_0$ is irreducible with lowest weight $-2\rho$ for the action of $\mathfrak{h}_C$ and
relative to the order deduced from $R^+(\mathfrak{m}'_C, \mathfrak{h}_C)$, such that $\underline{a}$ acts trivially on the
eigenspace of weight $-2\rho$.

(b) Extend the unitary representations $T_{\bar{\lambda}}^{M'_0}$ and $T_{\lambda_r}^{M'_0}$ of $M'_0$ whose “spaces” are denoted $\mathcal{H}$ and $V$, into continuous linear representations $\pi_{\bar{\lambda}}$ and $\pi_{\lambda_r}$ of $\mathcal{M}'$ by the conditions $\pi_{\bar{\lambda}}(\underline{a}) = S_{\bar{\lambda}}(\hat{a})$ and $\pi_{\lambda_r}(\underline{a}) = S_{\lambda_r}(\hat{a})$, where $S_{\bar{\lambda}}$ and $S_{\lambda_r}$ are the representations of $\mathcal{M}'(\tilde{\lambda})^{m'/\hbar}$ (which is equal to $\mathcal{M}'(\lambda_r)^{m'/\hbar}$) attached to $\tilde{\lambda}$ and $\lambda_r$ in Proposition 8.1 (b).

The representations of $\mathcal{M}'$ in $\mathcal{H}$ and $(V \otimes_{\mathbb{C}} F)^{\chi_{\mathfrak{M}'\mathbb{C}}^{\mathfrak{M}'\mathbb{C}}}$ are trace class and have the same character.

**Proof of the proposition.** (a) There exists — and we fix — an irreducible linear representation of $\mathfrak{m}'_C$ in a finite-dimensional complex Hilbert space $F$, of lowest weight $-2\rho$ for the action of $\mathfrak{h}_C$ relative to the order deduced from $R^+(\mathfrak{m}'_C, \mathfrak{h}_C)$. It integrates into a representation of “the” universal covering of $M'_0$ with a trivial central character, and then into a representation of $M'_0$ denoted $\pi_{F,0}$. Since $\underline{a}$ normalizes $\mathfrak{h}$ and fixes $-2\rho$, there exists a unique intertwining operator $\Phi$ from $(F, \text{int } \underline{a} \cdot \pi_{F,0})$ to $(F, \pi_{F,0})$ that acts trivially on the eigensubspace of $F$ of weight $-2\rho$ under the action of $\mathfrak{h}_C$, common to both representations and of dimension 1. Construct $\pi_F$ by extending $\pi_{F,0}$ and choosing $\pi_F(\underline{a})$ equal to $\Phi$.

(b) First, note that $\mathcal{M}'$ stabilizes $\mathcal{H}_0 := (V \otimes_{\mathbb{C}} F)^{\chi_{\mathfrak{M}'\mathbb{C}}^{\mathfrak{M}'\mathbb{C}}}$. By Lemma 12.1 and Proposition 12.3 (c), the representations of $\mathcal{M}'$ in $\mathcal{H}$ and $\mathcal{H}_0$ are trace class.

Fix a maximal compact subgroup $K_{M'_0}$ of $M'_0$ stable under int $a$ and whose Cartan involution normalizes $\mathfrak{h}$. For example, take $\exp(c_{M'} \cap \mathfrak{m}')$, where $c_{M'}$ is the Lie algebra of a maximal compact subgroup of the semidirect product of $\{1, c\}$ ($c$: conjugation of $\mathcal{M}'(\mathbb{C})$) by $\mathcal{M}'(\mathbb{C})$ that contains both $c$ and the subgroup generated by the product of the projection in $\mathcal{M}'(\mathbb{R})$ of $a$ with $\exp_{\mathcal{M}'(\mathbb{C})}(i\mathfrak{h}(\mathbb{R}))$. Denote by $\mathcal{H}^f$, $\mathcal{H}_0^f$ and $V^f$ the $(\mathfrak{m}'_C, K_{M'_0})$-modules stable under $\underline{a}$ associated with $\mathcal{H}$, $\mathcal{H}_0$ and $V$. By [Kna 86, Prop. 10.5 p. 336] with its proof (see [Kna 86, Cor. 8.8 p. 211]), the representations of $\mathcal{M}'$ in $\mathcal{H}$ and $\mathcal{H}_0$ have the same character if there exists an isomorphism of $(\mathfrak{m}'_C, K_{M'_0})$-modules from $\mathcal{H}^f$ onto $\mathcal{H}_0^f$ compatible with the action of $\underline{a}$.

From the proof of Proposition 8.1 (a), we know that $\mathcal{H}^f$ and $V^f$ are irreducible, isomorphic to $\mathcal{R}_{M'_0}^{q'}(\mathbb{C}_{i\lambda-\rho})$ and $\mathcal{R}_{M'_0}^{q'}(\mathbb{C}_{i\lambda_r-\rho})$ where $\mathcal{R}_{M'_0}^{q'}$ is the cohomological induction functor relative to $\mathfrak{b}_{M'}$, and $\mathbb{C}_{\Lambda}$ is for each $\Lambda \in i\mathfrak{h}^*$ the $(\mathfrak{h}_C, T_0)$-module $\mathbb{C}$ (see . 2 (a)) on which $\mathfrak{h}$ acts by $\Lambda$ id. From [KV 95, Th. 7.237 p. 544] and using the notation $\psi_l^{\prime}$ of [KV 95, (7.141) p. 493], we have

$$
\mathcal{R}_{M'_0}^{q'}(\psi_{i_{\lambda_r-\rho}}^{i_{\lambda-\rho}}(\mathbb{C}_{i_{\lambda_r-\rho}})) \simeq \psi_{i_{\lambda_r}}^{i_{\lambda}}(\mathcal{R}_{M'_0}^{q'}(\mathbb{C}_{i_{\lambda_r-\rho}})).
$$

Thus, the $(\mathfrak{m}'_C, K_{M'_0})$-modules $\mathcal{H}^f$ and $\mathcal{H}_0^f$ are isomorphic.

Given the characterization of $S_{\bar{\lambda}}(\hat{a})$ in Proposition 8.1 (b), Schur's Lemma in [Wal 88, Lem. 3.3.2 p.80] and the identification $H_{q'}(\mathfrak{n}_{M'}, \mathcal{H}^{\infty})^{*} = H_{q'}(\mathfrak{n}_{M'}, \mathcal{H}^{f})^{*}$ in [Duf 82a, Lem. 4 p. 165], to obtain the compatibility with the action of $\underline{a}$ of a fixed $(\mathfrak{m}'_C, K_{M'_0})$-module isomorphism from $\mathcal{H}^f$ onto $\mathcal{H}_0^f$, it suffices to show that $\underline{a}$ acts from $\pi_{\lambda_r}(\underline{a}) \otimes \pi_F(\underline{a})$ in $(H_{q'}(\mathfrak{n}_{M'}, \mathcal{H}_0^f)^*)_{-(i\lambda+\rho)}$ by $\rho_{\bar{\lambda}}^{m'/\hbar}(\hat{a})^{-1}$ id, that is to say by $\rho_{\lambda_r}^{m'/\hbar}(\hat{a})^{-1}$ id.
Bouaziz's idea basically consists in verifying the compatibility with the action of $\underline{a}$ of the isomorphism that would be obtained at the level of homology by taking $\mathbf{q}$ equal to the conjugate of $\mathfrak{b}_{M'}$ and $X$ equal to $V^f$ in [KV 95, Th. 7.242 p. 546]. Unfortunately, the assumption “$i\lambda_r$ is at least as singular as $i\lambda$” does not hold. Instead of going from $V^f$ to $\mathcal{H}_0^f$, we will go from $\mathcal{H}_0^f$ to $V^f$ using a property of the composite of two Zuckerman translation functors.

By [KV 95, Th. 7.220 p. 536], the homomorphism of $(\mathfrak{m}'_\mathcal{C}, K_{M'_0})$-modules compatible with the action of $\underline{a}$ from $(\mathcal{H}_0^f \otimes_{\mathbb{C}} F^*)^{\chi_{i_{\lambda_r}}^{U_{\mathfrak{h}_\mathcal{C}}}}$ to $V^f$ obtained by restriction from the canonical map $V^f \otimes_{\mathbb{C}} F \otimes_{\mathbb{C}} F^* \to V^f$ is nonzero and therefore surjective. The $Z(U\mathfrak{h}_\mathcal{C})$-primary components associated with $\chi_{i_{\lambda_r}}^{U_{\mathfrak{h}_\mathcal{C}}}$ in the exact sequence of homology $\mathfrak{h}_\mathcal{C}$-modules resulting from this surjection yield a homomorphism of $\mathfrak{h}_\mathcal{C}$-modules from $H_{q'}(\mathfrak{n}_{M'}, (\mathcal{H}_0^f \otimes_{\mathbb{C}} F^*)^{\chi_{i_{\lambda_r}}^{U_{\mathfrak{h}_\mathcal{C}}}})^{\chi_{i_{\lambda_r}}^{U_{\mathfrak{h}_\mathcal{C}}}}$ to $H_{q'}(\mathfrak{n}_{M'}, V^f)^{\chi_{i_{\lambda_r}}^{U_{\mathfrak{h}_\mathcal{C}}}}$ compatible with the action of $\underline{a}$, which is surjective according to [KV 95, Lem. 8.9 p. 553].

Let $v_0^*$ be a nonzero vector of $F^*$ of weight $2\rho$ for the action of $\mathfrak{h}_\mathcal{C}$. We have $\mathfrak{n}_{M'} \cdot v_0^* = \{0\}$ and $\underline{a} \cdot v_0^* = v_0^*$ according to (a). It remains to prove that each of the two homomorphisms of $\mathfrak{h}_\mathcal{C}$-modules compatible with the action of $\underline{a}$ deduced from the canonical injections from $(\mathcal{H}_0^f \otimes_{\mathbb{C}} F^*)^{\chi_{i_{\lambda_r}}^{U_{\mathfrak{h}_\mathcal{C}}}}$ and $\mathcal{H}_0^f \otimes_{\mathbb{C}} \mathcal{C}v_0^*$ into $\mathcal{H}_0^f \otimes_{\mathbb{C}} F^*$, defined respectively on $H_{q'}(\mathfrak{n}_{M'}, (\mathcal{H}_0^f \otimes_{\mathbb{C}} F^*)^{\chi_{i_{\lambda_r}}^{U_{\mathfrak{h}_\mathcal{C}}}})^{\chi_{i_{\lambda_r}}^{U_{\mathfrak{h}_\mathcal{C}}}}$ and $H_{q'}(\mathfrak{n}_{M'}, \mathcal{H}_0^f)^{\chi_{i_{\lambda_r}}^{U_{\mathfrak{h}_\mathcal{C}}}} \otimes_{\mathbb{C}} \mathcal{C}v_0^*$, and both with values in $H_{q'}(\mathfrak{n}_{M'}, \mathcal{H}_0^f \otimes_{\mathbb{C}} F^*)^{\chi_{i_{\lambda_r}}^{U_{\mathfrak{h}_\mathcal{C}}}}$, is bijective.

Indeed, in this case and with the notation $\mathbb{C}_\Lambda$ introduced earlier in this proof, the line formed by homomorphisms of $\mathfrak{h}_\mathcal{C}$-modules from $H_{q'}(\mathfrak{n}_{M'}, V^f)$ to $\mathbb{C}_{i_{\lambda_r+\rho}}$ will be injected in a way compatible with the action of $\underline{a}$ into that formed by $\mathfrak{h}_\mathcal{C}$-module homomorphisms from $H_{q'}(\mathfrak{n}_{M'}, \mathcal{H}_0^f)$ to $\mathbb{C}_{i_{\lambda_r+\rho}}$.

The first of the two preceding homomorphisms is bijective by [KV 95, (7.243) p. 547, see Prop. 7.166 p. 506]. For the second, set $W = F^* / \mathcal{C}v_0^*$.

The exact sequence of $\mathfrak{b}_{M'}$-modules $0 \to \mathcal{H}_0^f \otimes_{\mathbb{C}} \mathcal{C}v_0^* \to \mathcal{H}_0^f \otimes_{\mathbb{C}} F^* \to \mathcal{H}_0^f \otimes_{\mathbb{C}} W \to 0$ provides the following segment of the exact sequence of $\mathfrak{h}_\mathcal{C}$-modules in homology:
$$ H_{q'+1}(\mathfrak{n}_{M'}, \mathcal{H}_0^f \otimes_{\mathbb{C}} W) \to H_{q'}(\mathfrak{n}_{M'}, \mathcal{H}_0^f) \otimes_{\mathbb{C}} \mathcal{C}v_0^* \xrightarrow{H_{q'}(\mathfrak{n}_{M'}, \mathcal{H}_0^f \otimes_{\mathbb{C}} F^*)} H_{q'}(\mathfrak{n}_{M'}, \mathcal{H}_0^f \otimes_{\mathbb{C}} W). $$
We complete the proof by showing that $H_{\bullet}(\mathfrak{n}_{M'}, \mathcal{H}_0^f \otimes_{\mathbb{C}} W)^{\chi_{i_{\lambda_r}}^{U_{\mathfrak{h}_\mathcal{C}}}} = \{0\}$.

By Lie's theorem, there exists — and we fix — a sequence $\{0\} = W^{(-1)} \subseteq W^{(0)} \subseteq \dots \subseteq W^{(N)} = W$ of $\mathfrak{b}_{M'}$-submodules of $W$ with $\dim W^{(j)}/W^{(j-1)} = 1$ for $0 \le j \le N$. By [KV 95, Prop. 3.12 p. 188], for any $(\mathfrak{b}_{M'}, T_0)$-module $E$ and any $n \in \mathbb{N}$, the $\mathfrak{h}_\mathcal{C}$-module $H_n(\mathfrak{n}_{M'}, E)$ is isomorphic to $P_n(E)$, where $P_n$ is the $n^{\text{th}}$ derived functor from the right exact functor of the category of $(\mathfrak{b}_{M'}, T_0)$-modules to that of $(\mathfrak{h}_\mathcal{C}, T_0)$-modules denoted $P_{\mathfrak{b}_{M'}, T_0}^{\mathfrak{h}_\mathcal{C}, T_0}$ in [KV 95, (2.8) p. 104]. By [KV 95, Prop. D.57 (b) p. 887, see (2.123) p. 162], there exists — and we fix — a filtration $\{0\} = C_{\bullet}^{(-1)} \subseteq C_{\bullet}^{(0)} \subseteq \dots \subseteq C_{\bullet}^{(N)} = C_{\bullet}$ of a chain complex $C_{\bullet}$ into $(\mathfrak{h}_\mathcal{C}, T_0)$-modules null in (strictly) negative degree, of which each $\mathfrak{h}_\mathcal{C}$-module of homology $H_n(C_{\bullet})$ is isomorphic to $H_n(\mathfrak{n}_{M'}, \mathcal{H}_0^f \otimes_{\mathbb{C}} W)$ for $n \in \mathbb{N}$, and whose spectral sequence $(E^r)_{r \ge 0}$ provides $\mathfrak{h}_\mathcal{C}$-modules $E_{p,q}^1$ isomorphic to $H_{p+q}(\mathfrak{n}_{M'}, \mathcal{H}_0^f) \otimes_{\mathbb{C}} (W^{(p)}/W^{(p-1)})$ for $0 \le p \le N$ and $p+q \ge 0$. By [KV 95, Prop. 7.56 p. 460], the $Z(U\mathfrak{h}_\mathcal{C})$-primary components of each of these $\mathfrak{h}_\mathcal{C}$-modules $E_{p,q}^1$ are associated with characters of the form $\chi_{w_{i_{\lambda_r+\rho+\gamma}}^{U_{\mathfrak{h}_\mathcal{C}}}}$ with $w \in W(\mathfrak{m}'_\mathcal{C}, \mathfrak{h}_\mathcal{C})$, where $\gamma$ denotes the weight of $\mathfrak{h}_\mathcal{C}$ in $W^{(p)}/W^{(p-1)}$. Thus, given [KV 95, Prop. 7.166 p. 506], we have $(E_{p,q}^1)^{\chi_{i_{\lambda_r}}^{U_{\mathfrak{h}_\mathcal{C}}}} = \{0\}$. Furthermore, for $n \in \mathbb{N}$ and $q \in \mathbb{Z}$,
the $\mathfrak{h}_C$-module $E_{n-q,q}^\infty$ is a subquotient of $E_{n-q,q}^1$ isomorphic to the quotient of $\mathrm{pr}_{H_n(C_\bullet)}(H_n(C_\bullet^{(n-q)}))$ by $\mathrm{pr}_{H_n(C_\bullet)}(H_n(C_\bullet^{(n-q-1)}))$, where we have used the same notation $\mathrm{pr}_{H_n(C_\bullet)}$ for canonical maps with values in $H_n(C_\bullet)$. Hence for each $n \in \mathbb{N}$, the following relations between $\mathfrak{h}_C$-modules hold:

$$ H_n(\mathfrak{n}_{M'}, \mathcal{H}_0^f \otimes_{\mathbb{C}} W)^{\chi_{i_{\lambda_r+\rho}}^{U_{\mathfrak{h}_C}} \simeq \mathrm{pr}_{H_n(C_\bullet)}(H_n(C_\bullet^{(N)}))^{\chi_{i_{\lambda_r+\rho}}^{U_{\mathfrak{h}_C}}}} = \cdots = \mathrm{pr}_{H_n(C_\bullet)}(H_n(C_\bullet^{(-1)}))^{\chi_{i_{\lambda_r+\rho}}^{U_{\mathfrak{h}_C}} = \{0\}}. \quad \square $$

**Remark 13.2.** We retain the notation from the preceding proof. In particular, $K_{M'_0}$ is stable under $\mathrm{int} \, a$. Fix an intertwining operator $\Phi$ from $(\mathcal{H}, \mathrm{int} \, a \cdot T_{\tilde{\lambda}}^{M'_0})$ to $(\mathcal{H}, T_{\tilde{\lambda}}^{M'_0})$ and an isomorphism $\Psi$ of $(\mathfrak{m}'_C, K_{M'_0})$-modules from $\mathcal{H}^f$ onto $\mathcal{H}_0^f$. Let $\Phi^f$ denote the restriction of $\Phi$ to $\mathcal{H}^f$ and let $\pi_0(a)^f$ denote the operator by which $a$ acts on $\mathcal{H}_0^f$. According to Schur's Lemma, there exists $z \in \mathbb{C} \setminus \{0\}$ such that $\Phi^f = z \Psi^{-1} \circ \pi_0(a)^f \circ \Psi$. The proof of Proposition 13.1 (b) shows that the representation $S$ of Proposition 8.1 (b) satisfies $S(\hat{a}) = z^{-1} \Phi$. Furthermore, $K_{M'_0}$ is included in a maximal compact subgroup of $M'$ that contains $a$, and the maximal compact subgroups of $M'$ are conjugate under $M'_0$ (see [Hoc 65, Th. 3.1 p. 180 and lines before Th. 3.7 p. 186]). It's easy to deduce that $z$ is independent of the choice of $K_{M'_0}$. We could therefore have defined $S$ by returning to the regular semisimple case treated by Duflo (for values on elliptic elements) and making arrangements to ensure that $S(\exp X) = e^{-i\lambda(X)} T_{\tilde{\lambda}}^{M'_0}(\exp X)$ when $X \in \mathfrak{h}$. It would remain to use this new definition to prove that $S$ is a (unitary) representation of $M'(\tilde{\lambda})^{m'/\mathfrak{h}}$. ■

**End of the proof of the theorem.** It remains to prove the theorem for the representation $T_{\tilde{\lambda},\sigma}^{M'}$. The reference situation is that of [Zuc 77, Lem. 5.4 p. 304], in which $M'$ is replaced by $M_0$ and $e$ is replaced by a regular semisimple element of $M_0$.

Assume $e \in M'$. Let $\mathcal{V}_e^{M'}$, $d_e^{M'}$ and $D_e^{M'}$ denote the objects analogous to $\mathcal{V}_e$, $d_e$ and $D_e$ attached to $M'$. Choose $\hat{a} \in M'(\tilde{\lambda})^{m'/\mathfrak{h}}$ lying over an elliptical $a \in M'(\tilde{\lambda})$ such that $e \in a M'_0$. Associate it with $\underline{M'}$, $\pi_{\tilde{\lambda}}$ and $\pi_{\lambda_r}$ as in Proposition 13.1, so $a \in \underline{M'}$. We will use the notations of Proposition 12.3 (a), with $M'$ and $\underline{M'}$ instead of $G$ and $\underline{G}$. We introduce the element $\sigma_r$ of $X_{M'}^{\mathrm{irr}}(\lambda_r)$ such that:

$$ \sigma_r(\hat{u}) = \det(\mathrm{Ad}(u^\mathbb{C})_{n_{M'}} \sigma(\hat{u})) \quad \text{for} \quad \hat{u} \in M'(\tilde{\lambda})^{m'/\mathfrak{h}}. $$

$$ \text{It satisfies:} \quad \mathrm{tr}_{T_{\tilde{\lambda},\sigma}^{M'}} \Big|_{e^{M'_0}} = \mathrm{tr} \, \sigma(\hat{a}) \times (\mathrm{tr} \, \pi_{\tilde{\lambda}})_{aM'_0} \\ \text{and} \quad \mathrm{tr}_{T_{\lambda_r,\sigma_r}^{M'}} \Big|_{e^{M'_0}} = \mathrm{tr} \, \sigma_r(\hat{a}) \times (\mathrm{tr} \, \pi_{\lambda_r})_{aM'_0}. $$

To complete the proof, note that given Proposition 13.1, Proposition 12.3 will then allow us to link $(\mathrm{tr} \, T_{\tilde{\lambda},\sigma}^{M'})_e$ to $(\mathrm{tr} \, T_{\lambda_r,\sigma_r}^{M'})_e$.

We verify the character formula for $\mathrm{tr} \, T_{\tilde{\lambda},\sigma}^{M'}$ (in the version proposed by formula $(\mathbf{F}_{+,0})$ page 176) on the open set formed by $X \in \mathcal{V}_e^{M'}$ such that $e \exp X \in M_{ss \, reg}'$, included in $\mathfrak{m}'(e)_{ss \, reg}$ and of negligible complement, on which the two members of the equality are analytic functions.

Let $X \in \mathcal{V}_e^{M'}$ such that $x := e \exp X \in M_{ss \, reg}'$. Set $j_x := \mathfrak{m}'(x)$. Then $j_x = \mathfrak{m}'(e)(X) \in \mathrm{Car} \, \mathfrak{m}'(e)$ (given the dimensions) and $j := \mathrm{C}_{\mathfrak{m}'}(j_x) \in \mathrm{Car} \, \mathfrak{m}'$. Let $\Gamma$ denote the connected component of $X$ in $j(x) \cap \mathfrak{m}'(e)_{ss \, reg}$. To continue the calculation, consider $Y \in j(x)_1$ (see 12.3 (b) for $G=M'$) such that $X+Y \in \Gamma \cap \mathcal{V}_e^{M'}$.
According to [Bou 87, Th. 5.5.3 p. 52] and by 4.3 (b), the character formula is acquired for $\text{tr } T_{\lambda_r, \sigma_r}^{M'}$. Therefore (see 10.1 (a) (e) and $(\mathbf{F}_{+},0)$ page 176), we have:

$$ 
\begin{aligned}
& \text{tr } \sigma_r(\hat{a}) \times |D_{M'}(x \exp Y)|^{1/2} (\text{tr } \pi_{\lambda_r})_{aM'_0}(x \exp Y) \\
&= |\det (1 - \text{Ad } e)_{\mathfrak{m}'/\mathfrak{m}'(e)}|^{1/2} |\mathcal{D}_{\mathfrak{m}'(e)}(X+Y)|^{1/2} (\text{tr } T_{\lambda_r, \sigma_r}^{M'})_e(X+Y) \\
&\stackrel{(\star)}{=} i^{-d_e^{M'}} (D_e^{M'})^{-\frac{1}{2}} |\det (1 - \text{Ad } e)_{\mathfrak{m}'/\mathfrak{m}'(e)}|^{1/2} \sum_{\substack{\lambda_r' \in M'(e)_0 \setminus M'_0, \lambda_r' \cap \mathfrak{m}'(e)^* \\ \mathcal{O}(\hat{e}')_{(1-\text{Ad } e')(\mathfrak{m}'/\mathfrak{h})} / \mathcal{O}(B_{\lambda_r})_{(1-\text{Ad } e')(\mathfrak{m}'/\mathfrak{h})} \text{ tr } \sigma_r(\hat{e}) |\mathcal{D}_{\mathfrak{m}'(e)}(X+Y)|^{1/2} \hat{\beta}_{M'(e)_0, \lambda_r'}(X+Y}} \\
& \text{where } g \in M'_0 \text{ satisfies } \lambda_r' = g\lambda_r \text{ and } \hat{e}' \in M'(\tilde{\lambda})^{\mathfrak{m}'/\mathfrak{h}} \text{ lies over } e' = g^{-1}eg.
\end{aligned}
$$

Consider $g \in M'_0$ as above. Denote by $\text{pr}$ the canonical map from $M'$ to $M'(\mathbb{C})$. We associate with $g$ an element $y^g$ of $M'(\mathbb{C})(\text{pr}(e))_0$ that satisfies $j(x)_{\mathbb{C}} = y^g \cdot (gh)(e)_{\mathbb{C}}$. Fix a system of positive roots $R^+(\mathfrak{m}'(e)_{\mathbb{C}}, j(x)_{\mathbb{C}})$ of $R(\mathfrak{m}'(e)_{\mathbb{C}}, j(x)_{\mathbb{C}})$. According to Proposition 2.6 (b), there exist complex numbers $c_w^g$ independent of $Y$ and indexed by the $w \in W(\mathfrak{m}'(e)_{\mathbb{C}}, j(x)_{\mathbb{C}})$ such that we have

$$ 
\begin{aligned}
& \prod_{\alpha \in R^+(\mathfrak{m}'(e)_{\mathbb{C}}, j(x)_{\mathbb{C}})} \alpha(X+Y) \times \hat{\beta}_{M'(e)_0 \cdot g\lambda_r}(X+Y) \stackrel{(\star\star)}{=} \sum_{w \in W(\mathfrak{m}'(e)_{\mathbb{C}}, j(x)_{\mathbb{C}})} c_w^g e^{iwy^g g\lambda_r(X+Y)} \\
& \text{and} \\
& \quad \prod_{\alpha \in R^+(\mathfrak{m}'(e)_{\mathbb{C}}, j(x)_{\mathbb{C}})} \alpha(X) \times \hat{\beta}_{M'(e)_0 \cdot (g\tilde{\lambda})[e]}(X) = \sum_{w \in W(\mathfrak{m}'(e)_{\mathbb{C}}, j(x)_{\mathbb{C}})} c_w^g e^{iwy^g g\lambda(X)} \\
& \text{when } (g\tilde{\lambda})[e] \in \tilde{\mathfrak{m}}'(e)_{\text{reg}}^*, \text{ and } \sum_{w \in W(\mathfrak{m}'(e)_{\mathbb{C}}, j(x)_{\mathbb{C}})} c_w^g e^{iwy^g g\lambda(X)} = 0 \text{ if } (g\tilde{\lambda})[e] \notin \tilde{\mathfrak{m}}'(e)_{\text{reg}}^*.
\end{aligned}
$$

Furthermore, the function $|\mathcal{D}_{\mathfrak{m}'(e)}|^{1/2} \times \left( \prod_{\alpha \in R^+(\mathfrak{m}'(e)_{\mathbb{C}}, j(x)_{\mathbb{C}})} \alpha \right)^{-1}$ on $\Gamma$ has values in the set of fourth roots of unity, and is therefore constant. From the equalities $(\star)$ and $(\star\star)$, we derive a formula for $\text{tr } \sigma_r(\hat{a}) \times (\text{tr } \pi_{\lambda_r})_{aM'_0}(x \exp Y)$ that involves this constant.

According to Proposition 8.4 (a), the representation $\pi_{\lambda_r}$ of $\mathcal{M}'$ has the infinitesimal character $\chi_{i\lambda_r}^{\mathfrak{m}'\mathbb{C}}$. Apply Proposition 12.3 (c) to $\mathcal{M}'$ with $\pi = \pi_{\lambda_r}$, $\Lambda = i\lambda_r$ and $\Lambda_F = -2\rho$ ($\pi_F$ below). We have a representation $(\pi_{\lambda_r})_{Zuc}$ of $\mathcal{M}'$ and, given the above, a formula for $\text{tr } \sigma_r(\hat{a}) \times (\text{tr}(\pi_{\lambda_r})_{Zuc})_{aM'_0}(x \exp Y)$. By Proposition 13.1 (b), we also have $\text{tr}(\pi_{\lambda_r})_{Zuc} = \text{tr } \pi_{\tilde{\lambda}}$ by choosing $\pi_F$.

We now choose $Y=0$, and summarize. We see that $(\text{tr } T_{\tilde{\lambda},\sigma}^{M'})_e(X)$ can be deduced from the expression for $(\text{tr } T_{\lambda_r,\sigma_r}^{M'})_e(X)$ obtained above, by replacing the coefficient $c_w^g e^{iwy^g \lambda_r(X)}$ by $(\text{det}(\text{Ad } a^{\mathbb{C}})_{nM'})^{-1} \times \underline{x}_{l_F} c_w^g e^{iwy^g \lambda_r(X)}$ for each $w \in W(\mathfrak{m}'(e)_{\mathbb{C}}, j(x)_{\mathbb{C}})$, where $l_F := -2wy^g\rho$ ($\underline{x}_{l_F}$ is defined in 12.3 (c)).

Let $w \in W(\mathfrak{m}'(e)_{\mathbb{C}}, j(x)_{\mathbb{C}})$. We are going to calculate the term $\underline{x}_{l_F}$ associated with it. Choose a representative $\tilde{w}$ of $w$ in $M'(\mathbb{C})(\text{pr}(e))_0$. Let $\mathcal{M}'_{\mathbb{C}}$ denote the semidirect product of $\mathbb{Z}$ by $M'(\mathbb{C})_0$ such that $1 \in \mathbb{Z}$ acts on $M'(\mathbb{C})_0$ by int $\text{pr}(a)$, and let $\underline{\text{pr}}$ denote the canonical map from $\mathcal{M}'$ to $\mathcal{M}'_{\mathbb{C}}$. There is a unique holomorphic representation $\pi_{F,\mathbb{C}}$ of $\mathcal{M}'_{\mathbb{C}}$ in $F$ such that $\pi_F = \pi_{F,\mathbb{C}} \circ \underline{\text{pr}}$. Fix a nonzero vector $v_0$ of $F$ of weight $-2\rho$ for the action of $\mathfrak{h}_{\mathbb{C}}$ on $F$. We are interested in the nonzero vector $\tilde{w}y^g g \cdot v_0$ of $F$ which has weight $l_F$ under the action of $j(x)_{\mathbb{C}}$. Again, set $e' = g^{-1}eg$. In the semidirect product of $\mathbb{Z}$ by $M'(\mathbb{C})$ by means of int $\text{pr}(a)$, we have $\underline{\text{pr}}(\underline{x}) = \underline{\text{pr}}(\underline{a}(a^{-1}x))$ (see 12.3 (c)). We obtain:
$$ 
\begin{aligned}
\underline{\text{pr}}(\underline{x}) \tilde{w}y^g \underline{\text{pr}}(g) &= \underline{\text{pr}}(x) \tilde{w}y^g \underline{\text{pr}}(g) \underline{\text{pr}}(a)^{-1} \underline{\text{pr}}(\underline{a}) \\
&= \underline{\text{pr}}(\exp X) \tilde{w}y^g \underline{\text{pr}}(g) \underline{\text{pr}}(e'a^{-1}) \underline{\text{pr}}(\underline{a})
\end{aligned}
$$
then $\underline{x} \cdot (\tilde{w}y^g g \cdot v_0) = \exp X \cdot (\tilde{w}y^g g \cdot ((e'a^{-1}) \cdot v_0))$.

The property $e'a^{-1} \in M'_0(\tilde{\lambda}) = \exp \mathfrak{h}$ allows us to deduce that:

$$ \underline{x}_{l_F} = \det(\mathrm{Ad} a^{\mathbb{C}})_{n_{M'}} (\det(\mathrm{Ad} e^{\mathbb{C}})_{n_{M'}})^{-1} e^{-2 wy^g g \rho(X)}. $$

This leads to the conclusion. $\square$

Index of notation

⟨,⟩, 141, 145
$G(\tilde{\lambda})_0^{g/\mathfrak{h}}$, 156

a, 141
$\mathfrak{g}^*(x)$, 142
$\mathfrak{g}^*(X)$, 142

A, 141
$\tilde{\mathfrak{g}}_{\text{fund}}^*$, 143

B, 156
$\tilde{\mathfrak{g}}_{\text{fund},G}^*$, 143

b, 163
$\tilde{\mathfrak{g}}_{\text{fund}}^*(e)$, 150

M', 163
$\tilde{\mathfrak{g}}_I^*$, 143

Car, 141
$\tilde{\mathfrak{g}}_I^*(e)$, 150

c, 172
$\tilde{\mathfrak{g}}_I^*(e)$, 150

e', 172
$\tilde{\mathfrak{g}}_{I,G}^*$, 143

C, 143
$\tilde{\mathfrak{g}}_{\text{Inc}}^*$, 143

C, 143
$\tilde{\mathfrak{g}}_{\text{Inc}}^*(e)$, 150

d, 172
$\tilde{\mathfrak{g}}_{\text{Inc},G}^*$, 143

D, 172
$\tilde{\mathfrak{g}}_{\text{reg}}^*$, 142

D, 172
$\tilde{\mathfrak{g}}_{\text{reg}}^*$, 143

D, 147
$\tilde{\mathfrak{g}}_{\text{reg}}^*(e)$, 150

D, 172
$\tilde{\mathfrak{g}}_{\text{reg}}^*(e)$, 150

DL, 152
$\tilde{\mathfrak{g}}_{\text{reg},G}^*$, 143

F, 150
$\tilde{\mathfrak{g}}_{ss}^*$, 142

F, 150
$\tilde{\mathfrak{g}}_{ss,G}^*$, 143

F, 146
$\tilde{\mathfrak{g}}_{ssI}^*$, 142

g, 138
$\tilde{\mathfrak{g}}_{ssI,G}^*$, 143

G, 138
$\tilde{\mathfrak{g}}_{ssInc}^*$, 142

G, 138
$\tilde{\mathfrak{g}}_{ssInc,G}^*$, 143

G, 138
$\tilde{\mathfrak{g}}_{ss reg}$, 147

G, 143
$\tilde{\mathfrak{g}}_{ss reg}$, 142

G, 143
$\tilde{\mathfrak{g}}_{ss reg}$, 142

G, 143
$\tilde{\mathfrak{g}}_{ss reg}$, 172

G, 143
$\tilde{\mathfrak{g}}_{ss reg}$, 172

g, 142
$\tilde{\mathfrak{g}}_{(x)}$, 142

g, 142
$\tilde{\mathfrak{g}}_{(x)}$, 142

g, 142
$\tilde{\mathfrak{g}}_{(x)}$, 142

g, 142
$\tilde{\mathfrak{g}}_{(x)}$, 142

g, 144
$\tilde{\mathfrak{g}}(\tilde{\lambda})$, 144

g, 144
$\tilde{\mathfrak{g}}(\tilde{\lambda})$, 144

g, 144
$\tilde{\mathfrak{g}}(\lambda_+)^{g/\mathfrak{h}}$, 156

g, 144
$\tilde{\mathfrak{g}}(\lambda_+)^{g/\mathfrak{h}}_0$, 156

g, 144
$\tilde{\mathfrak{g}}(\tilde{\lambda})^{g/\mathfrak{g}}(\lambda)(i\rho_{\mathfrak{F}})$, 156

g, 144
$\tilde{\mathfrak{g}}(\tilde{\lambda})^{g/\mathfrak{g}}(\lambda)(i\rho_{\mathfrak{F}})$, 156

g, 144
$\tilde{\mathfrak{g}}(\tilde{\lambda})^{g/\mathfrak{g}}(\lambda)(i\rho_{\mathfrak{F}})$, 156

m, 161
$\mathcal{V}_e$, 172

M, 161
$\mathcal{V}^\chi$, 178

m', 163
$W(G, \mathfrak{h})$, 141

M', 163
$X_G^{\text{final},+}$, 159

M, 161
$X_G^{\text{final},+}(\tilde{\lambda}, \mathfrak{a}^{*+})$, 159

M, 163
$X_G^{\text{Ind}}$, 159

n, 153
$X_G^{\text{Ind}}(\tilde{\lambda})$, 160

n, 153
$X_G^{\text{Ind}}(\tilde{\lambda})$, 160

n, 163
$X_G^{\text{Ind}}(\tilde{\lambda})$, 160

n, 152
$X_G^{\text{Ind}}(\tilde{\lambda})$, 160

n, 152
$X_G^{\text{Ind}}(\tilde{\lambda})$, 160

n, 152
$X_G^{\text{Ind}}(\tilde{\lambda})$, 160

n, 152
$X_G^{\text{Ind}}(\tilde{\lambda})$, 160

n, 152
$X_G^{\text{Ind}}(\tilde{\lambda})$, 160

n, 152
$X_G^{\text{Ind}}(\tilde{\lambda})$, 160

n, 152
$X_G^{\text{Ind}}(\tilde{\lambda})$, 160

n, 152
$X_G^{\text{Ind}}(\tilde{\lambda})$, 160

n, 152
$X_G^{\text{Ind}}(\tilde{\lambda})$, 160

n, 152
$X_G^{\text{Ind}}(\tilde{\lambda})$, 160

n, 152
$X_G^{\text{Ind}}(\tilde{\lambda})$, 160

n, 152
$X_G^{\text{Ind}}(\tilde{\lambda})$, 160

n, 152
$X_G^{\text{Ind}}(\tilde{\lambda})$, 160

n, 152
$X_G^{\text{Ind}}(\tilde{\lambda})$, 160

n, 152
$X_G^{\text{Ind}}(\tilde{\lambda})$, 160

n, 152
$X_G^{\text{Ind}}(\tilde{\lambda})$, 160

n, 152
$X_G^{\text{Ind}}(\tilde{\lambda})$, 160

n, 152
$X_G^{\text{Ind}}(\tilde{\lambda})$, 160

n, 152
$X_G^{\text{Ind}}(\tilde{\lambda})$, 160

n, 152
$X_G^{\text{Ind}}(\tilde{\lambda})$, 160

n, 152
$X_G^{\text{Ind}}(\tilde{\lambda})$, 160

n, 152
$X_G^{\text{Ind}}(\tilde{\lambda})$, 160

n, 152
$X_G^{\text{Ind}}(\tilde{\lambda})$, 160

n, 152
$X_G^{\text{Ind}}(\tilde{\lambda})$, 160

n, 152
$X_G^{\text{Ind}}(\tilde{\lambda})$, 160

n, 152
$X_G^{\text{Ind}}(\tilde{\lambda})$, 160

n, 152
$X_G^{\text{Ind}}(\tilde{\lambda})$, 160

n, 152
$X_G^{\text{Ind}}(\tilde{\lambda})$, 160

n, 152
$X_G^{\text{Ind}}(\tilde{\lambda})$, 160
<table><tr><td>ρ<sub>F+</sub>, 144</td><td>ρ<sub>˜λ</sub><sup>g/g(λ)(iρ<sub>F+</sub>)</sup>, 158</td><td>τ<sub>+</sub>, 159</td><td>Φ, 153</td></tr><tr><td>ρ<sub>g,h</sub>, 141</td><td>ρ<sub>˜λ,a<sup>*+</sup></sub>, 158</td><td>τ<sub>M</sub>, 162</td><td>χ<sub>l</sub><sup>Ugc</sup>, 165</td></tr><tr><td>ρ<sub>L</sub>, 153</td><td></td><td></td><td></td></tr><tr><td>ρ<sub>λ<sub>+</sub></sub><sup>g/h</sup>, 157</td><td>τ, 159</td><td>τ<sub>M'</sub>, 164</td><td>χ<sub>˜λ</sub><sup>G</sup>, 160</td></tr></table>

References

[ABV 92] Adams, J., D. Barbasch, and D. A. Vogan, "The Langlands Classification and Irreducible Characters for Real Reductive Groups," Birkhäuser, Boston, 1992.

[B.. 72] Bernat, P., N. Conze, M. Duflo, M. Lévy-Nahas, M. Raïs, P. Renouard, and M. Vergne, "Représentations des groupes de Lie résolubles," Dunod, Paris, 1972.

[BW 80] Borel, A., and N. Wallach, "Continuous Cohomology, Discrete Subgroups, and Representations of Reductive Groups," Annals of Math. Studies 94, Princeton University Press, 1980. Second edition: Math. Surveys 67, Amer. Math. Soc., 2000.

[Bou 84] Bouaziz, A., *Sur les représentations des groupes de Lie réductifs non connexes*, Math. Ann. **268** (1984), 539-555.

[Bou 87] —, *Sur les caractères des groupes de Lie réductifs non connexes*, J. Funct. Anal. **70** (1987), 1-79.

[B 67] Bourbaki, N., "Théories spectrales, Chapitres 1 et 2," Hermann, Paris, 1967. Second edition: Springer, Cham, 2019.

[Cha 96] Charbonnel, J.-Y., *Orbites fermées et orbites tempérées*, II, J. Funct. Anal. **138** (1996), 213-222.

[Cow 88] Cowling, M., *On the characters of unitary representations*, J. Austral. Math. Soc. Ser. A **45** (1988), 62-65.

[Dix 64] Dixmier, J., "Les $C^*$-algèbres et leurs représentations," Gauthier-Villars, Paris, 1964. English translation: $C^*$-algebras," North-Holland Math. Library, Vol. 15.

[Dix 69] —, *Sur la représentation régulière d'un groupe localement compact connexe*, Ann. Sci. École Norm. Sup. **2** (1969), 423-436.

[Duf 82a] Duflo, M., *Construction de représentations unitaires d'un groupe de Lie*, In: "Harmonic analysis and group representations," Cours d'été du C.I.M.E., Cortona 1980, Liguori, Naples, 1982, 129-221

[Duf 82b] —, *Théorie de Mackey pour les groupes de Lie algébriques*, Acta Math. **149** (1982), 153-213.

[Duf 84] —, *On the Plancherel formula for almost algebraic real Lie groups*, In: "Lie group representations III," Lect. Notes in Math. **1077**, Springer-Verlag, Berlin etc., 1984, 101-165.

[DHV 84] Duflo, M., G. Heckman, and M. Vergne, *Projection d'orbites, formule de Kirillov et formule de Blattner*, Mém. Soc. Math. Fr. (N.S.) **15** (1984), 65-128.
[DV 88] Duflo, M., and M. Vergne, *La formule de Plancherel des groupes de Lie semi-simples réels*, In: "Representations of Lie groups," numéro 14 in Adv. Stud. Pure Math., Academic Press, 1988, 289-336.

[DV 93] —, *Cohomologie équivariante et descente*, Astérisque **215** (1993), 5-108.

[Har 65] Harish-Chandra, *Discrete series for semisimple Lie groups I. Construction of invariant eigendistributions*, Acta Math. **113** (1965), 241-318.

[Har 76] —, *Harmonic analysis on real reductive groups III. The Maass-Selberg relations and the Plancherel formula*, Ann. of Math. **104** (1976), 117-201.

[Her 83] Herb, R. A., *Discrete series characters and Fourier inversion on semisimple real Lie groups*, Trans. Amer. Math. Soc. **277** (1983), 241-262.

[Hoc 65] Hochschild, G., "The Structure of Lie Groups," Holden-Day, San Francisco 1965.

[Kna 86] Knapp, A. W., "Representation Theory of Semisimple Groups," Princeton University Press, 1986.

[Kna 96] —, "Lie Groups beyond an Introduction," Birkhäuser, Boston, 1996. Digital Second Edition: 2023.

[KV 95] Knapp, A. W., and D. A. Vogan, "Cohomological Induction and Unitary Representations," Princeton University Press, 1995.

[KZ 82] Knapp, A. W., and G. J. Zuckerman, *Classification of irreducible tempered representations of semisimple groups*, Ann. of Math. **116** (1982), 389-501. Typesetter's correction: Ann. of Math. **119** (1984), 639.

[Kos 59] Kostant, B., *The principal three-dimensional subgroup and the Betti numbers of a complex simple Lie group*, Amer. J. Math., **81** (1959), 973-1032.

[Mac 58] Mackey, G. W., *Unitary representations of group extension I*, Acta Math. **99** (1958), 265-311.

[Ran 72] Ranga Rao, R., *Orbital integrals in reductive groups*, Ann. of Math., **96** (1972), 505-510.

[Ros 80] Rossmann, W., *Limit characters of reductive Lie groups*, Invent. Math. **61** (1980), 53-66.

[Ros 82] —, *Limit orbit in reductive Lie algebras*, Duke Math. J. **49** (1982), 215-229.

[Spr 66] Springer, T. A., *Some arithmetical results on semi-simple Lie algebras*, Inst. Hautes Études Sci. **30** (1966), 115-141.

[Var 77] Varadarajan, V. S., Harmonic analysis on real reductive groups," Lect. Notes in Math. **576**, Springer-Verlag, Berlin etc., 1977.

[Ver 94] Vergne, M., *Geometric quantization and equivariant cohomology*, In: "First European Congress of Mathematics I, Paris 1992," Progr. Math. **119**, Birkhäuser, Boston, 1994, 249-295.

[Wal 88] Wallach, N., "Real Reductive Groups I," Academic Press, New York etc., 1988.
[Zuc 77] Zuckerman, G., *Tensor products of finite and infinite dimensional representations of semisimple Lie groups*, Ann. of Math. **106** (1977), 295–308.

Jean-Yves Ducloux
Université Paris 7
UFR de Mathématiques, UMR 7586
2 place Jussieu,
F-75251 Paris Cedex 05, France
ducloux@math.jussieu.fr

Received September 22, 2000
and in final form June 28, 2001