# The Orbit Method and Character Formulas for Tempered representations of a Nonconnected Reductive Real Algebraic Group

Jean-Yves Ducloux

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Jean-Yves Ducloux. The Orbit Method and Character Formulas for Tempered representations of a Nonconnected Reductive Real Algebraic Group. 2002. [hal-05429207v2](https://hal.science/hal-05429207v2)

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# The Orbit Method and Character Formulas for Tempered representations of a Nonconnected Reductive Real Algebraic Group(\*)

Jean-Yves Ducloux

Communicated by J. Faraut

**Abstract.** Let $G$ be a possibly disconnected reductive real Lie group. In this paper, I parametrize the set of irreducible tempered characters of $G$. I then describe these characters using certain “Kirillov’s formulas,” based on the descent method near each elliptic element in $G$.

If $G$ is linear and connected, the parameters I use are “final basic” parameters in the sense of Knapp and Zuckerman (see [KZ 82, p. 453]).

## Table of contents

<table><tr><td>Introduction and general notation</td><td>138</td></tr><tr><td>I. The “linear form” parameters</td><td>142</td></tr><tr><td>1. The parameters $\tilde{\lambda} \in \tilde{\mathfrak{g}}_{reg,G}^*$</td><td>142</td></tr><tr><td>2. The measures $\beta_{G,\tilde{\lambda}}$</td><td>146</td></tr><tr><td>3. Fixed points of an elliptic element</td><td>150</td></tr><tr><td>II. The “projective representation” parameters</td><td>151</td></tr><tr><td>4. About special-metalinear and metaplectic groups</td><td>151</td></tr><tr><td>5. The parameters $\tau \in X_G^{Ind}(\tilde{\lambda})$</td><td>156</td></tr><tr><td>6. Integrability condition</td><td>160</td></tr><tr><td>III. Construction of representations</td><td>161</td></tr><tr><td>7. The case where $G$ is connected</td><td>161</td></tr><tr><td>8. The representations $T_{\tilde{\lambda},a^{*+},\tau_{+}}^G$</td><td>163</td></tr><tr><td>9. The injection $G \cdot (\tilde{\lambda}, \tau) \mapsto T_{\tilde{\lambda},\tau}^G$ from $G \setminus X_G^{Ind}$ to $\hat{G}$</td><td>166</td></tr><tr><td>IV. Characters of representations</td><td>172</td></tr><tr><td>10. Character restriction formula</td><td>172</td></tr><tr><td>11. Transition from $T_{\tilde{\lambda},\tau_{M'}}^{M'}$ to $T_{\tilde{\lambda},a^{*+},\tau_{+}}^G$</td><td>176</td></tr><tr><td>12. Translation in the sense of Zuckerman</td><td>177</td></tr><tr><td>13. Transition from $T_{\tilde{\lambda},\sigma_r}^{M'}$ to $T_{\tilde{\lambda},\tau_{M'}}^{M'}$</td><td>182</td></tr></table>

ISSN: 0949-5932

(\*) In this translation written in December 2025, the following lines contain corrections: simplification of notation pages 143 and 187 ("$\mathfrak{g}_{ssfund,G}^*$" $\mapsto$ "$\mathfrak{g}_{ss,G}^*$"), line 31 p. 146 ("p. 221" $\mapsto$ "p. 217"), lines 26-29 p. 148 (brief ideas rewritten), line 4 p. 149 ("$\beta$" $\mapsto$ "$\hat{\beta}$" twice), lines 30 and 31 p. 150 and line 20 p. 151 (we do not ask $\mathcal{F}^{\prime+}$ to be in $C(\mathfrak{g}(\lambda), \mathfrak{h})_{reg}$), line 20 p. 153 ("$\theta_q$" $\mapsto$ "$\theta_n$"), line 24 p. 161 ("$R^+(\mathfrak{m}_C, \mathfrak{t}_C)$" $\mapsto$ "$R(\mathfrak{m}_C, \mathfrak{t}_C)$"), line 29 p. 166 ("(a)" $\mapsto$ "(c)"), lines 30 and 31 p. 168 ("is a multiple of" $\mapsto$ "has a common multiple with"), line 35 p. 172 ("(d)" $\mapsto$ "(e)").
<table><tr><td>Index of notation</td><td>187</td></tr><tr><td>References</td><td>188</td></tr></table>

**Introduction and general notation**

In this paper, I will describe the tempered dual of a real reductive Lie group using the orbit method, based on the following four works:

- the description of the tempered dual of a connected real reductive Lie group by Adams, Barbasch, and Vogan in [ABV 92, Ch. 11];

- the parametrization by Duflo of the equivalence classes of representations of a real reductive Lie group that are irreducible tempered with a regular infinitesimal character, obtained via “Mackey theory” in [Duf 82a, III];

- the “Kirillov’s character formula” of Bouaziz in [Bou 87], which expresses the character of these representations in terms of the Fourier transforms of regular semisimple coadjoint orbits;

- Rossmann’s formulas in [Ros 82] (where “principal” corresponds to “regular” and “regular” to “regular semisimple”), which relate the Fourier transforms of regular orbits to those of regular semisimple orbits.

A theorem of Duflo in [Duf 82b, p. 189] reduces the classification of unitary duals of real algebraic linear groups to the classification of unitary duals of “almost algebraic reductive groups with finite kernels.” It is therefore these reductive groups that interest me.

**Hypothesis.** Let $G$ be a real Lie group with Lie algebra $\mathfrak{g}$, a linear algebraic group $G$ defined over $\mathbb{R}$, and a Lie group homomorphism from $G$ to $G(\mathbb{R})$ with a finite central kernel whose image is open in the usual topology. Thus, every element of $G$ (respectively $\mathfrak{g}$) admits a real Jordan decomposition into elliptic, positively hyperbolic, and unipotent components (respectively infinitesimally elliptic, hyperbolic, and nilpotent components), as described in [DV 93, top of p. 36 and Lem. 31 p. 38]. Assume that $\mathfrak{g}$ is reductive with center consisting of semisimple elements.

Duflo parameterized in [Duf 82a, Lem. 8 p. 173] a subset of the unitary dual $\hat{G}$ of $G$ (specified above), in terms of regular semisimple coadjoint orbits, by reducing via induction to the discrete series representations of certain subgroups of $G$. When $G$ is connected, this subset of $\hat{G}$ was described by Harish-Chandra (see [Har 76, Th. 1 p. 198]). Here, I parameterize, in Theorem 9.6 (b) p. 169, in terms of generalized regular coadjoint orbits, a larger subset of $\hat{G}$, namely the tempered dual of $G$, obtained by replacing, at the initial step of the induction, the discrete series representations with limits of discrete series representations. When $G$ is connected, my statement reproduces part of Theorem 11.14 de [ABV 92, p. 131] (see also [Kna 86, Th. 14.76 p. 598]) with the following addition: the equality of two representations corresponds exactly to the conjugacy, under $G$, of the associated pairs of good parameters.

The expression “good parameters” refers to the constraints imposed on these parameters. The following is the heuristic principle (see [Duf 82a, Th. 1
p. 193] and [Duf 82b, Th. 19 p. 211]). An irreducible unitary representation $T$ of $G$ will, in favorable cases, be parameterized by the orbit under $G$ of an ordered pair $(\tilde{\lambda}, \tau)$. The term $\tilde{\lambda}$ must be a kind of linear form whose semisimple component $l$ corresponds to the character $\chi_{il}^{U_{\mathfrak{g}_c}}$ through which the set $(U_{\mathfrak{g}_c})^G$ of $G$-invariant elements of $U_{\mathfrak{g}_c}$ acts on the space of $C^\infty$ vectors of the representation space of $T$ (see 8.4 (a)). In this paper, this first parameter is related to a limit of discrete series, of which $T$ will, roughly speaking, be an induced representation. For this reason, it will be written in the form $\tilde{\lambda} = (\lambda, \mathcal{F}^+)$, where $\mathcal{F}^+$ is a Weyl chamber for imaginary roots (see [Zuc 77, Th. 5.7 p. 305]). Its “semisimple component” is the linear form $l := \lambda$, in a sense consistent with the definitions relevant to the case of coadjoint orbits (see 2.2 (a)). Let $G(\tilde{\lambda})$ denote the stabilizer of $\tilde{\lambda}$ in $G$. The term $\tau$ is to be a projective representation of $G(\tilde{\lambda})/G(\tilde{\lambda})_0$, lifted to a unitary representation of a degree 2 covering of $G(\tilde{\lambda})$, such that $d_1\tau = \text{id}$ and the central character of $\tau$ extends that of $T$ (see 8.4 (b)). In this paper, this second parameter is a unitary representation (possibly reducible) of a suitable analogue for $\tilde{\lambda}$ of Duflo's double cover of the stabilizer $G(f)$ in $G$ of an element $f$ of $\mathfrak{g}^*$. In fact, the constructions developed here will use a pair of parameters $((\tilde{\lambda}, \mathfrak{a}^{*+}), \tau_+)$ instead of $(\tilde{\lambda}, \tau)$, where $\mathfrak{a}^{*+}$ is a certain Weyl chamber for restricted roots, and $\tau_+$ is an irreducible unitary representation of a covering of $N_{G(\tilde{\lambda})}(\mathfrak{a}^{*+})$. This new pair of parameters still satisfies the conditions described above. The pair $(\tilde{\lambda}, \tau)$ will turn out to be “more canonical” once the character of $T$ has been calculated, but it will play only a secondary role. The $G$-orbits of $(\tilde{\lambda}, \mathfrak{a}^{*+})$ and $((\tilde{\lambda}, \mathfrak{a}^{*+}), \tau_+)$ are respectively identified with those of $\tilde{\lambda}$ and $(\tilde{\lambda}, \tau)$. In the statement of results below, I will use the set $X_G^{\text{Ind}}$ of pairs $(\tilde{\lambda}, \tau)$. It will be defined more precisely in Part II., in connection with the notion of “final character” (see 5.5 (c)).

The main result of this paper is Theorem 10.2 p. 173 that describes the
character of tempered representations $T$ of $G$ in terms of Fourier transforms of
canonical measures on certain regular coadjoint orbits associated with a parameter
$\tilde{\lambda}$ attached to $T$. These character formulas show the need to adapt the “nilpo-
tent orbits” perspective of the orbit method, disregarding the “coadjoint orbit”
aspect of $G \cdot \tilde{\lambda}$. For example, the coadjoint orbits associated with the two limits
of the discrete series representations of $SL(2, \mathbb{R})$, which correspond to the two
open nilpotent half-cones, have no fixed point under the action of the “rotation”
$\mathrm{Ad}^*\left(\begin{smallmatrix} 0 & -1 \\ 1 & 0 \end{smallmatrix}\right)$ (see Remark 3.3 (3)); they must instead be replaced by the sets of
half-lines contained in the interior of their respective convex hulls, up to multi-
plication by $\mathrm{i}$. Remark 6.2 will also show that this new viewpoint clarifies why
certain “admissible” nilpotent orbits in the sense of Duflo do not correspond to any
representation.

My two theorems can be summarized as follows (see the index of notation).

**Results.** There exists a “canonical” bijection $G \cdot (\tilde{\lambda}, \tau) \mapsto T_{\tilde{\lambda}, \tau}^G$ from $G \setminus X_G^{\text{Ind}}$ onto the tempered dual of $G$. It is determined by character formulas of the form:

$$
k_e(X) \operatorname{tr} T_{\tilde{\lambda}, \tau}^{G} (e \exp X) = \sum_{\substack{\tilde{\lambda}' \in G(e) \setminus G \cdot \tilde{\lambda} \cap \tilde{\mathfrak{g}}_{\text{reg}}(e) \\ \text{such that } \tilde{\lambda}'[e] \in \mathfrak{g}(e)^*_{\text{reg}}}} c_{\tilde{e}', \tilde{\lambda}} \operatorname{tr} \tau(\tilde{e}') \hat{\beta}_{G(e) \cdot \tilde{\lambda}'[e]}(X)
$$
where $(\tilde{\lambda}, \tau) \in X_G^{\text{Ind}}$, $e$ ranges over the set of elliptic elements in $G$, $X$ is an element of a certain neighborhood $\mathcal{V}_e$ of $0$ in $\mathfrak{g}(e)$ such that $e \exp X$ is regular semisimple in $G$ (in which case $X$ is regular semisimple in $\mathfrak{g}(e)$), $\hat{e}'$ is an element of the double cover $G(\tilde{\lambda})^{\mathfrak{g}(\tilde{\lambda})(i\rho_{\mathcal{F}^+})}$ (with $\tilde{\lambda} = (\lambda, \mathcal{F}^+)$) that projects to $e' = g^{-1}eg$ for a $g \in G$ such that $\tilde{\lambda}' = g\tilde{\lambda}$, and, $k_e(X)$ and $c_{\tilde{e}', \tilde{\lambda}}$ are certain nonzero complex numbers (see (F) p. 175).

The ideas that enabled me to obtain these results are as follows. Duflo's constructions, carried out step by step, yield a bijection from a set of orbits onto the tempered dual of $G$, by means of Lemma 9.3 (b) p. 166. To move from the connected case to the disconnected case, I will need a generalization of a result of Vogan used by Duflo in a homological construction. This can be found on page 555 of the book [KV 95] by Knapp and Vogan. Given the connected case (see [Ros 80, p. 64]), it is then natural to seek to recover the characters of the irreducible tempered representations by taking a limit from those associated with the regular semisimple orbits. In the disconnected case, the calculation of the character of the representations will be broken down into two steps, the first of which is rather cumbersome.

**Step 1.** We induce a limit of discrete series representation of a subgroup of $G$ to the neutral component of the Levi component $M'$ (which is generally outside the Harish-Chandra class) of a certain “parabolic subgroup” $M'U$ of $G$. The analogue of this induced representation in the connected case was simply a limit of discrete series representation of the Levi component $M$ of a cuspidal parabolic subgroup $MAN$ of $G$. The method here, as in the connected case, is to apply Zuckerman’s translation functor (adapted to the case of the disconnected group $M'$) to reduce to representations of $M'$ whose character is known. One difficulty is that the move from $M'_0$ to $M'$ involves a homological step. On the advice of Bouaziz, I drew inspiration from points (i) and (ii) on page 550 of his paper [Bou 84] to calculate the action of a group $M'(\tilde{\lambda})^{m/\beta}$ on a weight space of the homology of a $M'_0$-module translated in the sense of Zuckerman.

**Step 2.** We then induce from $M'U$ to $G$. The groundwork will have been laid in Part I. with a review and extension of Rossmann’s results for calculating the limits of Fourier transforms of canonical measures on regular semisimple orbits. In particular, in Section 3., I will have highlighted the effect of the passage from $G$ to $G(e)$ in terms of the parameters $\tilde{\lambda}$. The rest of this step consists in applying the methods and results of Bouaziz.

The following definitions and notations will allow me, first, to specify my general conventions, and second, to introduce the conventions related to the reductive groups that I will use most frequently.

**Conventions.** **1.** (a) Let $|A|$ denote the cardinality of a set $A$ and let $\partial$ denote the class of an element $a$ in $A$ modulo an equivalence relation $\sim$ on $A$, denote by $f|_{B'} : B' \to C$ the restriction of a map $f : B \to C$ to a subset $B'$ of $B$, by $\bar{F}$ the closure of a subset $F$ in a topological space $E$ (in Sections 2. and 12.), by $l^\mathbb{C}$ the complexification of a real linear map $l$. Let $\bar{w}$ and $\bar{W}$ denote the conjugate of a vector $w$ and of a vector subspace $W$ in the complexification $V_\mathbb{C}$
of a real vector space $V$ (in Sections 1., 3., 4., 5., 8. and 9.). Let $u_Y$ and $u_{X/Y}$ denote the endomorphisms induced on $Y$ and $X/Y$ by an endomorphism $u$ of a vector space $X$ leaving invariant a vector subspace $Y$ of $X$.

(b) Let $A$ be a real Lie group (we will always assume that its topology is Hausdorff with a countable base) and let $\mathfrak{a}$ be its Lie algebra. Let $1$ denote the neutral element of $A$, and $A_0$ the neutral component of $A$. Denote by $\exp_A$ (or $\exp$) the exponential map from $\mathfrak{a}$ to $A$, by $\mathrm{Ad}^*$ and $\mathrm{ad}^*$ the coadjoint representations of $A$ and of $\mathfrak{a}$, by $\mathrm{int}\,\mathfrak{a}$ the subgroup of the linear group of $\mathfrak{a}$ generated by the elements $\exp(\mathrm{ad} X)$ as $X$ ranges over $\mathfrak{a}$, by $Z(A)$ and $D(A)$ the center and the derived subgroup of $A$, by $Z(\mathfrak{a})$ and $D(\mathfrak{a})$ the center and the derived Lie algebra of $\mathfrak{a}$, by $Z(U\mathfrak{a}_C)$ the center of the enveloping algebra $U\mathfrak{a}_C$ of $\mathfrak{a}_C$, and by $\hat{A}$ the unitary dual of $A$. To every Lie subalgebra $\mathfrak{b}$ of $\mathfrak{a}$, we associate the Lie subgroups

$C_A(\mathfrak{b}) = \{x \in A \mid (\mathrm{Ad} x)|_b = \mathrm{id}\}$ and $N_A(\mathfrak{b}) = \{x \in A \mid \mathrm{Ad} x \cdot \mathfrak{b} \subseteq \mathfrak{b}\}$ of $A$, with Lie algebras

$C_{\mathfrak{a}}(\mathfrak{b}) = \{X \in \mathfrak{a} \mid (\mathrm{ad} X)|_b = 0\}$ and $N_{\mathfrak{a}}(\mathfrak{b}) = \{X \in \mathfrak{a} \mid \mathrm{ad} X \cdot \mathfrak{b} \subseteq \mathfrak{b}\}$.

(c) Let $M$ be a smooth manifold of dimension $m$ that is Hausdorff and second-countable. A $C^\infty$ density on $M$ is a complex Radon measure $\rho$ on $M$ that, in any chart of $M$ centered at a point $x$, takes the form $a dx_1 \cdots dx_m$, where $a$ is a $C^\infty$ function defined on an open neighborhood of $0$ in $\mathbb{R}^m$. It is “identified” with the family of maps $\rho(x)$ with $x \in M$, which send an element of $\Lambda^m T_x M \setminus \{0\}$ whose image under the chart is $\omega_0$ to $a(0) \mid (dx_1 \wedge \cdots \wedge dx_m)(\omega_0) \mid$. A generalized function on $M$ is a continuous linear form on the space of compactly supported $C^\infty$ densities on $M$, equipped with the Schwartz topology.

(d) Let $A$ be a real Lie group and $d_A$ be a left Haar measure on $A$. A continuous representation $T$ of $A$ in a complex Hilbert space is said to be trace class if the operators $T(\varphi d_A)$ with $\varphi \in C_c^\infty(A)$ are trace class; in this case $\mathrm{tr} T : \varphi d_A \mapsto \mathrm{tr} T(\varphi d_A)$ is a generalized function on $A$. For any continuous unitary representation $\pi$ of a closed subgroup $B$ of $A$ equipped with a left Haar measure $d_B$, let $\mathrm{Ind}_B^A \pi$ denote the unitary representation of $A$ “induced” from $\pi$ as in [B.. 72, p. 99].

(e) Let $V$ be a finite-dimensional real vector space. The Fourier transform of a tempered distribution $\mu$ on $V^*$ is the generalized function $\hat{\mu}$ on $V$ defined by the equality
$$ \hat{\mu}(v) = \int_{V^*} e^{il(v)} d\mu(l) \quad \text{of generalized functions in } v \in V. $$

**Conventions. 2.** (a) Let $\mathfrak{C}\mathfrak{g}$ denote the set of Cartan subalgebras of $\mathfrak{g}$. Let $\mathfrak{h} \in \mathfrak{C}\mathfrak{g}$. Fix a system of positive roots $R^+(\mathfrak{g}_C, \mathfrak{h}_C)$, arbitrarily (unless otherwise indicated), in the set $R(\mathfrak{g}_C, \mathfrak{h}_C)$ of roots of $\mathfrak{h}_C$ in $\mathfrak{g}_C$.

Denote by $W(G, \mathfrak{h})$ the finite group $N_G(\mathfrak{h})/C_G(\mathfrak{h})$, by $\mathfrak{t}$ and $\mathfrak{a}$ the infinitesimally elliptic and hyperbolic components of $\mathfrak{h}$, $\mathfrak{h}_{(\mathbb{R})} = \mathfrak{t} \oplus \mathfrak{a}$, $T_0 = \exp \mathfrak{t}$ and $A = \exp \mathfrak{a}$. Let $\rho_{\mathfrak{g},\mathfrak{h}}$ denote the half sum of the elements of $R^+(\mathfrak{g}_C, \mathfrak{h}_C)$, and let $H_\alpha$ denote the coroot of $\alpha$ in the root system $R(\mathfrak{g}_C, \mathfrak{h}_C)$ (such that $\alpha(H_\alpha) = 2$).

A root $\alpha \in R(\mathfrak{g}_C, \mathfrak{h}_C)$ is said to be complex (respectively real, imaginary, or compact) when its conjugate $\bar{\alpha} : X \in \mathfrak{h}_C \mapsto \bar{\alpha}(\bar{X})$ satisfies $\bar{\alpha} \notin \{\alpha, -\alpha\}$ (respectively $\bar{\alpha} = \alpha$, $\bar{\alpha} = -\alpha$, or $(\mathfrak{C}H_\alpha \oplus \mathfrak{g}_C^\alpha \oplus \mathfrak{g}_C^{-\alpha}) \cap \mathfrak{g} \simeq \mathfrak{su}(2)$).

(b) Fix a nondegenerate $G$-invariant bilinear form $\langle \cdot, \cdot \rangle$ on $\mathfrak{g}$ whose complexification (again denoted by $\langle \cdot, \cdot \rangle$) restricts to a scalar product on each $\mathfrak{h}_{(\mathbb{R})}$,
$\mathfrak{h} \in \text{Car} \mathfrak{g}$. Let $\perp^{(\langle, \rangle)}$ denote the relation of $\langle, \rangle$-orthogonality. Let $x \in G$ and $X \in \mathfrak{g}$ be semisimple. Let $G(x)$ (resp. $G(X))$ and $\mathfrak{g}(x)$ (resp. $\mathfrak{g}(X))$ denote the centralizers of $x$ (resp. $X$) in $G$ and $\mathfrak{g}$. The $G(x)$-module $\mathfrak{g}(x)^*$ (resp. the $G(X)$-module $\mathfrak{g}(X)^*$) is canonically isomorphic, via restriction, to the set $\mathfrak{g}^*(x)$ (resp. $\mathfrak{g}^*(X))$ of elements of $\mathfrak{g}^*$ fixed under $\text{Ad}^* x$ (resp. annihilated by $\text{ad}^* X$), i.e., vanishing everywhere on $\mathfrak{g}(x)^{\perp^{(\langle, \rangle)}}$ (resp. $\mathfrak{g}(X)^{\perp^{(\langle, \rangle)}}$).

(c) Let $f \in \mathfrak{g}^*$. Denote by $G(f)$ the stabilizer of $f$ in $G$ and by $\mathfrak{g}(f)$ the Lie algebra of $G(f)$. Whenever possible, $\mu$, $\nu$, $\lambda = \mu+\nu$, and $\xi$ will be used to denote the “infinitesimally elliptic, hyperbolic, semisimple, and nilpotent components of $f$” (concepts from $\mathfrak{g}$ using $\langle, \rangle$). We identify $\mathfrak{g}(\lambda)^*$ with the set of elements of $\mathfrak{g}^*$ that are vanishing everywhere on $\mathfrak{g}(\lambda)^{\perp^{(\langle, \rangle)}}$.

(d) Let $\mathfrak{m}$ be the centralizer in $\mathfrak{g}$ of a semisimple element $X$ of $\mathfrak{g}$. Given Haar measures $d_{\mathfrak{g}}$ and $d_{\mathfrak{m}}$ on $\mathfrak{g}$ and $\mathfrak{m}$, we set for all $f \in \mathfrak{m}^*$:

$$|\Pi_{\mathfrak{g},\mathfrak{m}}|(f) = \frac{1}{k!} |(B_f|_{[\mathfrak{g},X]^2})^k(\omega)| \times (d_{[\mathfrak{g},X]}(0)(\omega))^{-1} \quad (\text{see (b) and Conventions . 1 (c))},$$
with $k = \frac{1}{2} \dim[\mathfrak{g}, X]$, $B_f = f([\cdot, \cdot])$, $\omega \in \Lambda^{2k}[\mathfrak{g}, X] \setminus \{0\}$, and $d_{[\mathfrak{g},X]} = d_{\mathfrak{g}}/d_{\mathfrak{m}}$.

The other notations used in the statements can be found in the index of notations at the end of this paper.

# I. The “linear form” parameters

The orbit method, first proposed by Kirillov in the case of simply connected nilpotent Lie groups, consists of parameterizing the irreducible unitary representations of a Lie group using the orbits of its coadjoint representation. When the representation is trace class, its character in some neighborhood of the neutral element must be related to the Fourier transform (assuming it exists) of the canonical measure on the associated orbit.

A slight modification of this point of view will be necessary here. Each irreducible tempered representation of $G$ will be associated with an orbit $\tilde{\Omega}$ of $G$ in a certain set $\tilde{\mathfrak{g}}_{reg}^*$ other than $\mathfrak{g}^*$, and to this orbit $\tilde{\Omega}$ will be attached a finite sum of canonical measures on coadjoint orbits. In preparation for an upcoming paper, I also introduce below parts $\tilde{\mathfrak{g}}_I^*$ and $\tilde{\mathfrak{g}}_{Inc}^*$ of $\tilde{\mathfrak{g}}_{reg}^*$.

In this section, I will make use of certain results of Rossmann.

## 1. The parameters $\tilde{\lambda} \in \tilde{\mathfrak{g}}_{reg,G}^*$

In this section, we construct a set $\tilde{\mathfrak{g}}_{reg}^*$ equipped with an action of $G$ in which the set of $\lambda \in \mathfrak{g}^*$ regular semisimple embeds in a $G$-equivariant manner.

**Definition 1.1.** Let $\mathfrak{g}_{reg}^*$ denote the set of $f \in \mathfrak{g}^*$ regular (i.e., the set of $f \in \mathfrak{g}^*$ for which the dimension of $\mathfrak{g}(f)$ is equal to the rank of $\mathfrak{g}$), $\mathfrak{g}_{ss}^*$ the set of $f \in \mathfrak{g}^*$ semisimple and $\mathfrak{g}_{ss reg}^* = \mathfrak{g}_{ss}^* \cap \mathfrak{g}_{reg}^*$.

Also let $\mathfrak{g}_{ssI}^*$ (respectively: $\mathfrak{g}_{ssInc}^*$) denote the set of $\lambda \in \mathfrak{g}_{ss}^*$ such that $\mathfrak{g}(\lambda)$ has a Cartan subalgebra $\mathfrak{h}$ for which the roots of $(\mathfrak{g}(\lambda)_{\mathbb{C}}, \mathfrak{h}_{\mathbb{C}})$ are imaginary (respectively: noncompact imaginary).
Let $\lambda \in \mathfrak{g}_{ss}^*$ and $\mathfrak{h} \in \text{Car} \mathfrak{g}(\lambda)$. Let $C(\mathfrak{g}(\lambda), \mathfrak{h})$ denote the set of (open) chambers in $\mathfrak{h}_{(\mathbb{R})}^*$ for the imaginary roots of $(\mathfrak{g}(\lambda)_{\mathbb{C}}, \mathfrak{h}_{\mathbb{C}})$ and let $C(\mathfrak{g}(\lambda), \mathfrak{h})_{\text{reg}}$ denote the set of $\mathcal{F}^+ \in C(\mathfrak{g}(\lambda), \mathfrak{h})$ such that the imaginary roots of $(\mathfrak{g}(\lambda)_{\mathbb{C}}, \mathfrak{h}_{\mathbb{C}})$ that are simple relative to $\mathcal{F}^+$ are noncompact.

Set $\tilde{\mathfrak{g}}_{\text{reg}}^* = \{ (\lambda, \mathcal{F}^+) ; \lambda \in \mathfrak{g}_{ss}^*, \mathfrak{h} \in \text{Car} \, \mathfrak{g}(\lambda) \text{ and } \mathcal{F}^+ \in C(\mathfrak{g}(\lambda), \mathfrak{h})_{\text{reg}} \}$,
and $\tilde{\mathfrak{g}}_{\text{reg},G}^* = \{ (\lambda, \mathcal{F}^+) \in \tilde{\mathfrak{g}}_{\text{reg}}^* | \forall Z \in \text{Ker} \, \exp_{T_0} e^{(i\lambda+\rho_{\mathfrak{g},\mathfrak{h}})(Z)} = 1 \} \text{ (see } . \ 2 \text{ (a))}$

independently of the choice of a system of positive roots $R^+(\mathfrak{g}_C, \mathfrak{h}_C)$ associated with the element $\mathfrak{h}$ of $\text{Car} \mathfrak{g}(\lambda)$ attached to $\mathcal{F}^+$.

Also set (in the following “fundamental” means “without real root”):

$$
\begin{align*}
\tilde{\mathfrak{g}}_{\text{fund}}^* &= \left\{ (\lambda, \mathcal{F}^+) ; \lambda \in \mathfrak{g}_{ss}^*, \mathfrak{h} \in \text{Car} \, \mathfrak{g}(\lambda) \text{ fundamental and } \mathcal{F}^+ \in C(\mathfrak{g}(\lambda), \mathfrak{h})_{\text{reg}} \right\}, \\
\text{and } \mathfrak{g}_{ss,G}^* &= \left\{ \lambda \in \mathfrak{g}_{ss}^* \mid \forall Z \in \text{Ker} \, \exp_{T_0} e^{(i\lambda+\rho_{\mathfrak{g},G})(Z)} = 1 \right\}
\end{align*}
$$

independently of the choice of a fundamental $\mathfrak{h} \in \text{Car} \mathfrak{g}(\lambda)$ and of a system of positive roots $R^+(\mathfrak{g}_C, \mathfrak{h}_C)$, $\mathfrak{g}_{ssI,G}^* = \mathfrak{g}_{ssI}^* \cap \mathfrak{g}_{ss,G}^*$ and $\mathfrak{g}_{ssInc,G}^* = \mathfrak{g}_{ssInc}^* \cap \mathfrak{g}_{ss,G}^*$.

Finally, let $\tilde{\mathfrak{g}}_I^*$, $\tilde{\mathfrak{g}}_{\text{Inc}}^*$, $\tilde{\mathfrak{g}}_{\text{fund},G}^*$, $\tilde{\mathfrak{g}}_{I,G}^*$, and $\tilde{\mathfrak{g}}_{\text{Inc},G}^*$ denote the inverse images of $\mathfrak{g}_{ssI}^*$, $\mathfrak{g}_{ssInc}^*$, $\mathfrak{g}_{ss,G}^*$, $\mathfrak{g}_{ssI,G}^*$, and $\mathfrak{g}_{ssInc,G}^*$ under the first projection from $\tilde{\mathfrak{g}}_{\text{fund}}^*$ to $\mathfrak{g}_{ss}^*$.

**Remark 1.2.** (1) By applying [Ros 82, Th. p. 217] to $\mathfrak{g}(\lambda)$ for three Cartan subalgebras (one equal to $\mathfrak{h}$, one without imaginary roots, and one fundamental) and taking into account [Ros 82, Lem. A (a) ⇒ (c) p. 220 and Suppl. C p. 218], we see that the images of the canonical maps from $\tilde{\mathfrak{g}}_{\text{reg}}^*$ and $\tilde{\mathfrak{g}}_{\text{fund}}^*$ to $\mathfrak{g}^*$ consist of those $\lambda \in \mathfrak{g}_{ss}^*$ such that $\mathfrak{g}(\lambda)$ has a Cartan subalgebra without imaginary roots.

(2) According to [Kna 96, Th. 6.74 p. 341 and Th. 6.88 p. 344], each complex semisimple Lie algebra has, up to isomorphism, a unique real form $\mathfrak{g}_0$ that has a Cartan subalgebra $\mathfrak{h}_0$ for which the root system $R(\mathfrak{g}_0, \mathfrak{h}_0)$ has a basis consisting of noncompact imaginary roots. Given (1) above (or [Kna 96, Pb. 18 p. 369 and p. 557]), the image of the canonical map from $\tilde{\mathfrak{g}}_I^*$ to $\mathfrak{g}^*$ consists of those $\lambda \in \mathfrak{g}_{ss}^*$ such that each of the simple ideals of $\mathfrak{g}(\lambda)$ is isomorphic to one of the following Lie algebras:

su($p, p$) with $p \ge 1$ and su($p, p - 1$) with $p \ge 2$, so($p, p - 1$) with $p \ge 3$, sp($2n, \mathbb{R}$) (denoted sp($n, \mathbb{R}$) in [Kna 96]) with $n \ge 3$, so($p, p$) with $p$ even $\ge 4$ and so($p, p - 2$) with $p$ even $\ge 6$, EII, EV, EVIII, FI, G.

(3) The sum of two noncompact imaginary roots is never a noncompact root according to [Kna 96, (6.99) p. 352]. Thus, $\mathfrak{g}_{ssInc}^*$ consists of those semisimple $\lambda \in \mathfrak{g}^*$ such that the simple ideals of the reductive Lie algebra $\mathfrak{g}(\lambda)$ are isomorphic to $\mathfrak{sl}(2, \mathbb{R})$. $\blacksquare$

Starting from certain enriched parameters $(\tilde{\lambda}, \mathfrak{a}^{*+})$, I will canonically construct two systems of positive roots for $(\mathfrak{g}_C, \mathfrak{h}_C)$, which will have the same noncompact roots. The system of positive roots $R^+(\mathfrak{g}_C, \mathfrak{h}_C)$ is attached to a certain regular linear form $\lambda_+$. It will allow, by limiting processes (Section 11.) and translation (Section 13.), to construct representations in Part III. by reducing to the case of regular infinitesimal character. The system of positive roots $R_{\tilde{\lambda}, \mathfrak{a}^{*+}}^+$ has, as a subset of complex roots with negative conjugate, a part that is invariant under the action of the group $G(\tilde{\lambda})$ defined below. It will be used in Definition 5.5 (c) and
in Lemma 9.3 (b). We will see in Lemma 9.1 another parametrization in which the parameter $(\tilde{\lambda}, \mathfrak{a}^{*+})$ will be replaced by $\tilde{\lambda}$, and the role of $(R_{\tilde{\lambda}}^{+}, \mathfrak{a}^{*+}, \lambda_{+})$ will be played by a certain pair $(R_{\tilde{\lambda}}^{+}, \lambda_{can})$ obtained from $\tilde{\lambda}$.

**Definition 1.3.** Let $\lambda \in \mathfrak{g}_{ss}^*$, $\mathfrak{h} \in \text{Car}\,\mathfrak{g}(\lambda)$ and $\mathcal{F}^+ \in C(\mathfrak{g}(\lambda), \mathfrak{h})$ be given. Let $\mu$ and $\nu$ (resp. $\mathfrak{t}$ and $\mathfrak{a}$) denote the infinitesimally elliptic and hyperbolic components of $\lambda$ (resp. $\mathfrak{h}$). Set $\tilde{\lambda} = (\lambda, \mathcal{F}^+)$.

(a) Let $G(\tilde{\lambda})$ denote the normalizer of $\mathcal{F}^+$ in $G(\lambda)$ and let $\mathfrak{g}(\tilde{\lambda})$ denote its Lie algebra. Associate with $\tilde{\lambda}$ the half sum $\rho_{\mathcal{F}^+} \in \mathfrak{t}^*$ of the imaginary $\alpha \in R(\mathfrak{g}(\lambda)_{\mathbb{C}}, \mathfrak{h}_{\mathbb{C}})$ such that $\mathcal{F}^+(H_\alpha) \subseteq \mathbb{R}_+\{0\}$. (Thus $\mathfrak{h}$ is a Cartan subalgebra of $\mathfrak{g}(\lambda)(i\rho_{\mathcal{F}^+})$ without imaginary roots, and therefore $G \cdot (\lambda, i\rho_{\mathcal{F}^+})$ determines $G \cdot \tilde{\lambda}$.)

Fix, for the rest of this definition a chamber $\mathfrak{a}^{*+}$ of $(\mathfrak{g}(\lambda)(i\rho_{\mathcal{F}^+}), \mathfrak{a})$.

(b) Introduce the following system of positive roots:

$$
R^+(\mathfrak{g}(\lambda)(i\rho_{\mathcal{F}^+})_{\mathbb{C}}, \mathfrak{h}_{\mathbb{C}}) = \{ \alpha \in R(\mathfrak{g}(\lambda)(i\rho_{\mathcal{F}^+})_{\mathbb{C}}, \mathfrak{h}_{\mathbb{C}}) \mid \mathfrak{a}^{*+}(H_\alpha) \subseteq \mathbb{R}_+ \setminus \{0\} \}.
$$

Fix $\epsilon > 0$ small enough so that, upon setting $\nu_+ = \nu + \epsilon \rho_{\mathfrak{g}(\lambda)(i\rho_{\mathcal{F}^+}), \mathfrak{h}} \in \mathfrak{a}^*$ we have: $a \cdot \nu_+ \neq \nu_+$ for any automorphism $a$ of the root system $R(\mathfrak{g}_{\mathbb{C}}, \mathfrak{h}_{\mathbb{C}})$ such that $a \cdot \nu \neq \nu$. Choose $\epsilon$ so that the previous property remains valid when we replace $\epsilon$ by $t\epsilon$, $t \in [0, 1]$. We therefore have: $\mathfrak{g}(\nu_+) = \mathfrak{g}(\nu)(\rho_{\mathfrak{g}(\lambda)(i\rho_{\mathcal{F}^+}), \mathfrak{h}})$.

We will use the following systems of positive roots:

$$
R^+(\mathfrak{g}(\nu_+)_{\mathbb{C}}, \mathfrak{h}_{\mathbb{C}}) = \left\{ \alpha \in R(\mathfrak{g}(\nu_+)_{\mathbb{C}}, \mathfrak{h}_{\mathbb{C}}) \mid \text{i}\mu(H_{\alpha}) > 0 \text{ or } \left\{ \begin{array}{l} \text{i}\mu(H_{\alpha})=0 \\ \text{and} \\ \rho_{\mathcal{F}^+}(H_{\alpha})>0 \end{array} \right\} \right\}
$$

$$
\text{and } R^+(\mathfrak{g}_{\mathbb{C}}, \mathfrak{h}_{\mathbb{C}}) = \{ \alpha \in R(\mathfrak{g}_{\mathbb{C}}, \mathfrak{h}_{\mathbb{C}}) \mid \nu_+(H_{\alpha}) > 0 \} \cup R^+(\mathfrak{g}(\nu_+)_{\mathbb{C}}, \mathfrak{h}_{\mathbb{C}}).
$$

$$
\begin{align*}
\text{Set } \mu_+ = \mu - 2i\rho_{\mathfrak{g}(\nu_+), \mathfrak{h}} \in \mathfrak{t}^* \text{ and } \lambda_+ = \mu_+ + \nu_+ \\
(\text{therefore } R^+(\mathfrak{g}(\nu_+)_{\mathbb{C}}, \mathfrak{h}_{\mathbb{C}}) = \{ \alpha \in R(\mathfrak{g}(\nu_+)_{\mathbb{C}}, \mathfrak{h}_{\mathbb{C}}} \mid \text{i}\mu_+(H_\alpha) > 0 \}).
\end{align*}
$$

In some cases, we will write $\lambda_{\mathfrak{g}, \tilde{\lambda}, \mathfrak{a}^{*+}, \epsilon}$, $\mu_{\mathfrak{g}, \tilde{\lambda}, \mathfrak{a}^{*+}}$ and $\nu_{\mathfrak{g}, \tilde{\lambda}, \mathfrak{a}^{*+}, \epsilon}$ for $\lambda_+$, $\mu_+$ and $\nu_+$.

(c) Introduce the set

$$
R_{\tilde{\lambda}}^{+} = \left\{ \alpha \in R(\mathfrak{g}_{\mathbb{C}}, \mathfrak{h}_{\mathbb{C}}) \mid \nu(H_{\alpha}) > 0 \text{ or } \left\{ \begin{array}{l} \nu(H_{\alpha})=0 \\ \text{and} \\ i\mu(H_{\alpha})>0 \end{array} \right. \text{ or } \left\{ \begin{array}{l} \lambda(H_{\alpha})=0 \\ \text{and} \\ \rho_{\mathcal{F}^+}(H_{\alpha})>0 \end{array} \right\} \right\},
$$

the half sum $\rho_{can}$ of the elements of $R_{\tilde{\lambda}}^{+} \cap R(\mathfrak{g}(\nu)_{\mathbb{C}}, \mathfrak{h}_{\mathbb{C}})$, and the system of positive roots $R_{\tilde{\lambda}, \mathfrak{a}^{*+}} = R_{\tilde{\lambda}}^{+} \cup R^{+}(\mathfrak{g}(\lambda)(i\rho_{\mathcal{F}^{+}})_{\mathbb{C}}, \mathfrak{h}_{\mathbb{C}})$ of $R(\mathfrak{g}_{\mathbb{C}}, \mathfrak{h}_{\mathbb{C}})$.

Set $\mu_{can} = \mu - 2i\rho_{can} \in \mathfrak{t}^*$ and $\lambda_{can} = \mu_{can} + \nu$
(so that, using [KV 95, Cor. 4.69 p. 271], we find that $\mathfrak{g}(\lambda_{can}) = \mathfrak{g}(\lambda)(i\rho_{\mathcal{F}^+})$ and
$R_{\tilde{\lambda}}^+ \cap R(\mathfrak{g}(\nu)_{\mathbb{C}}, \mathfrak{h}_{\mathbb{C}}) = \{ \alpha \in R(\mathfrak{g}(\nu)_{\mathbb{C}}, \mathfrak{h}_{\mathbb{C}}) \mid i\mu_{can}(H_\alpha) > 0 \}$).

Here is the key lemma that will allow us to reduce $\tilde{\mathfrak{g}}_{\text{reg}}^*$ to $\mathfrak{g}_{\text{ss reg}}^*$.

**Lemma 1.4.** We place ourselves in the situation of the previous definition.
Let $t \in [0, 1]$. Set $\nu_t = \nu + t\epsilon \rho_{\mathfrak{g}(\lambda)(i\rho_{\mathcal{F}^+}), \mathfrak{h}}$, $\mu_t = \mu - 2it\rho_{\mathfrak{g}(\nu_+), \mathfrak{h}}$ and $\lambda_t = \mu_t + \nu_t$.

We have: $\mathfrak{g}(\tilde{\lambda}) = \mathfrak{h}$, $\lambda_t \in \mathfrak{g}_{ss \ reg}^*$ and $N_{G(\tilde{\lambda})}(\mathfrak{a}^{*+}) = G(\lambda_t)$.

Therefore $G(\lambda_+) = G(\tilde{\lambda})$ when $\mathfrak{a}^{*+} = \mathfrak{a}^*$ (for example, when $\tilde{\lambda} \in \tilde{\mathfrak{g}}_J^*$).

**Proof of the lemma.** We have $G(\tilde{\lambda}) \subseteq N_G(\mathfrak{h})$, therefore $\tilde{\mathfrak{g}}(\tilde{\lambda}) = \mathfrak{h}$.

By construction of $\epsilon$, we have $\mathfrak{g}(\nu_t) = \mathfrak{g}(\nu)(\rho_{\mathfrak{g}(\lambda)(i\rho_{\mathcal{F}^+}), \mathfrak{h}}) = \mathfrak{g}(\nu_+)$. Furthermore, $i\mu_t(H_\alpha) = i\mu(H_\alpha) + 2t\rho_{\mathfrak{g}(\nu_+), \mathfrak{h}}(H_\alpha) > 0$ holds for all $\alpha \in R^+(\mathfrak{g}(\nu_+)_{\mathbb{C}}, \mathfrak{h}_{\mathbb{C}})$. Therefore $\lambda_t \in \mathfrak{g}_{ss \ reg}^*$. A continuity argument in $t$ implies that:
$$
R^+(\mathfrak{g}_c, \mathfrak{h}_c) = \left\{ \alpha \in R(\mathfrak{g}_c, \mathfrak{h}_c) \mid \nu_t(H_\alpha) > 0 \text{ or } \left\{ \begin{array}{l} \nu_t(H_\alpha)=0 \\ \text{and} \\ i\mu_t(H_\alpha)>0 \end{array} \right\} \right\}.
$$

The inclusion $N_{G(\bar{\lambda})}(\mathfrak{a}^{*+}) \subseteq G(\lambda_t)$ is immediate. Let $g \in G(\lambda_t)$.

We have $g \cdot h = g \cdot g(\lambda_t) = h$ and $(\mathrm{Ad} g)|_h$ is an automorphism of the root system $R(\mathfrak{g}_c, \mathfrak{h}_c)$ such that $g\nu_t = \nu_t$, therefore $g(\nu_t - \nu) = (\nu_t - \nu)$ then $g\nu_+ = \nu_+$. Since $g$ stabilizes $R(\mathfrak{g}(\nu_+)_{\mathbb{C}}, \mathfrak{h}_{\mathbb{C}}) \cap R^+(\mathfrak{g}_{\mathbb{C}}, \mathfrak{h}_{\mathbb{C}})$, we have: $g\rho_{\mathfrak{g}(\nu_+), \mathfrak{h}} = \rho_{\mathfrak{g}(\nu_+), \mathfrak{h}}$ then $g\mu = \mu$. Next, since $g$ stabilizes the set of imaginary roots of $R(\mathfrak{g}(\lambda)_{\mathbb{C}}, \mathfrak{h}_{\mathbb{C}}) \cap R^+(\mathfrak{g}_{\mathbb{C}}, \mathfrak{h}_{\mathbb{C}})$, we have: $g\mathcal{F}^+ = \mathcal{F}^+$. Finally, since $g$ stabilizes the set of restrictions $\alpha|_{\mathfrak{a}}$ for $\alpha \in R(\mathfrak{g}(\lambda)(i\rho_{\mathcal{F}+})_{\mathbb{C}}, \mathfrak{h}_{\mathbb{C}}) \cap R^+(\mathfrak{g}_{\mathbb{C}}, \mathfrak{h}_{\mathbb{C}})$, we have: $g \cdot \mathfrak{a}^{*+} = \mathfrak{a}^{*+}$. $\square$

The following lemma will explain why, in certain formulas that I will write
later in another paper, and in which the absolute values of Pfaffians $|\Pi_{\mathfrak{g},C_{\mathfrak{g}(\mu)}}(\mathfrak{a})|$ appear as factors, only the elements of $\mathfrak{g}_{ssI}^*$ are involved.

To make these Pfaffians more amenable to calculation, I first specify a
formula from [DV 88, p. 301]. The bilinear form $\langle \cdot, \cdot \rangle$ from Conventions . 2 (b)
determines a Haar measure $d_V$ on any vector subspace $V$ of $\mathfrak{g}$ on which its restric-
tion is nondegenerate, by the condition $d_V(0)(v_1 \wedge \cdots \wedge v_n) = |\det((v_i, v_j))|_{1 \le i,j \le n}|^{\frac{1}{2}}$
for any basis $(v_1, \dots, v_n)$ of $V$. Let $\mathfrak{h} \in \text{Car } \mathfrak{g}$. Fix a regular $\lambda_0 = \mu_0 + \nu_0 \in \mathfrak{h}^*$ in
$\mathfrak{g}^*$. Associate it with the system of positive roots

$$
R_0^+(\mathfrak{g}_c, \mathfrak{h}_c) = \left\{ \alpha \in R(\mathfrak{g}_c, \mathfrak{h}_c) \mid \nu_0(H_\alpha) > 0 \text{ or } \left\{ \begin{array}{l} \nu_0(H_\alpha)=0 \\ \text{and} \\ i\mu_0(H_\alpha)>0 \end{array} \right\} \right\}.
$$

Let $d_{\mathfrak{g}}$ and $d_{\mathfrak{h}}$ be the Haar measures on $\mathfrak{g}$ and $\mathfrak{h}$ obtained from $\langle \cdot, \cdot \rangle$. For all $\lambda \in \mathfrak{h}^*$ and all $\omega_0 \in \Lambda^{2k}[\mathfrak{g}, \mathfrak{h}]\setminus\{0\}$ element of the orientation $\mathcal{o}(B_{\lambda_0})_{\mathfrak{g}/\mathfrak{h}}$ (see 4.2 (b) and 5.1 (a)) of $(\mathfrak{g}/\mathfrak{h}, B_{\lambda_0})$, we have

$$
(*) \quad \frac{1}{k!} (B_{\lambda}|_{[\mathfrak{g}, \mathfrak{h}]^2})^k (\omega_0) \times (d_{[\mathfrak{g}, \mathfrak{h}]}(0)(\omega_0))^{-1} = i^{n_0} \prod_{\alpha \in R_0^+(\mathfrak{g}_{\mathbb{C}}, \mathfrak{h}_{\mathbb{C}})} \langle \lambda, \alpha \rangle
$$

where $k = \frac{1}{2} \dim[\mathfrak{g}, \mathfrak{h}]$, $d_{[\mathfrak{g}, \mathfrak{h}]} = d_{\mathfrak{g}}/d_{\mathfrak{h}}$, $n_0 = |\{\alpha \in R_0^+(\mathfrak{g}_{\mathbb{C}}, \mathfrak{h}_{\mathbb{C}}) \mid \bar{\alpha} \notin R_0^+(\mathfrak{g}_{\mathbb{C}}, \mathfrak{h}_{\mathbb{C}})\}|$, denoting again by $\langle \cdot, \cdot \rangle$ the bilinear form on $\mathfrak{h}_{\mathbb{C}}^*$ dual to the restriction to $\mathfrak{h}_{\mathbb{C}}$ of $\langle \cdot, \cdot \rangle$.

**Lemma 1.5.** Let $f \in \mathfrak{g}^*$ be of infinitesimally elliptic, hyperbolic, semisimple, and nilpotent components $\mu, \nu, \lambda = \mu+\nu$, and $\xi$. Fix $\mathfrak{h} \in \text{Car } \mathfrak{g}$ such that $\lambda \in \mathfrak{h}^*$. Let $\mathfrak{a}$ denote the hyperbolic component of $\mathfrak{h}$.

(a) We have $f \in \mathfrak{g}_{\text{reg}}^*$ if and only if $\xi \in \mathfrak{g}(\lambda)^*_{\text{reg}}$.

(b) We have $\lambda \in \mathfrak{g}_{ssI}^*$ and $\mathfrak{h} \in \text{Car} \, \mathfrak{g}(\lambda)$ is fundamental, if and only if $|\Pi_{\mathfrak{g},C_{\mathfrak{g}(\mu)}(\mathfrak{a})}|(\lambda) \neq 0$ (see Conventions . 2 (d)).

**Proof of the lemma.** (a) It is clear that the equality $\dim \mathfrak{g}(f) = \text{rk } \mathfrak{g}$ is equivalent to $\dim \mathfrak{g}(\lambda)(\xi) = \text{rk } \mathfrak{g}(\lambda)$.

(b) For all $X \in \mathfrak{h}$ and $\lambda' \in \mathfrak{h}^*$, the complementary vector subspaces $[\mathfrak{g}, \mathfrak{g}(X)] = [\mathfrak{g}, X]$ and $[\mathfrak{g}(X), \mathfrak{h}] \subseteq \mathfrak{g}(X)$ of $[\mathfrak{g}, \mathfrak{h}]$ being orthogonal for both $B_{\lambda'}$ and $\langle \cdot, \cdot \rangle$, we have: $|\Pi_{\mathfrak{g},\mathfrak{h}}|(\lambda') = |\Pi_{\mathfrak{g},\mathfrak{g}(X)}|(\lambda') \times |\Pi_{\mathfrak{g}(X),\mathfrak{h}}|(\lambda')$. By choosing $X \in \mathfrak{h}$ such that $\mathfrak{g}(X) = C_{\mathfrak{g}(\mu)}(\mathfrak{a})$, it follows from (*) just before this lemma that:

$$
|\Pi_{\mathfrak{g}, C_{\mathfrak{g}(\mu)}(\mathfrak{a})}|(\lambda) = \prod_{\substack{\alpha \in R(\mathfrak{g}_{\mathbb{C}}, \mathfrak{h}_{\mathbb{C}}) \\ \alpha \notin R(C_{\mathfrak{g}(\mu)}(\mathfrak{a})_{\mathbb{C}}, \mathfrak{h}_{\mathbb{C}})}} |\langle \lambda, \alpha \rangle|^{\frac{1}{2}}.
$$

This gives the result.

$\square$
## 2. The measures $\beta_{G, \tilde{\lambda}}$

We will now associate to each orbit $\tilde{\Omega} \in G \backslash \tilde{\mathfrak{g}}_{reg}^*$ a Radon measure $\beta_{\tilde{\Omega}}$ on $\mathfrak{g}^*$ such that $\beta_{G, \tilde{\lambda}}$ coincides with the canonical measure on $G \cdot \lambda$ when $\lambda \in \mathfrak{g}_{ss \ reg}^*$, $\mathfrak{h} = \mathfrak{g}(\lambda)$ and $\tilde{\lambda} = (\lambda, \mathfrak{h}_{(\mathbb{R})}^*)$.

**Definition 2.1.** Let $\lambda \in \mathfrak{g}_{ss}^*$, $\mathfrak{h} \in \text{Car} \, \mathfrak{g}(\lambda)$ and $\mathcal{F}^+ \in \mathcal{C}(\mathfrak{g}(\lambda), \mathfrak{h})$.

(a) Let $\mathcal{F}_{\mathfrak{h}}^+$ denote $\mathfrak{a}^* + \mathfrak{t}^{*+}$, where $\mathfrak{t}$ and $\mathfrak{a}$ are the infinitesimally elliptic and hyperbolic components of $\mathfrak{h}$ and $\mathcal{F}^+$ is written $\mathcal{F}^+ = \mathfrak{a}^* + i\mathfrak{t}^{*+}$ with $\mathfrak{t}^{*+} \subseteq \mathfrak{t}^*$.

(b) Let $\text{Supp}_{\mathfrak{g}^*}(G \cdot (\lambda, \mathcal{F}^+))$ denote the union in $\mathfrak{g}_{reg}^*$ of $G \cdot (\omega + \{\lambda\})$, where $\omega$ ranges over the set of regular nilpotent orbits of $G(\lambda)$ in $\mathfrak{g}(\lambda)^*$ included in the closure of $G(\lambda) \cdot \mathcal{F}_{\mathfrak{h}}^+$ (see Lemma 1.5 (a) and Remark 2.4 (1)).

The following proposition is given in order to facilitate the use of the character formula 10.2. It will also, in some cases, make it possible to pass directly from the characters of the representations to the associated parameters (see 9.7).

**Proposition 2.2.** (a) The set $G \backslash \tilde{\mathfrak{g}}_{\text{fund}}^*$ is in bijection with $G \backslash \mathfrak{g}_{\text{reg}}^*$ via the map $R_G$ that sends $G \cdot \tilde{\lambda}$ to $\text{Supp}_{\mathfrak{g}^*}(G \cdot \tilde{\lambda})$.

(b) Let $\lambda \in \mathfrak{g}_{ss}^*$, $\mathfrak{h} \in \text{Car} \, \mathfrak{g}(\lambda)$ and $\mathcal{F}^+ \in \mathcal{C}(\mathfrak{g}(\lambda), \mathfrak{h})$. The part $\text{Supp}_{\mathfrak{g}^*}(G \cdot (\lambda, \mathcal{F}^+))$ of $\mathfrak{g}_{reg}^*$ is the union of $R_G(G \cdot (\lambda, \mathcal{F}^{'+}))$ with $\mathfrak{h}' \in \text{Car} \, \mathfrak{g}(\lambda)$ fundamental and $\mathcal{F}^{'+} \in \mathcal{C}(\mathfrak{g}(\lambda), \mathfrak{h}')_{\text{reg}}$ satisfying $G(\lambda) \cdot \rho_{\mathcal{F}^+} \cap \overline{\mathcal{F}}^{'+} \neq \emptyset$.

**Proof of the proposition.** (a) First, let $(\lambda, \mathcal{F}^+) \in \tilde{\mathfrak{g}}_{\text{fund}}^*$. According to [Ros 82, Th. p.217 and Suppl. A(b) p.218] applied to $G_0(\lambda)$, there exists a unique regular nilpotent orbit $\omega_0$ of $G_0(\lambda)$ in $\overline{G_0(\lambda)} \cdot \mathcal{F}_{\mathfrak{h}}^+$. The set $\omega := G(\lambda) \cdot \omega_0$ is therefore a regular nilpotent orbit of $G(\lambda)$ in $\overline{G(\lambda)} \cdot \mathcal{F}_{\mathfrak{h}}^+$. Any regular nilpotent orbit $\omega'$ of $G(\lambda)$ in $\overline{G(\lambda)} \cdot \mathcal{F}_{\mathfrak{h}}^+$ intersects $u \cdot \overline{G_0(\lambda)} \cdot \mathcal{F}_{\mathfrak{h}}^+$ for a certain $u \in G(\lambda)$, and therefore satisfies $\omega' \supseteq u \cdot \omega_0$ then $\omega' = \omega$. This allows us to define $R_G$.

We now abandon the notation used at the beginning of this proof.

Let $f \in \mathfrak{g}_{\text{reg}}^*$ be of semisimple and nilpotent components $\lambda$ and $\xi$.

Fix a fundamental $\mathfrak{h} \in \text{Car} \, \mathfrak{g}(\lambda)$. By Lemma 1.5 (a) and [Ros 82, Lem. A p. 220], we can go through a finite sequence of inverse Cayley transformations (see [Ros 82, p. 218]) from $\mathfrak{h}$ to a Cartan subalgebra of $\mathfrak{g}(\lambda)$ without imaginary roots. By [Ros 82, Suppl. A(a) and B and C p. 218, and Th. p. 217], there exists $\mathcal{F}^+ \in \mathcal{C}(\mathfrak{g}(\lambda), \mathfrak{h})_{\text{reg}}$ unique up to the action of $N_{G_0(\lambda)}(\mathfrak{h})$ for which $\omega_0 := G_0(\lambda) \cdot \xi$ is the regular nilpotent orbit of $G_0(\lambda)$ in $\overline{G_0(\lambda)} \cdot \mathcal{F}_{\mathfrak{h}}^+$. Thus, $G \cdot f$ is the image of $G \cdot (\lambda, \mathcal{F}^+)$ by $R_G$.

Let $G \cdot (\lambda', \mathcal{F}^{'+})$ be a preimage of $G \cdot f$ by $R_G$. We have $\mathcal{F}^{'+} \in \mathcal{C}(\mathfrak{g}(\lambda'), \mathfrak{h}')_{\text{reg}}$ for some fundamental $\mathfrak{h}' \in \text{Car} \, \mathfrak{g}(\lambda')$. Let $\omega'_0$ denote the regular nilpotent orbit of $G_0(\lambda')$ in $\overline{G_0(\lambda')} \cdot \mathcal{F}_{\mathfrak{h}'}^{'+}$. Given the beginning of this proof, there exists $g \in G$ such that $g\xi \in \omega'_0$ and $g\lambda = \lambda'$. Since the Cartan subalgebras $\mathfrak{h}$ and $g^{-1}\mathfrak{h}'$ of $\mathfrak{g}(\lambda)$ are fundamental, there exists $x \in G(\lambda)_0$ such that $g^{-1}\mathfrak{h}' = x\mathfrak{h}$. Therefore, $\omega_0$ which is equal to $g^{-1}\omega'_0$ is also the regular nilpotent orbit of $G_0(\lambda)$ in $\overline{G_0(\lambda)} \cdot x^{-1}g^{-1}\mathcal{F}_{\mathfrak{h}'}^{'+}$. Hence: $x^{-1}g^{-1} \cdot (\lambda', \mathcal{F}^{'+}) \in N_{G_0(\lambda)}(\mathfrak{h}) \cdot (\lambda, \mathcal{F}^+)$. It follows that $G \cdot (\lambda, \mathcal{F}^+)$ is the unique preimage of $G \cdot f$ by $R_G$.
(b) Using an argument from the beginning of the proof of (a), we see that $\text{Supp}_{g^*}(G \cdot (\lambda, \mathcal{F}^+))$ is the union of the $G$-orbits that intersect $\text{Supp}_{g^*}(G_0 \cdot (\lambda, \mathcal{F}^+))$. We are therefore reduced to proving (b) for $G_0$ with $\lambda = 0$. Let us consider this case. Fix a fundamental Cartan subalgebra $\mathfrak{h}'$ of $\mathfrak{g}$ that contains the infinitesimally elliptic linear form $-i \rho_{\mathcal{F}^+}$. Set $l_0 = -i \rho_{\mathcal{F}^+}$ in $\mathcal{F}_{\mathfrak{h}}^+$. We will briefly review the ideas of [Ros 82, Proof of Suppl. c p. 228].

From [Var 77, Th. 23 p.50] and [Ros 82, (L1)⇔(L2)⇔(L3) p.216], we have:

$$ \sum_{\Omega \in G_0 \setminus \text{Supp}_{g^*}(G_0 \cdot (0, \mathcal{F}^+))} \beta_{\Omega}(\varphi) = \lim_{t \to 0^+} \sum_{\Omega' \in L_t} \beta_{\Omega'}(\varphi) \quad \text{for all } \varphi \in C_c^{\infty}(\mathfrak{g}^*), $$

where we use 2.3 (c) and set $L_t = \lim_{\substack{l' \in \mathcal{F}_{\mathfrak{h}}^+ \cap \mathfrak{g}_{reg}^* \\ l' \to tl_0}} G_0 \cdot l' \subseteq G_0 \setminus \mathfrak{g}_{reg}^*$ when $t > 0$.

By [Ros 82, lines 2–9 p. 217], the canonical map from $G_0(l_0) \setminus \mathfrak{g}(l_0)_{reg}^*$ to $G_0 \setminus \mathfrak{g}_{reg}^*$ restricts to a bijection from $\lim_{\substack{l'' \in \mathcal{F}^+[l_0]_b \cap \mathfrak{g}(l_0)_{reg}^* \\ l'' \to tl_0}} G_0(l_0) \cdot l''$ onto $L_t$ for all $t > 0$, where the limit is taken in $G_0(l_0) \setminus \mathfrak{g}(l_0)_{reg}^*$ and $\mathcal{F}^+[l_0]$ is the element of $C(\mathfrak{g}(l_0), \mathfrak{h})$ containing $\mathcal{F}^+$.

The roots of $(\mathfrak{g}(l_0)_c, \mathfrak{h}_c)$ are nonimaginary and those of $(\mathfrak{g}(l_0)_c, \mathfrak{h}'_c)$ are nonreal. From [Ros 82, Suppl. c p. 218], we deduce that:

$$ \lim_{\substack{l'' \in \mathcal{F}^+[l_0]_b \cap \mathfrak{g}(l_0)_{reg}^* \\ l'' \to tl_0}} G_0(l_0) \cdot l'' = \bigcup_{\mathcal{F}'^+ \in W(G_0(l_0), \mathfrak{h}') \setminus \mathcal{E}} \lim_{\substack{l'' \in \mathcal{F}'^+[l_0]_{\mathfrak{h}'} \cap \mathfrak{g}(l_0)_{reg}^* \\ l'' \to tl_0}} G_0(l_0) \cdot l'' $$

where $\mathcal{E} := \{ \mathcal{F}'^+ \in C(\mathfrak{g}, \mathfrak{h}') \mid l_0 \in \overline{\mathcal{F}_{\mathfrak{h}'}^+} \}$ and $\mathcal{F}'^+[l_0]$ is the element of $C(\mathfrak{g}(l_0), \mathfrak{h}')$ containing $\mathcal{F}'^+$. Furthermore, the previous union is a union of pairwise disjoint sets, by [Ros 82, Suppl. b p. 218].

The Radon measures $\beta_{\Omega}$ with $\Omega \in G_0 \setminus \mathfrak{g}_{reg}^*$ (see 2.3 (c)) are linearly independent. Using [Ros 82, Th. p. 217], we find that $\text{Supp}_{g^*}(G_0 \cdot (0, \mathcal{F}^+))$ is the disjoint union of $R_{G_0}(G_0 \cdot (0, \mathcal{F}'^+))$ where $\mathcal{F}'^+ \in W(G_0(l_0), \mathfrak{h}') \setminus C(\mathfrak{g}, \mathfrak{h}')_{reg}$ and $\rho_{\mathcal{F}^+} \in \overline{\mathcal{F}'^+}$. This yields the result. $\square$

**Definition 2.3.** (a) Let $\mathcal{D}_{\mathfrak{g}}$ denote the function on $\mathfrak{g}$ whose value at $X \in \mathfrak{g}$ is the coefficient of $T^r$ in $\det(T \text{id} - \text{ad} X)$, where $r$ is the rank of $\mathfrak{g}$. It is polynomial and invariant under $\text{Ad} G$.

(b) Let $\mathfrak{g}_{ss reg}$ denote the set of semisimple $X \in \mathfrak{g}$ such that $\mathfrak{g}(X)$ is abelian. Therefore $\mathfrak{g}_{ss reg}$ is the dense open set (with negligible complement) of $\mathfrak{g}$ formed by the points where $\mathcal{D}_{\mathfrak{g}}$ does not vanish (see [Var 77, (2) and Lem. 1 p. 9]).

(c) Let $\Omega \in G \setminus \mathfrak{g}^*$. Let $\beta_{\Omega}$ denote the pushforward measure of the Liouville measure on the symplectic manifold $\Omega$ by the canonical injection $\Omega \hookrightarrow \mathfrak{g}^*$ (see [B.. 72, 2.6 p. 20]). This is a Radon measure on $\mathfrak{g}^*$ (see [Ran 72, Th. 2 p. 509]).

(d) Let $\tilde{\Omega}_0 \in G_0 \setminus \tilde{\mathfrak{g}}_{reg}^*$. Set $\beta_{\tilde{\Omega}_0} = \sum_{\Omega_0 \in G_0 \setminus \text{Supp}_{g^*}(\tilde{\Omega}_0)} \beta_{\Omega_0}$.

(e) Let $\tilde{\Omega} \in G \setminus \tilde{\mathfrak{g}}_{reg}^*$. Set $\beta_{\tilde{\Omega}} = \sum_{\tilde{\Omega}_0 \in G_0 \setminus \tilde{\Omega}} \beta_{\tilde{\Omega}_0}$ (see end of 2.6 (a)).

**Remark 2.4.** (1) Let $\tilde{\Omega} \in G \setminus \tilde{\mathfrak{g}}_{reg}^*$. From the beginning of the proof of Proposition 2.2 (b), the set $\text{Supp}_{g^*}(\tilde{\Omega})$ is the union of the orbits $\Omega \in G \setminus \mathfrak{g}_{reg}^*$ such that $\beta_{\tilde{\Omega}}(\Omega) \neq 0$.
It also follows from the above definition that $\beta_{\tilde{\Omega}} = \sum_{\Omega_0 \in G_0 \setminus \text{Supp}_{\mathfrak{g}^*}(\tilde{\Omega})} m_{\Omega_0} \beta_{\Omega_0}$ where $m_{\Omega_0}$ is the (nonzero) number of $\tilde{\Omega}_0 \in G_0 \setminus \tilde{\Omega}$ satisfying $\Omega_0 \subseteq \text{Supp}_{\mathfrak{g}^*}(\tilde{\Omega}_0)$. In particular: $\beta_{\tilde{\Omega}} = \beta_{R_G(\tilde{\Omega})}$ when $\tilde{\Omega} \in G \setminus \tilde{\mathfrak{g}}_{\text{fund}}^*$.

(2) The example where $G$ is equal to the semidirect product of $\mathbb{Z}/2\mathbb{Z}$ by $SL(2, \mathbb{R})^2$ where the nontrivial element of $\mathbb{Z}/2\mathbb{Z}$ acts on $SL(2, \mathbb{R})^2$ by permuting the coordinates, and where $\tilde{\Omega}$ is of the form $G \cdot (0, \mathcal{F}^+)$ for a chamber $\mathcal{F}^+$ associated to $\mathfrak{h} \in \text{Car g}$ neither fundamental nor split, shows that the coefficients $m_{\Omega_0}$ of (1) above can take values other than 1.

(3) From (1) and 2.2 (a), the map $\tilde{\Omega} \mapsto \beta_{\tilde{\Omega}}$ is injective on $G \setminus \tilde{\mathfrak{g}}_{\text{fund}}^*$.

When $G = SL(3, \mathbb{R})$, the two elements $\tilde{\Omega}$ of $G \setminus \tilde{\mathfrak{g}}_{\text{reg}}^*$ of the form $G \cdot (0, \mathcal{F}^+)$ (one of which lies in $G \setminus \tilde{\mathfrak{g}}_{\text{fund}}^*$) correspond to the same measure $\beta_{\tilde{\Omega}}$, equal to the Liouville measure of the unique regular nilpotent orbit of $G$ in $\mathfrak{g}^*$. $\blacksquare$

Here is a well-known result. Since (b) will not be used, I will give only an abbreviated proof of it, suggested by Charbonnel.

**Proposition 2.5.** (a) The function $|\mathcal{D}_{\mathfrak{g}}|^{-1/2}$ is locally integrable on $\mathfrak{g}$.

(b) The Radon measures $\beta_{\Omega}$ on $\mathfrak{g}^*$ with $\Omega \in G \setminus \mathfrak{g}^*$ are tempered.

(c) Let $\Omega \in G \setminus \mathfrak{g}^*$. The generalized function $\tilde{\beta}_{\Omega}$ is (modulo equality almost everywhere) a locally integrable function on $\mathfrak{g}$ and analytic on $\mathfrak{g}_{ss \text{ reg}}$.

**Proof of the proposition.** (a) This first assertion is proven in [Var 77, Prop. 15 p. 66].

(b) Consider a real linear algebraic group $H$. Let $\mathfrak{h}$ denote its Lie algebra. Let us consider a coadjoint orbit $\Omega$ of $H$ whose Liouville measure $\beta_{\Omega}$ is a Radon measure on $\mathfrak{h}^*$. We follow the arguments of [Cha 96, 3.2 p.220]. Let $W_0$ be an open set of $\mathbb{P}(\mathfrak{h}^* \times \mathbb{R})$ corresponding to the nonvanishing of a homogeneous coordinate, and let $U$ be a relatively compact open set in $W_0$ for the usual topology. We construct a certain regular function $q: U \to \mathbb{R}_+ \setminus \{0\}$. According to [Cha 96, p. 217], the meromorphic function $\Phi$ on $\mathbb{C}$ such that $\langle \Phi(s), \phi \rangle = \int_{\Omega \cap U} q(l)^s \phi(l) \, d\beta_{\Omega}(l)$ for $\phi \in C_c^{\infty}(\mathfrak{h}^* \cap U)$, when $\text{Re}(s)$ is large enough, is continuous for the topology of $S(\mathfrak{h}^*)$. Hence the measure $\beta_{\Omega}$ on $\mathfrak{h}^*$ is tempered by [Cha 96, line 3 p. 222].

(c) See [Var 77, Prop. 13 p.65, Th. 17 p.66, Th. 28 p.95, bottom of p.105]. $\square$

The following proposition will be of interest when attempting to reduce, by limiting processes or by translation, to regular semisimple linear forms.

**Proposition 2.6.** Let $\lambda \in \mathfrak{g}_{ss}^*$, $\mathfrak{h} \in \text{Car g}(\lambda)$, and $\mathcal{F}^+ \in C(\mathfrak{g}(\lambda), \mathfrak{h})$. Set $\tilde{\lambda} = (\lambda, \mathcal{F}^+)$. Let $\mathfrak{a}$ denote the hyperbolic component of $\mathfrak{h}$, and fix a chamber $\mathfrak{a}^{*+}$ of $(\mathfrak{g}(\lambda)(i\rho_{\mathcal{F}^+}), \mathfrak{a})$.

(a) The positive measures $\beta_{\tilde{\Omega}}$ with $\tilde{\Omega} \in G \setminus \tilde{\mathfrak{g}}_{\text{reg}}^*$ are tempered Radon measures. Let $\varphi \in S(\mathfrak{g}^*)$. We have

$$ \lim_{\substack{\lambda' \in \mathcal{F}_{\mathfrak{h}}^{+} \cap \mathfrak{g}_{\text{reg}}^* \\ \lambda' \to \lambda}} \left( |W(G, \mathfrak{h})(\lambda')| \beta_{G \cdot \lambda'}(\varphi) \right) = \begin{cases} |W(G, \mathfrak{h}) (\lambda_+)| & \text{if } \tilde{\lambda} \in \tilde{\mathfrak{g}}_{\text{reg}}^* \\ 0 & \text{otherwise.} \end{cases} $$
In particular $\beta_{G \cdot \bar{\lambda}}(\varphi) = \lim_{t \to 0^+} \beta_{G \cdot \lambda_t}(\varphi)$ when $\tilde{\lambda} \in \tilde{\mathfrak{g}}_{reg}^*$ (see 1.4).

(b) For all $X \in \mathfrak{g}_{ss\ reg}$, taking into account 2.5 (c) we have

$$ \lim_{\substack{\lambda' \in \mathcal{F}_h^+ \cap \mathfrak{g}_{reg}^* \\ \lambda' \to \lambda}} (|W(G,h)(\lambda')|) \hat{\beta}_{G \cdot \lambda'}(X) = \begin{cases} |W(G,h)(\lambda_+)| & \text{if } \tilde{\lambda} \in \tilde{\mathfrak{g}}_{reg}^* \\ 0 & \text{otherwise.} \end{cases} $$

In particular $\hat{\beta}_{G \cdot \bar{\lambda}}(X) = \lim_{t \to 0^+} \hat{\beta}_{G \cdot \lambda_t}(X)$ when $\tilde{\lambda} \in \tilde{\mathfrak{g}}_{reg}^*$ (see 1.4).

More precisely, given $j \in \text{Car} \, \mathfrak{g}$, $y \in \text{int}(\mathfrak{g}_{\mathbb{C}})$ for which $j_{\mathbb{C}} = yh_{\mathbb{C}}$, a system of positive roots $R^+(\mathfrak{g}_{\mathbb{C}}, j_{\mathbb{C}})$ of $R(\mathfrak{g}_{\mathbb{C}}, j_{\mathbb{C}})$, a connected component $\Gamma$ of $j \cap \mathfrak{g}_{ss\ reg}$, and a connected component $h^{*+}$ of $\mathcal{F}_h^+ \cap \mathfrak{g}_{reg}^*$ to which $\lambda$ is close to, there exists a family $(c_w)_{w \in W(\mathfrak{g}_{\mathbb{C}}, j_{\mathbb{C}})}$ of complex numbers such that, for all $X \in \Gamma$, we have

$$ \sum_{w \in W(\mathfrak{g}_{\mathbb{C}}, j_{\mathbb{C}})} c_w e^{iwy\lambda'(X)} = \prod_{\alpha \in R^+(\mathfrak{g}_{\mathbb{C}}, j_{\mathbb{C}})} \alpha(X) \times \hat{\beta}_{G_0 \cdot \lambda'}(X) \quad \text{when } \lambda' \in \mathfrak{h}^{*+} $$
and

$$ \sum_{w \in W(\mathfrak{g}_{\mathbb{C}}, j_{\mathbb{C}})} c_w e^{iwy\lambda'(X)} = \begin{cases} \prod_{\alpha \in R^+(\mathfrak{g}_{\mathbb{C}}, j_{\mathbb{C}})} \alpha(X) \times \hat{\beta}_{G_0 \cdot \bar{\lambda}}(X) & \text{if } \tilde{\lambda} \in \tilde{\mathfrak{g}}_{reg}^* \\ 0 & \text{otherwise.} \end{cases} $$

**Proof of the proposition.** (a) Let $\varphi \in C_c^\infty(\mathfrak{g}^*)$ be nonnegative. According to [Ros 82, (L1) $\Leftrightarrow$ (L2) $\Leftrightarrow$ (L3) p. 216, lines 9–12 p. 217, and Th. p. 217], we have

$$ \lim_{\substack{\lambda' \in \mathcal{F}_h^+ \cap \mathfrak{g}_{reg}^* \\ \lambda' \to \lambda}} \beta_{G_0 \cdot \lambda'}(\varphi) = \begin{cases} \beta_{G_0 \cdot \bar{\lambda}}(\varphi) & \text{if } \tilde{\lambda} \in \tilde{\mathfrak{g}}_{reg}^* \\ 0 & \text{otherwise.} \end{cases} $$

Fix a norm $\|\cdot\|$ on $\mathfrak{g}^*$. According to [Var 77, (i) p. 40], there exists $N \in \mathbb{N}$ such that the set formed by $\int_{\mathfrak{g}^*} (1 + \|l\|)^{-N} d\beta_{\Omega_0}(l)$ with $\Omega_0 \in G_0 \setminus \mathfrak{g}_{ss\ reg}^*$ is bounded.

It follows that $\beta_{G_0 \cdot \bar{\lambda}}$ is a tempered Radon measure when $\tilde{\lambda} \in \tilde{\mathfrak{g}}_{reg}^*$, and that the limiting process remains valid when the condition “$\varphi \in C_c^\infty(\mathfrak{g}^*)$ nonnegative” is replaced by “$\varphi \in S(\mathfrak{g}^*)$.”

We then obtain, for all $\varphi \in S(\mathfrak{g}^*)$:

$$ \lim_{\substack{\lambda' \in \mathcal{F}_h^+ \cap \mathfrak{g}_{reg}^* \\ \lambda' \to \lambda}} \sum_{\dot{x} \in G/G_0} \beta_{G_0 \cdot x\lambda'}(\varphi) = \begin{cases} \sum_{\dot{x} \in G/G_0} \beta_{G_0 \cdot x\bar{\lambda}}(\varphi) & \text{if } \tilde{\lambda} \in \tilde{\mathfrak{g}}_{reg}^* \\ 0 & \text{otherwise.} \end{cases} $$

We first transform the sum on the left for $\lambda' \in \mathcal{F}_h^+ \cap \mathfrak{g}_{reg}^*$:

$$ \sum_{\dot{x} \in G/G_0} \beta_{G_0 \cdot x\lambda'} = |G(\lambda')G_0/G_0| \quad \beta_{G \cdot \lambda'} = \frac{|W(G,h)(\lambda')|}{|C_G(h)/C_{G_0}(h)|} \quad \beta_{G \cdot \lambda'}. $$

Furthermore, the chambers of $(\mathfrak{g}(\lambda)(i\rho_{\mathcal{F}^+}), \mathfrak{a})$ are conjugate under $N_{G(\lambda)(i\rho_{\mathcal{F}^+})_0}(\mathfrak{a})$ and a *fortiori* under $G_0(\tilde{\lambda})$, which proves that $G(\tilde{\lambda}) = G_0(\tilde{\lambda}) N_{G(\tilde{\lambda})}(\mathfrak{a}^{*+})$.

When $\tilde{\lambda} \in \tilde{\mathfrak{g}}_{reg}^*$, we therefore have, using Lemma 1.4:

$$ \sum_{\dot{x} \in G/G_0} \beta_{G_0 \cdot x\bar{\lambda}} = |G(\bar{\lambda})G_0/G_0| \quad \beta_{G \cdot \bar{\lambda}} = \frac{|W(G,h)(\lambda_+)|}{|C_G(h)/C_{G_0}(h)|} \quad \beta_{G \cdot \bar{\lambda}}. $$

This proves (a).

(b) The penultimate equality in this statement is taken from [Var 77, Th. 4 p. 108]. It shows that $\hat{\beta}_{G_0 \cdot \lambda'}$ converges pointwise on $\mathfrak{g}_{ss\ reg}$ to a continuous $G_0$-invariant function when $\lambda' \to \lambda$ with $\lambda' \in \mathfrak{h}^{*+}$. Let $\psi \in C_c^\infty(\mathfrak{g}_{ss\ reg})$. From the proof in (a), we have
$$
\lim_{\substack{\lambda' \in \mathfrak{h}^{*+} \\ \lambda' \to \lambda}} \int_{\mathfrak{g}_{ss}^{\text{reg}}} \hat{\beta}_{G_0 \cdot \lambda'}(X) \psi(X) \, \mathrm{d}\mathfrak{g}(X) = \begin{cases} \hat{\beta}_{G_0 \cdot \tilde{\lambda}}(\psi \, \mathrm{d}\mathfrak{g}) & \text{if } \tilde{\lambda} \in \tilde{\mathfrak{g}}_{\text{reg}}^* \\ 0 & \text{otherwise.} \end{cases}
$$

In light of the upper bounds $|c_w e^{iwy\lambda'(X)}| \le |c_w|$ from [Var 77, Th. 7 p.111], we see that the $G_0$-invariant functions $|\mathcal{D}_{\mathfrak{g}}|^{1/2} \hat{\beta}_{G_0 \cdot \lambda'}$ on $\mathfrak{g}_{ss \text{reg}}$ with $\lambda' \in \mathfrak{h}^{*+}$ are uniformly bounded. Proposition 2.5 (a) therefore allows us to apply Lebesgue's dominated convergence theorem in the above limit. This yields the final equality in the statement.

The passage from $G_0$ to $G$ is carried out as in the proof of (a). $\square$

3. Fixed points of an elliptic element

Let us consider an elliptic element $e$ of $G$.

The Lie algebra $\mathfrak{g}(e)$ is reductive because $\mathfrak{g}(e)_{\mathbb{C}}$ has as real form $\mathfrak{c}(e)$,
where $\mathfrak{c}$ is the Lie algebra of a maximal compact subgroup of $\mathfrak{G}(\mathbb{C})$ that contains
the projection of $e$ into $\mathfrak{G}(\mathbb{R})$. Furthermore, any Cartan subalgebra $\mathfrak{h}_e$ of $\mathfrak{g}(e)$
intersects $\mathfrak{g}_{ss \ reg}$, which means that the centralizer $\mathfrak{h}$ of $\mathfrak{h}_e$ in $\mathfrak{g}$ is a Cartan
subalgebra of $\mathfrak{g}$ (see [Bou 87, Lem. 1.4.1 p. 6]). Therefore $G(e)$ satisfies the
hypothesis stated in the introduction concerning $G$ (see [DV 93, bottom of p.38]).

**Definition 3.1.** Let $\mathfrak{g}_{reg}^*(e)$ (respectively $\tilde{\mathfrak{g}}_I^*(e)$, $\tilde{\mathfrak{g}}_{Inc}^*(e)$, $\tilde{\mathfrak{g}}_{fund}^*(e)$, and $\tilde{\mathfrak{g}}_{reg}^*(e)$) denote the set of elements of $\mathfrak{g}_{reg}^*$ (respectively $\tilde{\mathfrak{g}}_I^*$, $\tilde{\mathfrak{g}}_{Inc}^*$, $\tilde{\mathfrak{g}}_{fund}^*$, and $\tilde{\mathfrak{g}}_{reg}^*$) that are fixed under the action of $e$.

Among the results of the following lemma, only point (a) is essential.
Point (b) will be used in Remark 9.7 and point (c) in a later paper.

**Lemma 3.2.** Let $\lambda \in \mathfrak{g}_{ss}^*$, $\mathfrak{h} \in \text{Car} \, \mathfrak{g}(\lambda)$ with infinitesimally elliptic and hyperbolic components $\mathfrak{t}$ and $\mathfrak{a}$, and let $\mathcal{F}^+ \in C(\mathfrak{g}(\lambda), \mathfrak{h})$ such that $e$ fixes $\tilde{\lambda} := (\lambda, \mathcal{F}^+)$. Assume there exists a chamber $\mathfrak{a}^{*+}$ of $(\mathfrak{g}(\lambda)(i\rho_{\mathcal{F}^+}), \mathfrak{a})$ stable under $e$ (we will refer to this condition by saying that "$\tilde{\lambda}[e]$ exists", see (a) below).

(a) We have $\lambda \in \mathfrak{g}(e)_{ss}^*$, $\mathfrak{h}(e) \in \text{Car} \, \mathfrak{g}(e)(\lambda)$, and the element $\rho_{\mathcal{F}^+}$ of $\mathfrak{h}(e)_{(\mathbb{R})}^*$ is regular for the imaginary roots of $(\mathfrak{g}(e)(\lambda)_{\mathbb{C}}, \mathfrak{h}(e)_{\mathbb{C}})$. By definition, $\mathcal{F}^+[e]$ denote the element of $C(\mathfrak{g}(e)(\lambda), \mathfrak{h}(e))$ containing $\rho_{\mathcal{F}^+}$, and $\tilde{\lambda}[e] = (\lambda, \mathcal{F}^+[e])$.

(b) There exists $e_0 \in e \exp \mathfrak{t}(e)$ such that: if $\mathcal{F}^{'+} \in C(\mathfrak{g}(\lambda), \mathfrak{h})$ and $\rho_{\mathcal{F}^{'+}} \in \mathcal{F}^{'+}[e_0]$, then $\mathcal{F}^{'+} = \mathcal{F}^{+}$. When $\tilde{\lambda} \in \tilde{\mathfrak{g}}_I^*$ we can choose $e_0$ such that $\tilde{\lambda}[e_0] \in \overline{\mathfrak{g}(e_0)}_{I}^{*}$.

(c) Assume that $\tilde{\lambda} \in \tilde{\mathfrak{g}}_I^*$. The cardinality of the set of $\tilde{\lambda}' = (\lambda', \mathcal{F}^{'+})$ with $\mathcal{F}^{'+} \in C(\mathfrak{g}(\lambda), \mathfrak{h})$ satisfying $e\tilde{\lambda}' = \tilde{\lambda}'$ and $\tilde{\lambda}'[e] = \tilde{\lambda}[e]$ is equal to $\frac{|W(\mathfrak{g}(\lambda)_{\mathbb{C}}, \mathfrak{h}_{\mathbb{C}})(\text{Ad}^*e)|}{|W(\mathfrak{g}(e)(\lambda)_{\mathbb{C}}, \mathfrak{h}(e)_{\mathbb{C}})|}$, where $W(\mathfrak{g}(\lambda)_{\mathbb{C}}, \mathfrak{h}_{\mathbb{C}})(\text{Ad}^*e)$ denotes the centralizer of $(\text{Ad}^*e^{\mathbb{C}})_{\mathfrak{h}_{\mathbb{C}}^*}$ in $W(\mathfrak{g}(\lambda)_{\mathbb{C}}, \mathfrak{h}_{\mathbb{C}})$.

**Proof of the lemma.** (a) By Lemma 1.4, we have: $\lambda_+ \in \mathfrak{g}(e)_{ss \ reg}^*$, then $\mathfrak{h}(e) = \mathfrak{g}(e)(\lambda_+) \in \text{Car} \, \mathfrak{g}(e)(\lambda)$. Let $\alpha' \in R(\mathfrak{g}(e)(\lambda)_{\mathbb{C}}, \mathfrak{h}(e)_{\mathbb{C}})$ be imaginary. There exists $\alpha \in R(\mathfrak{g}(\lambda)_{\mathbb{C}}, \mathfrak{h}_{\mathbb{C}})$ such that $\alpha' = \alpha_{|\mathfrak{h}(e)_{\mathbb{C}}}$.
We have: $\langle \nu_+, \alpha \rangle = \langle \nu_+, \alpha' \rangle = 0$
and therefore $\alpha \in R(\mathfrak{g}(\lambda)_{\mathbb{C}}, \mathfrak{h}_{\mathbb{C}}) \cap R(\mathfrak{g}(\nu_+)_{\mathbb{C}}, \mathfrak{h}_{\mathbb{C}})$. Furthermore $\langle \rho_{\mathcal{F}+}, \alpha' \rangle = \langle \rho_{\mathcal{F}+}, \alpha \rangle$,
which gives $\langle \rho_{\mathcal{F}+}, \alpha' \rangle \neq 0$ (see 1.3 (b)).

(b) Let $B$ be the base of $R(\mathfrak{C}_g(\mathfrak{a})(\lambda)_{\mathbb{C}}, \mathfrak{h}_{\mathbb{C}})$ associated with $\mathcal{F}^+$. For each $\alpha \in B$, let $m_{\hat{\alpha}}$ denote the cardinality of the orbit $\hat{\alpha}$ of $\alpha$ under the action of
the subgroup $\langle e \rangle$ of $G$ generated by $e$ on $B$, and choose $t_{\dot{\alpha}} \in \mathbb{R}$ such that
$$((\mathrm{Ad} e^{\mathbb{C}})^{m_{\dot{\alpha}}})_{\mathfrak{g}_{\mathbb{C}}^{\alpha}} = e^{i t_{\dot{\alpha}}} \mathrm{id}.$$ Take $e_0 = e \exp(X_0)$, where $X_0 \in \mathfrak{h}_{\mathbb{C}} \cap \mathrm{D}(\mathrm{C}_{\mathfrak{g}}(\mathfrak{a})(\lambda)_{\mathbb{C}})$ is such that $\alpha(X_0) = -i t_{\dot{\alpha}}/m_{\dot{\alpha}}$ for all $\alpha \in B$. It is clear that $\tilde{\lambda}[e_0]$ exists.
Each $\alpha|_{\mathfrak{b}(e)_{\mathbb{C}}}$ with $\alpha \in B$ is a weight of $\mathfrak{h}(e)_{\mathbb{C}}$ in $\mathfrak{g}(e_0)_{\mathbb{C}}$ whose eigenspace contains $(1+e_0+\cdots+e_0^{m_{\dot{\alpha}}-1}) \cdot \mathfrak{g}_{\mathbb{C}}^{\alpha}$. Let $\mathcal{F}'^+ \in \mathcal{F}(\mathfrak{h})(e)_{\mathbb{C}}$ be such that $\rho_{\mathcal{F}'^+} \in \mathcal{F}^+[e_0]$, hence $e_0\mathcal{F}'^+ = \mathcal{F}'^+$. For all $\alpha \in B$, we have: $\langle \rho_{\mathcal{F}^+}, \alpha|_{\mathfrak{b}(e)_{\mathbb{C}}} \rangle = \langle \rho_{\mathcal{F}^+}, \alpha \rangle > 0$ and $\langle \rho_{\mathcal{F}'^+}, \alpha \rangle = \langle \rho_{\mathcal{F}'^+}, \alpha|_{\mathfrak{b}(e)_{\mathbb{C}}} \rangle$, so the root $\alpha|_{\mathfrak{b}(e)_{\mathbb{C}}}$ of $R(\mathrm{C}_{\mathfrak{g}(e_0)}(\mathfrak{a}(e))(\lambda)_{\mathbb{C}}, \mathfrak{h}(e)_{\mathbb{C}})$ is positive with respect to $\mathcal{F}^+[e_0]$, therefore $\langle \rho_{\mathcal{F}'^+}, \alpha \rangle > 0$. It follows that $\mathcal{F}'^+ = \mathcal{F}^+$.

Now suppose that $\tilde{\lambda} \in \tilde{\mathfrak{g}}_I^*(e)$ is a restriction of $\lambda$ with $\tilde{\lambda}[e] \in \tilde{\mathfrak{g}}_I^*(e)_{\mathbb{C}}^*$.
Every root of $(\mathfrak{g}(e_0)(\lambda)_{\mathbb{C}}, \mathfrak{h}(e)_{\mathbb{C}})$ is a restriction of a root of $(\mathfrak{g}(\lambda)_{\mathbb{C}}, \mathfrak{h}_{\mathbb{C}})$. The roots $\alpha' = \alpha|_{\mathfrak{b}(e)_{\mathbb{C}}}$ with $\alpha \in \langle e \rangle \setminus B$ therefore describe as $\tilde{\lambda}$ varies (injectively because $\dim \mathfrak{g}(e_0)_{\mathbb{C}}^{\alpha'} = 1$), the set of simple roots with respect to $\mathcal{F}^+[e_0]$ in $R(\mathfrak{g}(e_0)(\lambda)_{\mathbb{C}}, \mathfrak{h}(e)_{\mathbb{C}})$. These roots $\alpha'$ are noncompact because $\alpha(\langle \cdot, \cdot \rangle)$ is nonnegative on the vector subspace $\mathfrak{g}(e_0)_{\mathbb{C}}^{\alpha'}$ of the orthogonal direct sum $\mathfrak{g}_{\mathbb{C}}^{\alpha} \oplus e_0 \mathfrak{g}_{\mathbb{C}}^{\alpha} \oplus \cdots \oplus e_0^{m_{\dot{\alpha}}-1} \mathfrak{g}_{\mathbb{C}}^{\alpha}$.

(c) The map $\pi_e : W(\mathfrak{g}(\lambda)_{\mathbb{C}}, \mathfrak{h}_{\mathbb{C}}) \cdot \rho_{\mathcal{F}^+} \cap \mathfrak{h}(e)_{\mathbb{C}}^* \to C(\mathfrak{g}(e)(\lambda), \mathfrak{h}(e))$ that sends a linear form to the chamber that contains it, commutes with the action of the group $W(\mathfrak{g}(e)(\lambda)_{\mathbb{C}}, \mathfrak{h}(e)_{\mathbb{C}})$. The groups $W(\mathfrak{g}(\lambda)_{\mathbb{C}}, \mathfrak{h}_{\mathbb{C}})(\mathrm{Ad}^* e)$ and $W(\mathfrak{g}(e)(\lambda)_{\mathbb{C}}, \mathfrak{h}(e)_{\mathbb{C}})$ act simply transitively on the domain and codomain of $\pi_e$ respectively. The fibers of $\pi_e$ therefore have cardinality $\frac{|W(\mathfrak{g}(\lambda)_{\mathbb{C}}, \mathfrak{h}_{\mathbb{C}})(\mathrm{Ad}^* e)|}{|W(\mathfrak{g}(\lambda)(e)_{\mathbb{C}}, \mathfrak{h}(e)_{\mathbb{C}})|}$. The set of $\tilde{\lambda}' = (\lambda', \mathcal{F}'^+)$ with $\mathcal{F}'^+ \in C(\mathfrak{g}(\lambda), \mathfrak{h})$ such that $e^{\tilde{\lambda}'} = \tilde{\lambda}'$ and $\tilde{\lambda}'[e] = \tilde{\lambda}[e]$, whose elements satisfy $\lambda' = \lambda$ and $\mathfrak{g}(\tilde{\lambda}') = \mathfrak{C}_{\mathfrak{g}}(\mathfrak{g}(e)(\tilde{\lambda}'[e])) = \mathfrak{h}$, is in bijection with $\pi_e^{-1}(\{\mathcal{F}^+[e]\})$ via the map $(\lambda', \mathcal{F}'^+) \mapsto \pi_1^{-1}(\mathcal{F}'^+)$. This provides its cardinal. $\square$

**Remark 3.3.** (1) It can happen that $\tilde{\lambda} \in \tilde{\mathfrak{g}}_I^*(e)$ satisfies $\tilde{\lambda}[e] \notin \tilde{\mathfrak{g}}(e)_I^*$. (However, the condition $\tilde{\lambda} \in \tilde{\mathfrak{g}}_{\mathrm{Inc}}^*(e)$ implies $\tilde{\lambda}[e] \in \tilde{\mathfrak{g}}(e)_{\mathrm{Inc}}^*$.) For example: $G = \mathrm{Sp}(4, \mathbb{R})$, $\tilde{\lambda} = (0, \mathcal{F}^+)$ such that $\mathfrak{h} := \mathfrak{g}(\tilde{\lambda})$ is equal to its infinitesimally elliptic component, and the system of positive roots of $(\mathfrak{g}_{\mathbb{C}}, \mathfrak{h}_{\mathbb{C}})$ associated with $\mathcal{F}^+$ is written $\{ \alpha_1, \alpha_2, \alpha_1+\alpha_2, 2\alpha_1+\alpha_2 \}$ with the only compact root $\alpha_1+\alpha_2$. Taking $e = \exp E$, where $E \in \mathfrak{h}$ is determined by $\alpha_1(E) = -\alpha_2(E) = i\pi$, the Lie algebra $\mathfrak{g}(e)$ is isomorphic to $\mathfrak{u}(2)$ and therefore $C(\mathfrak{g}(e), \mathfrak{h}(e))_{\mathrm{reg}} = \emptyset$.

(2) We can also find $\tilde{\lambda}_e \in \tilde{\mathfrak{g}}(e)_{\mathrm{reg}}^*$ that is not of the form $\tilde{\lambda}[e]$ for a $\tilde{\lambda} \in \tilde{\mathfrak{g}}_{\mathrm{reg}}^*(e)$. For example: $G = \mathrm{SU}(2)$, $e = \begin{pmatrix} 0 & -1 \\ 1 & 0 \end{pmatrix}$ and $\tilde{\lambda}_e = (0, \mathrm{iso}(2)^*)$.

(3) For certain $\tilde{\lambda} \in \tilde{\mathfrak{g}}_{\mathrm{fund}}^*(e)$ there is no $f \in \mathfrak{g}_{\mathrm{reg}}^*(e)$ such that $G \cdot f = R_G(G \cdot \tilde{\lambda})$. This is the case, with $G = \mathrm{SL}(2, \mathbb{R})$ and $e = \begin{pmatrix} 0 & -1 \\ 1 & 0 \end{pmatrix}$, for elements of $\tilde{\mathfrak{g}}_{\mathrm{fund}}^*(e)$ of the form $(0, \mathcal{F}^+)$ such that $\mathcal{F}^+ \subseteq \mathrm{iso}(2)^*$. $\blacksquare$

## II. The “projective representation” parameters

The parameters in this section are based on those introduced by Duflo, which were suitable for the case of regular semisimple coadjoint orbits of $G$.

## 4. About special-metalinear and metaplectic groups

My reference regarding the metaplectic group and the orientation functions of Duflo and Vergne attached to it is their paper [DV 93]. Like them, I choose to
avoid using the character of the metaplectic representation, following the viewpoint of [Ver 94, Conj. p. 291] (except for the sign of the symplectic form on coadjoint orbits).

**Definition 4.1.** Consider a finite-dimensional real vector space $V$.

(a) Denote by $DL(V) \to SL(V)$ “the” double cover of $SL(V)$, unique up to isomorphism of coverings, which is connected when $\dim V \ge 2$. (Item (b) below will provide a canonical description of such a cover, as a set of ordered pairs consisting of an element of $SL(V)$ and an orientation of a certain vector subspace of $V$.)

(b) Let $\hat{a} \in DL(V)$ lie over an elliptic $a \in SL(V)$.

When $\dim V \ge 2$, set

$$ \mathcal{O}(\hat{a})_{(1-a) \cdot V} = (-1)^{\frac{\alpha_1 + \dots + \alpha_p}{2\pi}} \operatorname{sg}(\sin(\frac{\beta_1}{2}) \dots \sin(\frac{\beta_q}{2})) \times \mathbb{R}_+ \setminus \{0\} (w_1 \wedge \dots \wedge w_{2q}), $$

where “sg” denotes the sign function on $\mathbb{R} \setminus \{0\}$, independently of the choice of an infinitesimally elliptic $A \in sl(V)$ such that $\hat{a} = \exp_{DL(V)} A$, and of a basis $(v_1, \dots, v_{2p}, w_1, \dots, w_{2q})$ of $A \cdot V$ in which the matrix of the restriction of $A$ has the form

$$ (**)^{\begin{pmatrix} (0 & -\alpha_1) \\ \alpha_1 & 0 \end{pmatrix} \cdots (\begin{pmatrix} 0 & -\alpha_p \\ \alpha_p & 0 \end{pmatrix} \cdots (\begin{pmatrix} 0 & -\beta_1 \\ \beta_1 & 0 \end{pmatrix} \cdots (\begin{pmatrix} 0 & -\beta_q \\ \beta_q & 0 \end{pmatrix})} $$

with $\alpha_1, \dots, \alpha_p \in 2\pi\mathbb{Z} \setminus \{0\}$ and $\beta_1, \dots, \beta_q \in \mathbb{R} \setminus 2\pi\mathbb{Z}$.

When $\dim V \le 1$, $\mathcal{O}(\hat{a})_{\{0\}}$ is equal to $\mathbb{R}_+ \setminus \{0\}$ or $\mathbb{R}_- \setminus \{0\}$ depending on whether $\hat{a}$ is trivial or not.

(c) Let $A \in sl(V)$ be infinitesimally elliptic. Set

$$ \mathcal{O}(A)_{A \cdot V} = \operatorname{sg}(\beta_1 \dots \beta_q) \times \mathbb{R}_+ \setminus \{0\} (w_1 \wedge \dots \wedge w_{2q}), $$

independently of the choice of a basis $(w_1, \dots, w_{2q})$ of $A \cdot V$ in which the matrix of the restriction of $A$ has the form

$$ \begin{pmatrix} (0 & -\beta_1) \\ \beta_1 & 0 \end{pmatrix} \cdots \begin{pmatrix} 0 & -\beta_q \\ \beta_q & 0 \end{pmatrix} $$

with $\beta_1, \dots, \beta_q \in \mathbb{R} \setminus \{0\}$.

**Definition 4.2.** Consider a finite-dimensional real vector space $V$ equipped with a nondegenerate alternating bilinear form $B$. We also denote by $B$ the complex bilinear extension of this bilinear form to $V_C$.

(a) Denote by $Mp(V)$ the inverse image of $Sp(V)$ in $DL(V)$.

(b) Denote by $\mathcal{O}(B)_V$ the orientation of $V$ (which means some $\mathbb{R}_+ \setminus \{0\} \omega_0$ with $\omega_0 \in \Lambda^{\max} V \setminus \{0\}$) on which $B^{\frac{1}{2} \dim V}$ is positive. A “symplectic basis” of $(V, B)$ is any basis $(P_1, \dots, P_n, Q_1, \dots, Q_n)$ of $V$ such that $B(P_i, Q_j) = \delta_{i,j}$ (Kronecker symbol) and $B(P_i, P_j) = B(Q_i, Q_j) = 0$ for $1 \le i, j \le n$.

(c) A “Lagrangian” of $(V_C, B)$ is any complex vector subspace of $V_C$ equal to its orthogonal with respect to $B$. A “positive Lagrangian” of $(V_C, B)$ is any Lagrangian of $(V_C, B)$ on which the Hermitian form $(v, w) \mapsto i B(v, \bar{w})$ is nonnegative.
Let $\mathcal{L}$ be a Lagrangian of $(V_{\mathbb{C}}, B)$. Let $Mp(V)_{\mathcal{L}}$ denote the normalizer of $\mathcal{L}$ in $Mp(V)$. To each $\hat{x} \in Mp(V)_{\mathcal{L}}$ lying over an element $x$ of $Sp(V)$, associate the number $n_{\mathcal{L}}(x)$ of eigenvalues counted with multiplicity in $]1, +\infty[$ of the restriction of $x^{\mathcal{C}}$ to $\mathcal{L}$, and the number $q_{\mathcal{L}}(x)$ of (strictly) negative coefficients in the matrix of the Hermitian form $(v, w) \mapsto iB(v, \bar{w})$ on $(1-x^{\mathcal{C}}) \cdot \mathcal{L}$ with respect to an orthogonal basis.

(d) Let $\mathcal{L}$ be a Lagrangian of $(V_{\mathbb{C}}, B)$. For every $\hat{x} \in Mp(V)_{\mathcal{L}}$ whose elliptic component is $\hat{x}_e$ (in the sense of [DV 93, Lem. 31 p. 38]) lying over an element $x$ of $Sp(V)$ whose elliptic component is $x_e$, set

$$ \rho_{\mathcal{L}}(\hat{x}) = (-1)^{q_{\mathcal{L}}(x_e)} \frac{\mathcal{O}(\hat{x}_e)_{(1-x_e) \cdot V}}{\mathcal{O}(B)_{(1-x_e) \cdot V}} \prod_{1 \le k \le n} (\sqrt{r_k} e^{i \theta_k / 2}), $$

where $r_1 e^{i \theta_1}, \dots, r_n e^{i \theta_n}$ are the eigenvalues counted with multiplicity of the restriction of $x^{\mathcal{C}}$ to $\mathcal{L}$, with $r_1, \dots, r_n \in \mathbb{R}_+ \setminus \{0\}$ and $\theta_1, \dots, \theta_n \in ]-2\pi, 0]$.

(e) Let $\hat{x} \in Mp(V)$ be of elliptic component $\hat{x}_e$ lying over a semisimple element $x$ of $Sp(V)$ with elliptic component $x_e$. Set

$$ \delta(\hat{x}) = \frac{\mathcal{O}(\hat{x}_e)_{(1-x_e) \cdot V}}{\mathcal{O}(B)_{(1-x_e) \cdot V}} \prod_{1 \le k \le n} e^{i \theta_k / 2}, $$

independently (given the proof of (c) of the proposition below) of the choices of an infinitesimally elliptic $E \in \mathfrak{sp}(V)$ such that $x_e = \exp_{\mathfrak{Sp}(V)} E$, and of a symplectic basis $(P_1, \dots, P_n, Q_1, \dots, Q_n)$ of $(V, B)$ for which the matrix of $E$ relative to $(P_1, Q_1, \dots, P_n, Q_n)$ has the form

$$ \begin{pmatrix} \begin{pmatrix} 0 & \theta_1 \\ -\theta_1 & 0 \end{pmatrix} & \cdots & \begin{pmatrix} 0 & \theta_n \\ -\theta_n & 0 \end{pmatrix} \end{pmatrix} $$

with $\theta_1, \dots, \theta_n \in ]-2\pi, 0]$.

(f) Let $\hat{x} \in Mp(V)$ be of elliptic component $\hat{x}_e$ lying over a semisimple element $x$ of $Sp(V)$ with elliptic component $x_e$. We set

$$ \Phi(\hat{x}) = \frac{\mathcal{O}(\hat{x}_e)_{(1-x_e) \cdot V}}{\mathcal{O}(B)_{(1-x_e) \cdot V}} i^{-\frac{1}{2} \dim(1-x_e) \cdot V} |\det (1-x)_{(1-x) \cdot V}|^{-1/2}. $$

We will now see, as Vergne suggests in [Ver 94, Prop. p. 289] (with the Weil representation contragredient to the one used in [DHV 84], derived from that of [DHV 84] by replacing, at will $B$ with $-B$ or the central character of the Heisenberg group with its conjugate), that the functions introduced in points (d), (e), (f) are the functions $\rho_{\mathcal{L}}$ and $\delta$ of [Duf 82a] and the function $\Phi$ of [DHV 84].

**Proposition 4.3.** Consider a finite-dimensional real vector space $V$ endowed with a nondegenerate alternating bilinear form $B$.

(a) The cover $Mp(V) \to Sp(V)$ is connected when $V \neq \{0\}$.

(b) Let $\mathcal{L}$ be a Lagrangian of $(V_{\mathbb{C}}, B)$. The function $\rho_{\mathcal{L}}$ is a Lie group homomorphism from $Mp(V)_{\mathcal{L}}$ to $\mathbb{C} \setminus \{0\}$ such that $d_1 \rho_{\mathcal{L}} = \frac{1}{2} \text{tr}(\cdot^{\mathcal{C}})_{\mathcal{L}}$. For every $\hat{x} \in Mp(V)_{\mathcal{L}}$ lying over a semisimple element $x$ of $Sp(V)$, we have

$$ \Phi(\hat{x}) = (-1)^{n_{\mathcal{L}}(x) + q_{\mathcal{L}}(x)} \rho_{\mathcal{L}}(\hat{x}) (\det (1 - x^{\mathcal{C}})_{(1-x^{\mathcal{C}}) \cdot \mathcal{L}})^{-1}. $$

(c) Let $\hat{x} \in Mp(V)$ lie over an $x \in Sp(V)$ with elliptic component $x_e$. There exists a positive Lagrangian of $(V_{\mathbb{C}}, B)$ stable under $x^{\mathcal{C}}$. For every Lagrangian $\mathcal{L}$ of $(V_{\mathbb{C}}, B)$ stable under $x^{\mathcal{C}}$, we have
$$ \rho_{\mathcal{L}}(\hat{x}) |\rho_{\mathcal{L}}(\hat{x})|^{-1} = \delta(\hat{x}) \prod_{z \in \text{Sp}(x_e)} z^{q_z}, $$

where $\text{Sp}(x_e)$ is the spectrum of $x_e$ and, for each $z \in \text{Sp}(x_e)$, $(p_z, q_z)$ is the signature $(n_+, n_-)$ of the Hermitian form $(v, w) \mapsto i B(v, \bar{w})$ on the eigenspace of $(x_e^\mathbb{C})_\mathcal{L}$ associated with $z$.

**Proof of the proposition.** (a) Assume $V$ is nonzero. Fix a symplectic basis $(P_1, \dots, P_n, Q_1, \dots, Q_n)$ of $(V, B)$. Let $T_{Sp}$ denote the maximal torus of $Sp(V)$ consisting of endomorphisms of $V$ whose matrices in the basis $(P_1, Q_1, \dots, P_n, Q_n)$ are block diagonal with blocks in $SO(2)$, and let $t$ denote its Lie algebra. The torus $T_{Mp} := \exp_{Mp(V)} t$ is a maximal torus of $DL(V)$.

Choose an element $A$ of $\text{Ker} \exp_{T_{Sp}}$ such that $\mathcal{o}(\exp_{DL(V)} A)_{\{0\}} = \mathbb{R}_- \setminus \{0\}$ (it exists). The path $t \in [0, 1] \mapsto \exp_{Mp}(tA)$ connects, in $Mp(V)$ the two points of the kernel of the canonical group homomorphism from $Mp(V)$ to $Sp(V)$.

(b) Let $\hat{x} \in Mp(V)_\mathcal{L}$ lie over a semisimple element $x$ of $Sp(V)$ with elliptic component $x_e$.

Let $\tilde{\rho}_\mathcal{L}$ denote the Lie group homomorphism from $Mp(V)_\mathcal{L}$ to $\mathbb{C} \setminus \{0\}$ defined in [Duf 84, p. 107]. According to [Duf 84, (10) p. 108], we have

$$ |\tilde{\rho}_{\mathcal{L}}(\hat{x})| = |\det(x^{\mathbb{C}})_{\mathcal{L}}|^{1/2} = |\rho_{\mathcal{L}}(\hat{x})| \quad \text{and} \quad d_1 \tilde{\rho}_{\mathcal{L}} = \frac{1}{2} \text{tr}(\cdot^{\mathbb{C}})_{\mathcal{L}}. $$

These equalities, in view of [DV 93, end of Lem. 30 p. 37], allow us to prove that the maps $\tilde{\rho}_\mathcal{L}$ and $\rho_\mathcal{L}$ commute with taking elliptic, positively hyperbolic, and unipotent components on the groups $Mp(V)_\mathcal{L}$ and $GL(\mathbb{C})$, and that they agree on each of the subsets of $Mp(V)_\mathcal{L}$ formed by its positively hyperbolic elements or its unipotent elements.

Let $\tilde{\Phi}$ denote the function (independent of $\mathcal{L}$) from the set of semisimple elements of $Mp(V)$ to $\mathbb{C} \setminus \{0\}$ defined in [DHV 84, p. 102]. It is written as

$$ \tilde{\Phi}(\hat{x}) = (-1)^{n_{\mathcal{L}}(x) + q_{\mathcal{L}}(x)} \tilde{\rho}_{\mathcal{L}}(\hat{x}) (\det (1 - x^{\mathbb{C}})_{(1-x^{\mathbb{C}}) \cdot \mathcal{L}})^{-1}. $$

Furthermore, the map $\tilde{v} \mapsto B(v, \cdot)$ from $(1-x^{\mathbb{C}}) \cdot V_{\mathbb{C}} / (1-x^{\mathbb{C}}) \cdot \mathcal{L}$ to $((1-x^{\mathbb{C}}) \cdot \mathcal{L})^*$ is a linear bijection that commutes with the action of $1-x$. It follows that

$$ |\tilde{\Phi}(\hat{x})| = |\det (1 - x)_{(1-x) \cdot V}|^{-1/2} = |\Phi(\hat{x})|. $$

To show that $\tilde{\Phi} = \Phi$ (respectively $\tilde{\rho}_\mathcal{L} = \rho_\mathcal{L}$), given [Duf 84, (9) p. 108] it remains to prove that $\tilde{\Phi}/|\tilde{\Phi}|$ and $\Phi/|\Phi|$ (respectively $\tilde{\rho}_\mathcal{L}$ and $\rho_\mathcal{L}$) coincide at one of the two points of $Mp(V)$ lying over $x$ (respectively $x_e$).

Let $V^z$ (respectively $V_{\mathbb{C}}^z$) denote the eigenspace of $x$ (respectively $x^{\mathbb{C}}$) associated with an eigenvalue $z \in \mathbb{R} \setminus \{0\}$ (respectively $z \in \mathbb{C} \setminus \{0\}$), and let $\perp^B$ denote the relation of $B$-orthogonality. Now consider an eigenvalue $z \in \mathbb{C} \setminus \{0\}$ of $x^{\mathbb{C}}$ such that $\text{Im } z \ge 0$ and $|z| \ge 1$. We will associate to it a certain symplectic basis $B_z$ of $((V_{\mathbb{C}}^z + V_{\mathbb{C}}^{z-1} + V_{\mathbb{C}}^{\bar{z}} + V_{\mathbb{C}}^{\bar{z}-1}) \cap V, B)$.

If $z \in \mathbb{R}$ and $|z| \ge 1$: when $|z| = 1$, fix a symplectic basis $B_z = (P_1^0, \dots, P_{n^0}^0, Q_1^0, \dots, Q_{n^0}^0)$ of $V^z$; when $|z| > 1$, fix a basis $(P_1^0, \dots, P_{n^0}^0)$ of $V^z$, and deduce from it a unique symplectic basis $B_z = (P_1^0, \dots, P_{n^0}^0, Q_1^0, \dots, Q_{n^0}^0)$ of $V^z \oplus V^{z-1}$ such that $Q_1^0, \dots, Q_{n^0}^0 \in V^{z-1}$, because $V^{z-1}$ identifies with $(V^z)^*$ by means of $B$ since $V^z \perp^B V^z$ and $V^{z-1} \perp^B V^{z-1}$.

If $\text{Im } z > 0$ and $|z| > 1$: fix a basis $(P_1 + iP_2, \dots, P_{2n-1} + iP_{2n})$ of $V_{\mathbb{C}}^z$ such that $P_1, \dots, P_{2n} \in V$; we deduce from it a unique symplectic basis $(P_1 + iP_2, \dots, P_{2n-1} + iP_{2n}, Q_1 - iQ_2, \dots, Q_{2n-1} - iQ_{2n})$ of $V_{\mathbb{C}}^z \oplus V_{\mathbb{C}}^{z-1}$ endowed with
$B/2$ such that $Q_1-iQ_2, \dots, Q_{2n-1}-iQ_{2n} \in V_C^{z-1}$ and $Q_1, \dots, Q_{2n} \in V$, because $V^{z-1}$ identifies with $(V^z)^*$ by means of $B$ since $V_C^z \perp^B V_C^z$ and $V_C^{z-1} \perp^B V_C^{z-1}$; we then obtain the symplectic basis $B_z := (P_1, \dots, P_{2n}, Q_1, \dots, Q_{2n})$ of $(V_C^z \oplus V_C^{z-1} \oplus V_C^{\bar{z}} \oplus V_C^{\bar{z}-1}) \cap V$, because $V_C^z \perp^B V_C^{\bar{z}}$, $V_C^{z-1} \perp^B V_C^{\bar{z}-1}$ and $V_C^z \perp^B V_C^{\bar{z}}$.

If $\operatorname{Im} z > 0$ and $|z| = 1$: fix a basis $(P'_1+iQ'_1, \dots, P'_p+iQ'_p, P''_1-iQ''_1, \dots, P''_q-iQ''_q)$ of $V_C^z$ such that $P'_1, Q'_1, \dots, P'_p, Q'_p, P''_1, Q''_1, \dots, P''_q, Q''_q \in V$, in which the nondegenerate Hermitian form $(v, w) \mapsto iB/2(v, \bar{w})$ has matrix $\begin{pmatrix} I_p & 0 \\ 0 & -I_p \end{pmatrix}$; it yields the symplectic basis $B_z := (P'_1, \dots, P'_p, P''_1, \dots, P''_q, Q'_1, \dots, Q'_p, Q''_1, \dots, Q''_q)$ of $(V_C^z \oplus V_C^{\bar{z}}) \cap V$, because $V_C^z \perp^B V_C^{\bar{z}}$ and $V_C^{\bar{z}} \perp^B V_C^{\bar{z}}$.

Note that the vector subspace $\mathcal{L}_x^+$ (respectively $\mathcal{L}_{x,e}^+$) of $V_C$ spanned by the various vectors $P_1^0, \dots, P_{n^0}^0$, and, $P_1, \dots, P_{2n}$, and, $P'_1 + iQ'_1, \dots, P'_p + iQ'_p, P''_1 + iQ''_1, \dots, P''_q + iQ''_q$ (respectively $P_1^0 + iQ_1^0, \dots, P_{n^0}^0 + iQ_{n^0}^0$, and, $P_1 + iQ_1, \dots, P_{2n} + iQ_{2n}$, and, $P'_1 + iQ'_1, \dots, P'_p + iQ'_p, P''_1 + iQ''_1, \dots, P''_q + iQ''_q$) as $z$ varies, is a positive Lagrangian of $(V_C, B)$ stable under $x^\mathbb{C}$ (respectively $x_e^\mathbb{C}$).

Let $E$ (respectively $H$) denote the element of $\mathfrak{sp}(V)$ that stabilizes the vector spaces $(V_C^z + V_C^{z-1} + V_C^{\bar{z}} + V_C^{\bar{z}-1}) \cap V$ with $z = re^{i\theta}$, $r \ge 1$, $-2\pi < \theta \le 0$, and whose restriction to such a space has, in the basis $B_z$, a block diagonal matrix whose blocks are the matrices

$(\begin{smallmatrix} 0 & \theta \\ -\theta & 0 \end{smallmatrix})$ (resp. $(\begin{smallmatrix} \ln r & 0 \\ 0 & -\ln r \end{smallmatrix})$) with respect to $(P_k^0, Q_k^0)$,

or, $(\begin{smallmatrix} 0 & \theta \\ -\theta & 0 \end{smallmatrix})$ (resp. $(\begin{smallmatrix} \ln r & \ln r \\ \ln r & -\ln r \end{smallmatrix})$) with respect to $(P_{2k-1}, P_{2k}, Q_{2k-1}, -Q_{2k})$,

or, $(\begin{smallmatrix} 0 & \theta \\ -\theta & 0 \end{smallmatrix})$ (resp. $(\begin{smallmatrix} 0 & 0 \\ 0 & 0 \end{smallmatrix})$) with respect to $(P'_k, Q'_k)$ and $(\begin{smallmatrix} 0 & -\theta \\ \theta & 0 \end{smallmatrix})$ (resp. $(\begin{smallmatrix} 0 & 0 \\ 0 & 0 \end{smallmatrix})$) with respect to $(P''_l, Q''_l)$.

The elements $\hat{x}_{E,H} = \exp_{Mp(V)} E \exp_{Mp(V)} H$ and $\hat{x}_E = \exp_{Mp(V)} E$ of $Mp(V)$ lie respectively over $x$ and $x_e$. By [Duf 84, (15) p. 109], we have:

$$ \frac{\tilde{\rho}_{\mathcal{L}_x^+}}{|\tilde{\rho}_{\mathcal{L}_x^+}|} (\hat{x}_{E,H}) = \tilde{\rho}_{\mathcal{L}_{x,e}^+} (\exp_{Mp(V)} E) = e^{\frac{1}{2} \operatorname{tr}(E^\mathbb{C})_{\mathcal{L}_{x,e}^+}}. $$

Hence we have the following equalities, which yield the result:

$$ \frac{\tilde{\Phi}}{|\tilde{\Phi}|}(\hat{x}_{E,H}) = (-1)^{n_{\mathcal{L}_x^+}(x)} e^{\frac{1}{2} \operatorname{tr}(E^\mathbb{C})_{\mathcal{L}_{x,e}^+}} \left( \frac{\det (1-x^\mathbb{C})_{(1-x^\mathbb{C}) \cdot \mathcal{L}_x^+}}{\det (1-x^\mathbb{C})_{(1-x^\mathbb{C}) \cdot \mathcal{L}_x^+}} \right)^{-1} = \dots = \frac{\Phi}{|\Phi|}(\hat{x}_{E,H}) \\ \text{and } \tilde{\rho}_{\mathcal{L}}(\hat{x}_E) = (-1)^{q_{\mathcal{L}}(x_e)} \frac{\Phi}{|\Phi|}(\hat{x}_E) \frac{\det (1-\exp E^\mathbb{C})_{(1-\exp E^\mathbb{C}) \cdot \mathcal{L}}}{\det (1-\exp E^\mathbb{C})_{(1-\exp E^\mathbb{C}) \cdot \mathcal{L}}} = \dots = \rho_{\mathcal{L}}(\hat{x}_E). $$

(c) The existence of a positive Lagrangian of $(V_C, B)$ stable under $x^\mathbb{C}$ is proven in [B.. 72, Cor. p. 82]. Assume $V$ is nonzero and denote by $\hat{x}_e$ the elliptic component of $\hat{x}$. By definition $\delta(\hat{x})$ is equal to $\rho_{\mathcal{C}(P_1+iQ_1)+\dots+\mathcal{C}(P_n+iQ_n)}(\hat{x}_e)$, with the notation of 4.2 (e). The equality [Duf 84, (15) p. 109] proves that this complex number is independent of the choice of $(P_1, \dots, P_n, Q_1, \dots, Q_n)$.

Consider a Lagrangian $\mathcal{L}$ of $(V_C, B)$ stable under $x^\mathbb{C}$ and a complement $\mathcal{S}$ of $\mathcal{L} \cap \bar{\mathcal{L}}$ in $\mathcal{L}$ that is stable under $x_e^\mathbb{C}$. Let $W$ denote the symplectic subspace of $(V, B)$ equal to the orthogonal complement of $\mathcal{S} + \bar{\mathcal{S}}$ in $V$. Fix a basis $(P_1^0, \dots, P_{n^0}^0, P_1^\mathbb{R}, \dots, P_{2m}^\mathbb{R})$ of the Lagrangian $\mathcal{L} \cap \bar{\mathcal{L}}$ of $(W_C, B)$, consisting of vectors of $V$, such that $P_1^0, \dots, P_{n^0}^0$ are eigenvectors of $x_e$ with real eigenvalues and $P_1^\mathbb{R}+iP_2^\mathbb{R}, \dots, P_{2m-1}^\mathbb{R}+iP_{2m}^\mathbb{R}$ are eigenvectors of $x_e^\mathbb{C}$ with nonreal eigenvalues. It extends to a symplectic basis $(P_1^0, \dots, P_{n^0}^0, P_1^\mathbb{R}, \dots, P_{2m}^\mathbb{R}, Q_1^0, \dots, Q_{n^0}^0, 2Q_1^\mathbb{R}, \dots, 2Q_{2m}^\mathbb{R})$ of $(W, B)$ such that $Q_1^0, \dots, Q_{n^0}^0$ and $Q_1^\mathbb{R}+iQ_2^\mathbb{R}, \dots, Q_{2m-1}^\mathbb{R}+iQ_{2m}^\mathbb{R}$ are eigen-
vectors of $x_e^\mathcal{C}$. Indeed, we can construct $(Q_1^\mathbb{R}, \dots, Q_{2m}^\mathbb{R})$ by induction on $m = \frac{1}{2} \dim(x_e^2 - 1) \cdot W$, associating to the vector $P := P_{2m-1}^\mathbb{R} + iP_{2m}^\mathbb{R}$ of $(x_e^2 - 1) \cdot (\mathcal{L} \cap \bar{\mathcal{L}})$ an eigenvector $Q = Q_{2m-1}^\mathbb{R} - iQ_{2m}^\mathbb{R}$ of $x_e^\mathcal{C}$ with $Q_{2m-1}^\mathbb{R}, Q_{2m}^\mathbb{R} \in (x_e^2 - 1) \cdot W$ such that $B(P, Q) = 1$ and $B(Q, \bar{Q}) = 0$ (the latter equality is achieved by replacing $Q$ with $Q - \frac{1}{2}B(Q, \bar{Q})\bar{P}$). Also fix a basis $(P'_1 + iQ'_1, \dots, P'_p + iQ'_p, P''_1 - iQ''_1, \dots, P''_q - iQ''_q)$ of $S$ consisting of eigenvectors of $x_e^\mathcal{C}$ with $P'_1, Q'_1, \dots, P'_p, Q'_p, P''_1, Q''_1, \dots, P''_q, Q''_q \in V$, in which the nondegenerate $x_e^\mathcal{C}$-invariant Hermitian form $(v, w) \mapsto iB/2(v, \bar{w})$ has matrix $\begin{pmatrix} I_p & 0 \\ 0 & -I_p \end{pmatrix}$.

The basis of $\mathcal{L}$ consisting of the eigenvectors $P_1^0, \dots, P_n^0, P_1^\mathbb{R} + iP_2^\mathbb{R}, P_1^\mathbb{R} - iP_2^\mathbb{R}, \dots, P_{2m-1}^\mathbb{R} + iP_{2m}^\mathbb{R}, P_{2m-1}^\mathbb{R} - iP_{2m}^\mathbb{R}$ and $P'_1 + iQ'_1, \dots, P'_p + iQ'_p, P''_1 - iQ''_1, \dots, P''_q - iQ''_q$ of $x_e^\mathcal{C}$ yields an expression for $\rho_{\mathcal{L}}(x_e)$. We calculate $\delta(\hat{x})$ using the symplectic basis $(P_1^0, \dots, P_n^0, P_1^+, \dots, P_{2m}^+, P_1', \dots, P'_p, P_1'', \dots, P''_q, Q_1^0, \dots, Q_{n0}^0, Q_1^+, \dots, Q_{2m}^+, Q_1', \dots, Q_p', Q_1'', \dots, Q_q'')$ of $(V, B)$, where $P_{2k-1}^+ + iQ_{2k-1}^+ := (P_{2k-1}^\mathbb{R} + iP_{2k}^\mathbb{R}) + i(Q_{2k-1}^\mathbb{R} + iQ_{2k}^\mathbb{R})$ and $P_{2k}^+ - iQ_{2k}^+ := -i(P_{2k-1}^\mathbb{R} + iP_{2k}^\mathbb{R}) - (Q_{2k-1}^\mathbb{R} + iQ_{2k}^\mathbb{R})$ for $1 \le k \le m$.

From this we can easily deduce the result. $\square$

**5. The parameters $\tau \in X_G^{\text{Ind}}(\tilde{\lambda})$**

In the following two sections, fix $\tilde{\lambda} = (\lambda, \mathcal{F}^+) \in \tilde{\mathfrak{g}}_{\text{reg}}^*$. Set $\mathfrak{h} = \mathfrak{g}(\tilde{\lambda})$. Let $\mathfrak{a}$ denote the hyperbolic component of $\mathfrak{h}$ and fix a chamber $\mathfrak{a}^{*+}$ of $(\mathfrak{g}(\lambda)(i\rho_{\mathcal{F}^+}), \mathfrak{a})$. Let $\mu$ and $\nu$ denote the infinitesimally elliptic and hyperbolic components of $\lambda$.

In order to provide a canonical parametrization of a subset of $\hat{G}$ that allows us to describe the Plancherel formula, Duflo introduced the objects in (a) of the definition below. When $\lambda \in \mathfrak{g}_{\text{ss reg}}^*$, these objects will be compatible with mine, because we can take $t \in [0, 1]$ by setting $\lambda_0 = \lambda$ in Lemma 5.2, and we will have $X_G^{\text{Ind}}(\tilde{\lambda}) = X_G^{\text{irr}}(\lambda)$ in Definition 5.5 (c).

**Definition 5.1.** (a) Let $f \in \mathfrak{g}^*$. We denote by $B_f$ the nondegenerate alternating bilinear form $(\dot{X}, \dot{Y}) \mapsto f([X, Y])$ on $\mathfrak{g}/\mathfrak{g}(f)$, $G(f)^{\mathfrak{g}/\mathfrak{g}(f)}$ the Lie subgroup of $G(f) \times DL(\mathfrak{g}/\mathfrak{g}(f))$ consisting of the ordered pairs $(x, \hat{a})$ such that $(\text{Ad } x)_{\mathfrak{g}/\mathfrak{g}(f)}$ is the image of $\hat{a}$ in $\text{SL}(\mathfrak{g}/\mathfrak{g}(f))$ (see 4.3 (a)), $\{1, \iota\}$ and $G(f)_0^{\mathfrak{g}/\mathfrak{g}(f)}$ the inverse images of $\{1\}$ and $G(f)_0$ under the canonical surjective Lie group homomorphism from $G(f)^{\mathfrak{g}/\mathfrak{g}(f)}$ to $G(f)$, $X_G^{\text{irr}}(f)$ the finite set of isomorphism classes of irreducible unitary representations $\tau$ of $G(f)^{\mathfrak{g}/\mathfrak{g}(f)}$ such that $\tau(\iota) = -\text{id}$ and $\tau(\exp X) = e^{i f(X)}$ id for $X \in \mathfrak{g}(f)$ (in this case $\tau$ is finite-dimensional).

(b) Denote by $G(\tilde{\lambda})^{\mathfrak{g}/\mathfrak{g}(\lambda)(i\rho_{\mathcal{F}^+})}$ the inverse image of $G(\tilde{\lambda})$ in $G(\lambda_{\text{can}})^{\mathfrak{g}/\mathfrak{g}(\lambda_{\text{can}})}$. We will also set $G(\tilde{\lambda})_0^{\mathfrak{g}/\mathfrak{h}} = G(\lambda_+)_0^{\mathfrak{g}/\mathfrak{h}}$ (see $G(\tilde{\lambda})_0 = G(\lambda_+)_0$ and $\mathfrak{g}(\lambda_+) = \mathfrak{h}$).

When $\mathfrak{a}^{*+} = \mathfrak{a}^*$, we have $\mathfrak{g}(\lambda)(i\rho_{\mathcal{F}^+}) = \mathfrak{h}$ and $G(\tilde{\lambda}) = G(\lambda_+)$, and in this case we will denote $G(\tilde{\lambda})^{\mathfrak{g}/\mathfrak{g}(\lambda)(i\rho_{\mathcal{F}^+})}$ and $G(\lambda_+)^{\mathfrak{g}/\mathfrak{h}}$ by $G(\tilde{\lambda})^{\mathfrak{g}/\mathfrak{h}}$.

In what follows, we will identify complex functions on $G(\lambda_+)$ with complex functions on the cover $G(\lambda_+)^{\mathfrak{g}/\mathfrak{h}}$ that are constant on the fibers.
**Lemma 5.2.** *Fix a real number $t \in ]0, 1]$. We use the positive root systems from Definition 1.3 (b) and the notation $\lambda_t$ from Lemma 1.4.*

We set $\mathcal{L}_{\lambda_t} = \mathfrak{h}_C \oplus \sum_{\alpha \in R^+(\mathfrak{g}_C, \mathfrak{h}_C)} \mathfrak{g}_C^\alpha$ and, $\mathfrak{m}' = \mathfrak{g}(\nu_+)$ in anticipation of Part III.

(a) We have $G(\lambda_t)^{\mathfrak{g}/\mathfrak{h}} = G(\lambda_+)^{\mathfrak{g}/\mathfrak{h}}$ and $\mathcal{L}_{\lambda_t}/\mathfrak{h}_C$ is a Lagrangian of $(\mathfrak{g}_C/\mathfrak{h}_C, B_{\lambda_t})$ stable under $G(\lambda_t)$. The complex function $\delta_{\lambda_t}^{\mathfrak{g}/\mathfrak{h}}$ on $G(\lambda_+)^{\mathfrak{g}/\mathfrak{h}}$ and the Lie group homomorphism $\rho_{\lambda_t}^{\mathfrak{g}/\mathfrak{h}}$ from $G(\lambda_+)^{\mathfrak{g}/\mathfrak{h}}$ to $\mathbb{C}\setminus\{0\}$ deduced from it (see 4.2 (e) and 4.3 (b)) are independent of $t$, and therefore equal to $\delta_{\lambda_+}^{\mathfrak{g}/\mathfrak{h}}$ and $\rho_{\lambda_+}^{\mathfrak{g}/\mathfrak{h}}$ respectively.
We have $d_1\rho_{\lambda_+}^{\mathfrak{g}/\mathfrak{h}} = \rho_{\mathfrak{g},\mathfrak{h}}$.

(b) Let $\hat{e} \in G(\lambda_+)^{\mathfrak{g}/\mathfrak{h}}$ lie over an elliptic element $e$ of $G(\lambda_+)$. We have

$$ \rho_{\lambda_+}^{\mathfrak{g}/\mathfrak{h}}(\hat{e}) = \delta_{\lambda_+}^{\mathfrak{g}/\mathfrak{h}}(\hat{e}) \times \det (\mathrm{Ad} e^{\mathbb{C}}) \sum_{\alpha' \in R_K^+(\mathfrak{g}_C, \mathfrak{h}_C)} \mathfrak{g}_C^{\alpha'} \times \prod_{\mathcal{O} \in \langle e \rangle \setminus \tilde{R}_C^+(\mathfrak{m}'_C, \mathfrak{h}_C)} (-1)^{m_\alpha - 1} u_{\mathcal{O}} $$

where $R_K^+(\mathfrak{g}_C, \mathfrak{h}_C)$ is the set of compact $\alpha' \in R^+(\mathfrak{g}_C, \mathfrak{h}_C)$, $\tilde{R}_C^+(\mathfrak{m}'_C, \mathfrak{h}_C)$ is the set of classes of complex $\beta \in R^+(\mathfrak{m}'_C, \mathfrak{h}_C)$ modulo the identification of $\beta$ with $-\bar{\beta}$, and, for each orbit $\mathcal{O}$ of an element $\{\beta, -\bar{\beta}\}$ of $\tilde{R}_C^+(\mathfrak{m}'_C, \mathfrak{h}_C)$ under the action of the subgroup $\langle e \rangle$ of $G$ generated by $e$, we denote by $m_\mathcal{O}$ the cardinality of $\mathcal{O}$ and by $u_\mathcal{O}$ the unique eigenvalue of $(\mathrm{Ad} e^{\mathbb{C}})^{m_\mathcal{O}}$ associated with an eigenvector that can be written $X_\beta - \bar{X}_{-\beta}$ for at least one $X_\beta \in \mathfrak{g}_C^\beta$ and one $X_{-\beta} \in \mathfrak{g}_C^{-\beta}$ satisfying $[X_\beta, X_{-\beta}] = H_\beta$.

**Proof of the lemma.** (a) The first equality follows from Lemma 1.4. We can see immediately that $\mathcal{L}_{\lambda_t}/\mathfrak{h}_C$ is a Lagrangian of $(\mathfrak{g}_C/\mathfrak{h}_C, B_{\lambda_t})$ stable under $G(\lambda_t)$.

Let $\varphi_t$ denote the Hermitian form $(v, w) \mapsto i B_{\lambda_t}(v, \bar{w})$ on $(\mathfrak{g}/\mathfrak{h})_C$. We will use the notation from statement (b). The vector space $\mathcal{L}_{\lambda_t}/\mathfrak{h}_C$ is the direct sum of the projections of the following vector spaces, which are pairwise orthogonal for $\varphi_t$ with the signature of the restricted Hermitian form $(n_+, n_-)$ specified in parentheses: the $\mathfrak{g}_C^{\alpha'}$ with $\alpha' \in R_K^+(\mathfrak{g}_C, \mathfrak{h}_C)$ (signature $(0,1)$), the $\mathfrak{g}_C^{\alpha''}$ with $\alpha'' \in R^+(\mathfrak{g}_C, \mathfrak{h}_C)$ imaginary noncompact (signature $(1,0)$), the $\mathfrak{g}_C^\beta \oplus \mathfrak{g}_C^{-\beta}$ with $\{\beta, -\bar{\beta}\} \in \tilde{R}_C^+(\mathfrak{m}'_C, \mathfrak{h}_C)$ (signature $(1,1)$), and the $\mathfrak{g}_C^\gamma$ with $\gamma \in R^+(\mathfrak{g}_C, \mathfrak{h}_C)$ outside $R^+(\mathfrak{m}'_C, \mathfrak{h}_C)$ (signature $(0,0)$).

Let $e$ be an elliptic element of $G(\lambda_+)$. By the above, the Hermitian form $\varphi_t$ is nondegenerate on $(\mathcal{L}_{\lambda_t} \cap \mathfrak{m}'_C)/\mathfrak{h}_C$ and vanishes everywhere on its orthogonal. Furthermore, $(\mathcal{L}_{\lambda_t} \cap \mathfrak{m}'_C)/\mathfrak{h}_C$ is an orthogonal direct sum, with respect to $\varphi_t$, of the image and kernel of $1 - \mathrm{Ad} e^{\mathbb{C}}$. Therefore, the restriction of $\varphi_t$ to each of them is nondegenerate. A continuity argument then proves that the signatures of these restrictions, and, a fortiori $q_{\mathcal{L}_{\lambda_t}}(e)$, are independent of $t$. Likewise, the orientation $\mathcal{o}(B_{\lambda_t})_{(1-\mathrm{Ad} e)^{(\mathfrak{g}/\mathfrak{h})}$ is independent of $t$. This can be clarified as in the proof of 5.4. We then use Definition 4.2 (d) and Proposition 4.3 (c).

The formula for $d_1\rho_{\lambda_+}^{\mathfrak{g}/\mathfrak{h}}$ is deduced from Proposition 4.3 (b).

(b) It is a matter of applying Proposition 4.3 (c) taking into account the proof in (a). We use the notation $\varphi_t$ from that proof.

Let $\mathcal{O}$ be the orbit under $\langle e \rangle$ of a $\{\beta, -\bar{\beta}\} \in \tilde{R}_C^+(\mathfrak{m}'_C, \mathfrak{h}_C)$. Fix $(X_\beta, X_{-\beta}) \in \mathfrak{g}_C^\beta \times \mathfrak{g}_C^{-\beta}$ such that $[X_\beta, X_{-\beta}] = H_\beta$. The condition "$X_\beta - \bar{X}_{-\beta}$ is an eigenvector of $(\mathrm{Ad} e^{\mathbb{C}})^{m_\mathcal{O}}$" is satisfied (for a unique eigenvalue of $(\mathrm{Ad} e^{\mathbb{C}})^{m_\mathcal{O}}$) when $e^{m_\mathcal{O}}\beta = \beta$, and is realized (again with uniqueness of the eigenvalue) by replacing $X_\beta$ and $X_{-\beta}$ with certain multiples of them when $e^{m_\mathcal{O}}\beta = -\bar{\beta}$. Under this condition, the
eigenvectors $\sum_{0 \le k \le m_O - 1} \zeta^{-k} (\mathrm{Ad} e^C)^k (X_\beta - \bar{X}_{-\beta})$ of $\mathrm{Ad} e^C$, where $\zeta$ ranges over the set of $m_O^{\mathrm{th}}$ roots of $u_O$, have projections in $\mathcal{L}_{\lambda_t}/\mathfrak{h}_C$ that are pairwise orthogonal for $\varphi_t$ with (strictly) negative “squares.” This leads to the stated formula. $\square$

**Remark 5.3.** Let $f \in \mathrm{Supp}_{\mathfrak{g}(\lambda)^*} (G(\lambda)_0 \cdot \tilde{\lambda})$.

Let $\xi$ denote the nilpotent component of $f$. There exists a unique Borel subalgebra $\mathfrak{b}$ of $\mathfrak{g}(\lambda)_C$ on which $\xi$ is zero. Indeed, by Lemma 1.5 (a) and [Kos 59, Cor. 5.3 p. 997 and Cor. 5.6 p. 1001], the element of $\mathrm{D}(\mathfrak{g}(\lambda))$ with which $\xi$ is identified using the bilinear form $\langle \cdot, \cdot \rangle$ belongs to a unique maximal subalgebra consisting of nilpotent elements of $\mathfrak{g}(\lambda)_C$, whose orthogonal complement with respect to $\langle \cdot, \cdot \rangle$ is suitable. The uniqueness of $\mathfrak{b}$ shows that $\bar{\mathfrak{b}} = \mathfrak{b}$.

Let $\mathfrak{n}_\lambda$ denote the sum of the eigenspaces for the action of $Z(\mathfrak{g}(\lambda))_C$ on $\mathfrak{g}_C$, whose weight $\alpha \in Z(\mathfrak{g}(\lambda))^*_C$ (always a restriction of a weight of $\mathfrak{h}_C$) satisfies: $\langle \nu, \alpha \rangle > 0$ or $(\langle \nu, \alpha \rangle = 0$ and $\mathrm{i}\langle \mu, \alpha \rangle > 0)$. Set $\mathcal{L}_f = \mathfrak{b} \oplus \mathfrak{n}_\lambda$. Note that $\mathcal{L}_f$ is a Borel subalgebra of $\mathfrak{g}_C$ by decomposing $\mathcal{L}_f$ into eigenspaces relative to a Cartan subalgebra of $\mathfrak{g}_C$ included in $\mathfrak{b}$.

The inclusions $[\mathfrak{b}, \mathfrak{b}] \subseteq \mathfrak{b} \subseteq \mathrm{Ker} \, \xi$ and $[\mathcal{L}_f, \mathfrak{n}_\lambda] \subseteq \mathfrak{n}_\lambda \subseteq (\mathfrak{g}(\lambda)_C)^{\perp(\cdot, \cdot)}$ provide the equalities $f([\mathfrak{b}, \mathfrak{b}]) = 0$ and $f([\mathcal{L}_f, \mathcal{L}_f]) = 0$. Therefore $\mathfrak{g}(f)_C \subseteq \mathfrak{b}$ and $\mathcal{L}_f/\mathfrak{g}(f)_C$ is a Lagrangian of $(\mathfrak{g}_C/\mathfrak{g}(f)_C, B_f)$, clearly stable under $G(f)$. By 4.3 (b) the Lie group homomorphism $\rho_f^{\mathfrak{g}/\mathfrak{g}(f)}$ from $G(f)^{\mathfrak{g}/\mathfrak{g}(f)}$ to $\mathbb{C}\setminus\{0\}$ deduced from it satisfies $\mathrm{d}_1\rho_f^{\mathfrak{g}/\mathfrak{g}(f)} = \frac{1}{2}\mathrm{tr}(\mathrm{ad} \cdot \mathbb{C})_{\mathcal{L}_f/\mathfrak{g}(f)_C}$. $\blacksquare$

I will now adapt Duflo's parameters to the situation that interests me.

**Lemma 5.4.** (a) The vector space $\mathcal{L}_{\tilde{\lambda}, \mathfrak{a}^{*+}} = \mathfrak{h}_C \oplus \sum_{\alpha \in R_{\tilde{\lambda}, \mathfrak{a}^{*+}}^+} \mathfrak{g}_C^\alpha$ (see 1.3 (c)) yields the Lagrangian $\mathcal{L}_{\tilde{\lambda}, \mathfrak{a}^{*+}}/\mathfrak{h}_C$ of $(\mathfrak{g}_C/\mathfrak{h}_C, B_{\lambda_+})$. Let $\rho_{\tilde{\lambda}, \mathfrak{a}^{*+}}^{\mathfrak{g}/\mathfrak{h}}$ denote the complex character of $G(\lambda_+)^{\mathfrak{g}/\mathfrak{h}}$ associated with it (see 4.3 (b)). The restriction of $\rho_{\tilde{\lambda}, \mathfrak{a}^{*+}}^{\mathfrak{g}/\mathfrak{h}}(\delta_{\lambda_+}^{\mathfrak{g}/\mathfrak{h}})^{-1}$ to $\mathrm{C}_{G_0}(\mathfrak{h})$ is independent of $\mathfrak{a}^{*+}$.

(b) The restriction of $B_{\lambda_+}$ to the vector subspace $\mathfrak{g}(\lambda)(\mathrm{i}\rho_{\mathfrak{F}^+})/\mathfrak{h}$ of $\mathfrak{g}/\mathfrak{h}$ is nondegenerate. The orthogonal complement of $\mathfrak{g}(\lambda)(\mathrm{i}\rho_{\mathfrak{F}^+})/\mathfrak{h}$ in $\mathfrak{g}/\mathfrak{h}$ endowed with $B_{\lambda_+}$ is the projection in $\mathfrak{g}/\mathfrak{h}$ of the intersection of $\mathfrak{g}$ with the sum of $\mathfrak{g}_C^\alpha$ as $\alpha$ ranges over $R(\mathfrak{g}_C, \mathfrak{h}_C) \setminus R(\mathfrak{g}(\lambda)(\mathrm{i}\rho_{\mathfrak{F}^+})_C, \mathfrak{h}_C)$. We will identify it with $\mathfrak{g}/\mathfrak{g}(\lambda)(\mathrm{i}\rho_{\mathfrak{F}^+})$.

Let $e \in G(\lambda_+)$ be elliptic. The orientations $\mathcal{O}(B_{\lambda_+})(1-\mathrm{Ad} e)(\mathfrak{g}/\mathfrak{g}(\lambda)(\mathrm{i}\rho_{\mathfrak{F}^+}))$ and $\mathcal{O}(B_{\lambda_{\mathrm{can}}})(1-\mathrm{Ad} e)(\mathfrak{g}/\mathfrak{g}(\lambda)(\mathrm{i}\rho_{\mathfrak{F}^+}))$ are equal.

(c) The vector space $\mathcal{L}_{\tilde{\lambda}} = \mathfrak{g}(\lambda)(\mathrm{i}\rho_{\mathfrak{F}^+})_C \oplus \sum_{\alpha \in R_{\tilde{\lambda}}^+} \mathfrak{g}_C^\alpha$ (see 1.3 (c)) yields the Lagrangian $\mathcal{L}_{\tilde{\lambda}}/\mathfrak{g}(\lambda)(\mathrm{i}\rho_{\mathfrak{F}^+})_C$ of $\mathfrak{g}_C/\mathfrak{g}(\lambda)(\mathrm{i}\rho_{\mathfrak{F}^+})_C$, both for $B_{\lambda_{\mathrm{can}}}$ and for $B_{\lambda_+}$. Let $\rho_{\tilde{\lambda}}^{\mathfrak{g}/\mathfrak{g}(\lambda)(\mathrm{i}\rho_{\mathfrak{F}^+})}$ denote the complex character of $G(\tilde{\lambda})^{\mathfrak{g}/\mathfrak{g}(\lambda)(\mathrm{i}\rho_{\mathfrak{F}^+})}$ associated with $(\mathfrak{g}_C/\mathfrak{g}(\lambda)(\mathrm{i}\rho_{\mathfrak{F}^+})_C, B_{\lambda_{\mathrm{can}}})$ (see 4.3 (b)). It extends the complex character of the inverse image of $G(\lambda_+)$ in $G(\tilde{\lambda})^{\mathfrak{g}/\mathfrak{g}(\lambda)(\mathrm{i}\rho_{\mathfrak{F}^+})}$ associated with $(\mathfrak{g}_C/\mathfrak{g}(\lambda)(\mathrm{i}\rho_{\mathfrak{F}^+})_C, B_{\lambda_+})$. **Proof of the lemma.** (a) Let $e \in \mathrm{C}_{G_0}(\mathfrak{h})$ be elliptic. The independence of $(\rho_{\tilde{\lambda}, \mathfrak{a}^{*+}}^{\mathfrak{g}/\mathfrak{h}}(\delta_{\lambda_+}^{\mathfrak{g}/\mathfrak{h}})^{-1})(e)$ with respect to $\mathfrak{a}^{*+}$ is obtained by resuming the proof of Lemma 5.2 (b).

(b) At the outset, we use the fact that $\mathfrak{g}_C/\mathfrak{h}_C$ is the $B_{\lambda_+}$-orthogonal direct sum of the projections of the vector subspaces $\mathfrak{g}_C^\alpha \oplus \mathfrak{g}_C^{-\alpha}$ of $\mathfrak{g}_C$ with $\alpha \in \mathbb{R}^+(\mathfrak{g}_C, \mathfrak{h}_C)$.
Set $V = \mathfrak{g}/\mathfrak{g}(\lambda)(i\rho_{\mathcal{F}^+})$ and $V(e) = \mathfrak{g}(e)/\mathfrak{g}(e)(\lambda)(i\rho_{\mathcal{F}^+})$. The vector space $V$ is the direct sum of the vector subspaces $V(e)$ and $(1-\mathrm{Ad}e)(V)$, which are simultaneously orthogonal for $B_{\lambda_{\mathrm{can}}}$ and for $B_{\lambda_+}$. For each $\alpha \in \mathbb{R}^+(\mathfrak{g}_{\mathbb{C}}, \mathfrak{h}_{\mathbb{C}})$, fix $X_{\alpha} \in \mathfrak{g}_{\mathbb{C}}^{\alpha}$ and $X_{-\alpha} \in \mathfrak{g}_{\mathbb{C}}^{-\alpha}$ such that $[X_{\alpha}, X_{-\alpha}] = H_{\alpha}$. Note that the orientations $\mathcal{o}(B_{\lambda_+})_V$ and $\mathcal{o}(B_{\lambda_{\mathrm{can}}})_V$ are both oriented by the exterior product of the following vectors: the $X_{\alpha} \wedge X_{-\alpha}$ with real $\alpha \in \mathbb{R}_{\lambda}^{+}$, the $i(X_{\alpha} \wedge X_{-\alpha})$ with imaginary $\alpha \in \mathbb{R}_{\lambda}^{+}$, and the $X_{\alpha} \wedge X_{-\alpha} \wedge \overline{X_{\alpha}} \wedge \overline{X_{-\alpha}}$ where $\dot{\alpha}$ ranges over the classes of complex $\beta \in R(\mathfrak{g}_{\mathbb{C}}, \mathfrak{h}_{\mathbb{C}}) \setminus R(\mathfrak{g}(\lambda)(i\rho_{\mathcal{F}^+})_{\mathbb{C}}, \mathfrak{h}_{\mathbb{C}})$ modulo the action of the group generated by conjugation and sign change. Likewise, we can prove that the orientations $\mathcal{o}(B_{\lambda_+})_{V(e)}$ and $\mathcal{o}(B_{\lambda_{\mathrm{can}}})_{V(e)}$ are equal, using the root system $R(\mathfrak{g}(e)_{\mathbb{C}}, \mathfrak{h}(e)_{\mathbb{C}})$ and the ordered pair $(\lambda, \rho_{\mathcal{F}^+})$. Therefore $\mathcal{o}(B_{\lambda_+})_{(1-\mathrm{Ad}e)(V)} = \mathcal{o}(B_{\lambda_{\mathrm{can}}})_{(1-\mathrm{Ad}e)(V)}$.

(c) Let $e \in G(\lambda_+)$ be elliptic and let $\hat{e} \in G(\tilde{\lambda})\mathfrak{g}/\mathfrak{g}(\lambda)(i\rho_{\mathcal{F}^+})$ lie over $e$. Let $\varphi_{\mathrm{can}}$ and $\varphi_+$ denote the Hermitian forms $(v, w) \mapsto i B_{\lambda_{\mathrm{can}}}(v, \bar{w})$ and $(v, w) \mapsto i B_{\lambda_+}(v, \bar{w})$ on $\mathfrak{g}_{\mathbb{C}}/\mathfrak{g}(\lambda)(i\rho_{\mathcal{F}^+})_{\mathbb{C}}$. The vector space $\mathcal{L} := \mathcal{L}_{\tilde{\lambda}}/\mathfrak{g}(\lambda)(i\rho_{\mathcal{F}^+})_{\mathbb{C}}$ is the direct sum of the vector subspaces $\mathcal{L}(e) := (\mathcal{L}_{\tilde{\lambda}} \cap \mathfrak{g}(e)_{\mathbb{C}})/\mathfrak{g}(e)(\lambda)(i\rho_{\mathcal{F}^+})_{\mathbb{C}}$ and $(1-\mathrm{Ad}e)(\mathcal{L})$, which are simultaneously orthogonal for $\varphi_{\mathrm{can}}$ and for $\varphi_+$.

We follow the beginning of the proof of Lemma 5.2 (b). The number of (strictly) negative coefficients in the matrix of either of the Hermitian forms $\varphi_{\mathrm{can}}$ and $\varphi_+$ on $\mathcal{L}$, with respect to an orthogonal basis, is equal to: half the number of compact roots in $R(\mathfrak{g}_{\mathbb{C}}, \mathfrak{h}_{\mathbb{C}}) \setminus R(\mathfrak{g}(\lambda)(i\rho_{\mathcal{F}^+})_{\mathbb{C}}, \mathfrak{h}_{\mathbb{C}})$ plus one quarter of the number of complex roots in $R(\mathfrak{g}(\nu)_{\mathbb{C}}, \mathfrak{h}_{\mathbb{C}}) \setminus R(\mathfrak{g}(\lambda)(i\rho_{\mathcal{F}^+})_{\mathbb{C}}, \mathfrak{h}_{\mathbb{C}})$. An analogous property holds for $\mathcal{L}(e)$. By (b) and Definition 4.2 (d), the functions $\rho$ associated with $B_{\lambda_{\mathrm{can}}}$ and $B_{\lambda_+}$ therefore take the same values at $\hat{e}$. $\square$

**Definition 5.5.** (a) Let $\alpha \in R(\mathfrak{g}_{\mathbb{C}}, \mathfrak{h}_{\mathbb{C}})$ be real. Set $n_{\alpha} = \frac{1}{2} \sum_{\beta} \beta(H_{\alpha})$ where the sum ranges over $\beta \in R(\mathfrak{g}_{\mathbb{C}}, \mathfrak{h}_{\mathbb{C}})$ such that $\beta + \bar{\beta} \in \mathbb{R}_{+} \alpha$. To two vectors $X_{\alpha} \in \mathfrak{g}_{\mathbb{C}}^{\alpha} \cap \mathfrak{g}$ and $X_{-\alpha} \in \mathfrak{g}_{\mathbb{C}}^{-\alpha} \cap \mathfrak{g}$ such that $[X_{\alpha}, X_{-\alpha}] = H_{\alpha}$, we associate the elliptic element $\gamma_{\alpha} = \exp(\pi(X_{\alpha} - X_{-\alpha}))$ of $C_{G_0}(\mathfrak{h})$. (We know that $n_{\alpha} \in \mathbb{N}$ and that the set $\{\gamma_{\alpha}, \gamma_{\alpha}^{-1}\}$ is independent of the choice of $(X_{\alpha}, X_{-\alpha})$, see [DV 88, p. 327].)

(b) Let $X_G^{irr,+}(\tilde{\lambda}, \mathfrak{a}^{*+})$ denote the finite set of isomorphism classes of irreducible unitary representations $\tau_+$ of $G(\lambda_+)^{\mathfrak{g}/\mathfrak{h}}$ such that $\tau_+(\iota) = -\mathrm{id}$ and $\tau_+(\exp X) = e^{i\lambda(X)}$ id for $X \in \mathfrak{h}$ (such a $\tau_+$ is of finite-dimensional), and let $X_G^{final,+}(\tilde{\lambda}, \mathfrak{a}^{*+})$ denote the set of $\tau_+ \in X_G^{irr,+}(\tilde{\lambda}, \mathfrak{a}^{*+})$ "final" in the sense that for every real root $\alpha \in R(\mathfrak{g}(\lambda)_{\mathbb{C}}, \mathfrak{h}_{\mathbb{C}})$ the endomorphism $(\delta_{\lambda_+}^{\mathfrak{g}/\mathfrak{h}}\tau_+)(\gamma_{\alpha})$ does not admit the eigenvalue $(-1)^{n_{\alpha}}$. (When $G$ is connected, the proof of 9.3 (b) will show that this condition can be replaced by $(\delta_{\lambda_+}^{\mathfrak{g}/\mathfrak{h}}\tau_+)(\gamma_{\alpha}) \neq (-1)^{n_{\alpha}}$ id.)

Let $X_G^{final,+}$ denote the set of $(\tilde{\lambda}', \mathfrak{a}'^{*+}, \tau_+')$ with $\tilde{\lambda}' = (\lambda', \mathcal{F}'^+) \in \tilde{\mathfrak{g}}_{\mathrm{reg}}^*, \mathfrak{a}'^{*+}$ is a chamber of $(\mathfrak{g}(\lambda')(i\rho_{\mathcal{F}'^+}), \mathfrak{a}')$ where $\mathfrak{a}'$ is the hyperbolic component of $\mathfrak{h}' := \mathfrak{g}(\tilde{\lambda}')$, and $\tau_+ \in X_G^{final,+}(\tilde{\lambda}', \mathfrak{a}'^{*+})$.

(c) Let $X_G(\tilde{\lambda})$ (resp. $X_G^{irr}(\tilde{\lambda})$) denote the set of isomorphism classes of unitary (resp. irreducible unitary) representations $\tau$ of $G(\tilde{\lambda})^{\mathfrak{g}/\mathfrak{g}(\lambda)(i\rho_{\mathcal{F}^+})}$ such that $\tau(\iota) = -\mathrm{id}$ and $\tau(\exp X) = e^{i\lambda(X)}$ id for $X \in \mathfrak{h}$, and let $X_G^{Ind}(\tilde{\lambda})$ denote the set (independent of the choice of $\mathfrak{a}^{*+}$) consisting of the isomorphism classes of unitary representations $\tau$ of $G(\tilde{\lambda})^{\mathfrak{g}/\mathfrak{g}(\lambda)(i\rho_{\mathcal{F}^+})}$ such that
$$ \frac{\rho_{\tilde{\lambda}}^{g/g(\lambda)(i\rho_{F^+})}}{|\rho_{\tilde{\lambda}}^{g/g(\lambda)(i\rho_{F^+})}|} \tau = \mathrm{Ind}_{G(\lambda_+)}^{G(\tilde{\lambda})} \left( \frac{\rho_{\tilde{\lambda}, \mathfrak{a}^{*+}}^{g/\mathfrak{h}}}{|\rho_{\tilde{\lambda}, \mathfrak{a}^{*+}}^{g/\mathfrak{h}}|} \tau_+ \right) $$

for some $\tau_+ \in X_G^{\mathrm{final},+}(\tilde{\lambda}, \mathfrak{a}^{*+})$ (hence $X_G^{\mathrm{Ind}}(\tilde{\lambda}) \subseteq X_G(\tilde{\lambda})$).

We also define: $X_G^{\mathrm{Ind}} = \{ (\tilde{\lambda}', \tau') ; \tilde{\lambda}' \in \tilde{\mathfrak{g}}_{\mathrm{reg}}^* \text{ and } \tau' \in X_G^{\mathrm{Ind}}(\tilde{\lambda}') \}$.

# 6. Integrability condition

We retain the notation from the previous section.

The following lemma is an adaptation of [Duf 82a, Rem. 2 p. 154]. It will not be used in the remainder of this paper.

**Lemma 6.1.** *The following properties are equivalent:*

(i) $X_G^{\mathrm{irr},+}(\tilde{\lambda}, \mathfrak{a}^{*+}) \neq \emptyset$;

(ii) there exists a (unique) unitary character $\chi_{\tilde{\lambda}}^G$ of $G(\tilde{\lambda})_0^{g/\mathfrak{h}}$ such that: $\chi_{\tilde{\lambda}}^G(t) = -1$ and $d_1\chi_{\tilde{\lambda}}^G = i\lambda|_{\mathfrak{h}}$;

(iii) $\tilde{\lambda} \in \tilde{\mathfrak{g}}_{\mathrm{reg},G}^*$.

**Proof of the lemma.** (i) $\Leftrightarrow$ (ii) The implication “(i) $\Rightarrow$ (ii)” is clear.

Assume (ii). The restriction to $G(\tilde{\lambda})_0^{g/\mathfrak{h}}$ of the unitary representation $\mathrm{Ind}_{G(\lambda_+)^{g/\mathfrak{h}}_0}^{G(\lambda_+)^{g/\mathfrak{h}}}$ of $G(\lambda_+)^{g/\mathfrak{h}}$ is nonzero (since it is induced from a nonzero space) and a multiple of $\chi_{\tilde{\lambda}}^G$. Theorem 8.5.2 of [Dix 64, p. 153] therefore shows the existence of an element of $X_G^{\mathrm{irr},+}(\tilde{\lambda}, \mathfrak{a}^{*+})$.

(ii) $\Leftrightarrow$ (iii) According to Lemma 5.2 (a), by multiplying $\chi_{\tilde{\lambda}}^G$ by $\rho_{\lambda_+}^{g/\mathfrak{h}}$ condition (ii) is equivalent to the existence of a complex character of the Lie group $G(\lambda_+)_0$ with differential $i\lambda+\rho_{g,\mathfrak{h}}$. This property can be written as: $\tilde{\lambda} \in \tilde{\mathfrak{g}}_{\mathrm{reg},G}^*$. $\square$

**Remark 6.2.** Let $f \in \mathrm{Supp}_{\mathfrak{g}(\lambda)_*}(G(\lambda)_0 \cdot \tilde{\lambda})$. In particular, we have $f \in \mathfrak{g}_{\mathrm{reg}}^*$.

The real Lie group $G(f)_0$ is abelian (see [B.. 72, Th. p. 17]). The stabilizer of the nilpotent component of $f$ in the adjoint group of $G(\lambda)_0$ is a unipotent linear algebraic subgroup by [Spr 66, Th. 5.9 (b) p. 138]. The maximal torus of $G(f)_0$ is therefore equal to that of $Z(G(\lambda)_0)$. Given Remark 5.3, the arguments of the proof of the previous lemma yield:

$$ X_G^{\mathrm{irr}}(f) \neq \emptyset \iff \forall Z \in \mathrm{Ker} \exp_{Z(G(\lambda)_0)} e^{(i\lambda + \rho_{g,\mathfrak{h}})(Z)} = 1. $$

Therefore, the condition “$X_G^{\mathrm{final},+}(\tilde{\lambda}, \mathfrak{a}^{*+}) \neq \emptyset$” implies the condition “$X_G^{\mathrm{irr}}(f) \neq \emptyset$.”

The converse is false, as shown by the case where $G = \mathrm{PSL}(2, \mathbb{R})$, $\lambda = 0$ and $\mathfrak{h} = \mathfrak{so}(2)$, therefore $X_G^{\mathrm{final},+}(\tilde{\lambda}, \{0\}) = \emptyset$, $X_G^{\mathrm{irr}}(f) \neq \emptyset$, and also $X_G^{\mathrm{irr}}(-f) \neq \emptyset$. This reflects the fact that $\mathrm{PSL}(2, \mathbb{R})$ has no limit of discrete series representation outside its discrete series representations. To generalize the orbit method proposed by Duflo, it was natural to imagine that the regular coadjoint orbits of $G$ that could be the orbit $G \cdot l$ associated with a representation $T_{l,\tau}^G$ of zero infinitesimal character would be $G \cdot f$ and $G \cdot (-f)$ (see 8.4 (a), replacing $(\tilde{\lambda}, \mathfrak{a}^{*+})$ and $\lambda$ with $l$). This generalization would have created difficulties, because $G \cdot f \neq G \cdot (-f)$ and the unitary dual of $G$ has only one element with zero infinitesimal character. ■
### III. Construction of representations

This section closely follows the paper [Duf 82a] by Duflo, pp. 160–180.

Given $\tilde{\lambda} = (\lambda, \mathcal{F}^+) \in \tilde{\mathfrak{g}}_{reg}^*$. Set $\mathfrak{h} = \mathfrak{g}(\tilde{\lambda})$. Let $\mu$ and $\nu$ (resp. $t$ and $a$) denote the infinitesimally elliptic and hyperbolic components of $\lambda$ (resp. of $\mathfrak{h}$). We also fix a chamber $a^{*+}$ of $(\mathfrak{g}(\lambda)(i\rho_{\mathcal{F}^+}), a)$ and $\tau_+ \in X_G^{final,+}(\tilde{\lambda}, a^{*+})$. This provides $R^+(\mathfrak{g}_C, \mathfrak{h}_C)$ and $\lambda_+ = \mu_++\nu_+$ (see 1.3 (b)). Set $\tilde{\mu} = (\mu, \mathcal{F}^+ \cap it^*)$.

## 7. The case where $G$ is connected

Throughout this section, we assume $G$ is connected.

The inclusion $\text{Ad}G \subseteq \text{int}\,\mathfrak{g}_C$ ensures that $G(\lambda_+)$ is equal to $C_G(\mathfrak{h})$, and that $\delta_{\lambda_+}^{g/\mathfrak{h}}$ is a unitary character of $G(\lambda_+)^{g/\mathfrak{h}}$ (see 4.3, and also [Duf 84, top of p. 118]). Indeed, for every connected component $a_0^{*+}$ of the set of elements of $a^*$ that do not vanish on any $H_\alpha$ with nonimaginary $\alpha \in R(\mathfrak{g}_C, \mathfrak{h}_C)$, the vector space

$$ \mathcal{L}^+ := \mathfrak{h}_C \oplus \sum_{\substack{\alpha \in R^+(\mathfrak{g}_C, \mathfrak{h}_C) \\ \alpha \text{ noncompact imaginary}}} \mathfrak{g}_C^\alpha \oplus \sum_{\substack{\alpha \in R^+(\mathfrak{g}_C, \mathfrak{h}_C) \\ \alpha \text{ compact}}} \mathfrak{g}_C^{-\alpha} \oplus \sum_{\substack{\alpha \in R(\mathfrak{g}_C, \mathfrak{h}_C) \\ a_0^{*+}(H_\alpha) \subseteq \mathbb{R}_+\{0\}}} \mathfrak{g}_C^\alpha $$

yields the positive Lagrangian $\mathcal{L}^+/\mathfrak{h}_C$ of $(\mathfrak{g}_C/\mathfrak{h}_C, B_{\lambda_+})$ stable under $G(\lambda_+)$.

Let $M$ denote the intersection of the kernels of the positive real characters of $C_G(a)$ and let $\mathfrak{m}$ be its Lie algebra (thus, we have $\tilde{\mu} \in \tilde{\mathfrak{m}}_{\tilde{\lambda},M}$ by 7.3 (a) and $X_M^{\text{irr}}(\tilde{\mu}) = X_M^{\text{final,+}}(\tilde{\mu}, \{0\})$), $R^+(\mathfrak{m}_C, \mathfrak{t}_C) = R(\mathfrak{m}_C, \mathfrak{t}_C) \cap R^+(\mathfrak{g}_C, \mathfrak{h}_C)$ (Definition 1.3 (b) relative to $\mathfrak{m}$ and $(\tilde{\mu}, \{0\})$), $\mu_{+,\mathfrak{m}} = \mu_{\mathfrak{m},\tilde{\mu},\{0\}}$ the element $\mu - 2i\rho_{\mathfrak{m},\mathfrak{t}}$ of $t^*$ (thus, we have $M(\tilde{\mu}) = C_M(t)$ and $\delta_{\mu_{+,\mathfrak{m}}}^{\mathfrak{m}/\mathfrak{t}}$ is a unitary character of $M(\tilde{\mu})^{\mathfrak{m}/\mathfrak{t}}$),

$$ \mathfrak{n}_M = \sum_{\alpha \in R^+(\mathfrak{m}_C, \mathfrak{t}_C)} \mathfrak{g}_C^\alpha \text{ and } \mathfrak{b}_M = \mathfrak{t}_C \oplus \mathfrak{n}_M, $$

$$ q = |\{\alpha \in R^+(\mathfrak{m}_C, \mathfrak{t}_C) \mid \alpha \text{ compact}\}|, $$

$$ \mathfrak{k}_M = \mathfrak{t} \oplus \left( \sum_{\substack{\alpha \in R(\mathfrak{m}_C, \mathfrak{t}_C) \\ \alpha \text{ compact}}} \mathfrak{g}_C^\alpha \right) \cap \mathfrak{g} \text{ and } K_{M_0} = \exp \mathfrak{k}_M. $$

**Definition 7.1.** (a) We denote by $T_{\tilde{\mu}}^{M_0}$ the “limit of discrete series representations of $M_0$” belonging to $\tilde{M}_0$, whose space of $K_{M_0}$-finite vectors is isomorphic to the image of the unitary character of $T_0$ (see . 2 (a)) with differential $i\mu - \rho_{\mathfrak{m},\mathfrak{t}}$ under the “cohomological induction” functor $\mathcal{R}_{M_0}^q := (\mathfrak{u}\mathcal{R}_{\mathfrak{b}_M, T_0}^{\mathfrak{m}_C, K_{M_0}})^q (\bullet \otimes \Lambda^{\max} \mathfrak{n}_M)$ described in [KV 95, (11.73) p. 677]. (It is denoted $\pi_{i\mu, b_M}$ in [KV 95, bottom of p. 734], taking into account [KV 95, Prop. 11.180 p. 733].)

(b) We set $T_{\tilde{\mu},\sigma}^M = \text{Ind}_{C_M(\mathfrak{m}),M_0}^M (\delta_{\mu_{+,\mathfrak{m}}}^{\mathfrak{m}/\mathfrak{t}} \sigma \otimes T_{\tilde{\mu}}^{M_0})$ for every $\sigma \in X_M^{\text{irr}}(\tilde{\mu})$, where we can define a representation $\delta_{\mu_{+,\mathfrak{m}}}^{\mathfrak{m}/\mathfrak{t}} \sigma \otimes T_{\tilde{\mu}}^{M_0}$ of $C_M(\mathfrak{m})M_0$ by the equality $(\delta_{\mu_{+,\mathfrak{m}}}^{\mathfrak{m}/\mathfrak{t}} \sigma \otimes T_{\tilde{\mu}}^{M_0})(xy) = (\delta_{\mu_{+,\mathfrak{m}}}^{\mathfrak{m}/\mathfrak{t}} \sigma)(x) \otimes T_{\tilde{\mu}}^{M_0}(y)$ for $x \in C_M(\mathfrak{m})$ and $y \in M_0$ (in which $(\delta_{\mu_{+,\mathfrak{m}}}^{\mathfrak{m}/\mathfrak{t}} \sigma)(x) = \sigma(x, 1)$), because the group $Z(M_0)$ operates in “the space” of $T_{\tilde{\mu}}^{M_0}$ via the unitary character $(\delta_{\mu_{+,\mathfrak{m}}}^{\mathfrak{m}/\mathfrak{t}} \sigma)|_{Z(M_0)}$ by [KV 95, (11.184c) p. 734].

**Remark 7.2.** Fix a maximal compact subgroup $K$ of $G$ whose Cartan involution normalizes $\mathfrak{h}$. Therefore $K_M := K \cap M$ is a maximal compact subgroup of
$M$ containing $C_M(\mathfrak{m})K_{M_0}$. Every $\pi \in (C_M(\mathfrak{m})M_0)^\wedge$ is related to certain representations $\pi_1 \in (C_M(\mathfrak{m}))^\wedge$ and $\pi_2 \in (M_0)^\wedge$ by the equalities $\pi(xy) = \pi_1(x) \otimes \pi_2(y)$ for $x \in C_M(\mathfrak{m})$ and $y \in M_0$ (see [Dix 64, Prop. 13.1.8 p. 251]). Let $\sigma \in X_M^{irr}(\tilde{\mu})$. By [KV 95, (11.187) p.735 and proof of Prop. 11.192 (a) p.737] and 5.2 (b) to pass from $M_0$ to $C_M(\mathfrak{m})M_0$, and [KV 95, Prop. 11.57 p. 672 and end of (5.8) p. 332] to pass from the “parabolic subgroup” $C_M(\mathfrak{m})M_0$ of $M$ to $M$, the $(\mathfrak{m}_C, K_M)$-module associated with $T_{\tilde{\mu},\sigma}^M$ is isomorphic to $\mathcal{R}_M^q((\rho_{\mu_{+,m}}^{m/t})^{-1}\sigma)$, where $\mathcal{R}_M^q$ is the cohomological induction functor relative to $\mathfrak{b}_M$. ■

**Lemma 7.3.** (a) We have $G(\lambda_+) = C_M(\mathfrak{m}) \exp \mathfrak{h}$ and $M(\tilde{\mu}) = C_M(\mathfrak{m}) \exp \mathfrak{t}$. There therefore exists a unique $\tau_M \in X_M^{irr}(\tilde{\mu})$ such that $\delta_{\mu_{+,m}}^{m/t} \tau_M = (\delta_{\lambda_+}^{g/\mathfrak{h}} \tau_+)|_{M(\tilde{\mu})}$. (This notation does not take into account the fact that $\tau_M$ depends on $\mathfrak{a}^{*+}$.)

(b) The representation class $\text{Ind}_{M.A.N}^G(T_{\tilde{\mu},\tau_M}^M \otimes e^{i\nu\circ\text{Log}} \otimes \mathbf{1}_N)$ constructed from the choice of a subgroup $N$ of $G$ which is the unipotent radical of a parabolic subgroup of $G$ with split component $A$, belongs to $\tilde{G}$ and is independent of $N$.

**Proof of the lemma.** (a) We have $G(\lambda_+) = C_G(\mathfrak{h}) \subseteq C_G(\mathfrak{a}) = MA$ and $M(\tilde{\mu}) = C_M(\mathfrak{t})$. Given [Var 77, Th. 17 p. 199], it follows that $M(\tilde{\mu}) = C_M(\mathfrak{m}) \exp \mathfrak{t}$, and then $G(\lambda_+) = C_M(\mathfrak{m}) \exp \mathfrak{h}$.

(b) Let $N$ be the unipotent radical of a parabolic subgroup of $G$ with split component $A$. By [Var 77, Th. 18 p. 289], there exists a connected component $\mathfrak{a}^{*+}_0$ of the set of elements of $\mathfrak{a}^*$ that do not vanish on any $H_\alpha$ with nonimaginary $\alpha \in R(\mathfrak{g}_C, \mathfrak{h}_C)$, such that

$$N = \exp \mathfrak{n} \text{ with } \mathfrak{n} = \left( \sum_{\alpha \in R(\mathfrak{g}_C, \mathfrak{h}_C) \text{ and } \mathfrak{a}^{*+}_0(H_\alpha) \subseteq \mathbb{R}_+ \setminus \{0\}} \mathfrak{g}_C^\alpha \right) \cap \mathfrak{g}.$$

Let $\nu_0 \in \mathfrak{a}^{*+}_0$ and $\lambda_0 \in (\mathfrak{t}^+ + \nu_0) \cap \mathfrak{g}^{\text{reg}}$. We use Proposition 11.1 (a) with $M'$ built from $\lambda_0$ instead of $\lambda$ (in which case “$(M', U)$” becomes “$(C_M(\mathfrak{m})M_0 A, N)$” and $\pi$ equal to $(\delta_{\mu_{+,m}}^{m/t} \tau_M \otimes T_{\tilde{\mu}}^{M_0}) \otimes e^{i\nu\circ\text{Log}}$. Since the class of $\text{Ind}_{M.A.N}^G(T_{\tilde{\mu},\tau_M}^M \otimes e^{i\nu\circ\text{Log}} \otimes \mathbf{1}_N)$ is determined by its character (see [Cow 88, p. 64]), it does not depend on $N$.

We choose $z = 1$ and $\Lambda = (\Lambda^{\text{can}}, R_{i\mathbb{R}}^+, R_{\mathbb{R}}^+)$ in [ABV 92, line 7 bottom of p. 121], where $\Lambda^{\text{can}} \otimes \rho(R^+(\mathfrak{g}_C, \mathfrak{h}_C)) = \rho_{\lambda_+}^{g/\mathfrak{h}} \tau_+$ (see [ABV 92, lines 9–10 bottom of p. 129]) and the systems of positive roots $R_{i\mathbb{R}}^+$ and $R_{\mathbb{R}}^+$ are included in $R^+(\mathfrak{g}_C, \mathfrak{h}_C)$. By Lemma 5.2 (b), the representation $\tilde{\Lambda}$ of [ABV 92, p. 123] is $\tilde{\Lambda} = \delta_{\lambda_+}^{g/\mathfrak{h}} \tau_+$. Given Proposition 11.1 (a), we have $\pi(\Lambda) = \text{Ind}_{M.A.N}^G(T_{\tilde{\mu},\tau_M}^M \otimes e^{i\nu\circ\text{Log}} \otimes \mathbf{1}_N)$ in [ABV 92, (11.2)(e) p. 122]. The ideas in the proof of Lemma 5 of [DV 88, p. 335] allow us to show that $\Lambda$ is “final” in the sense of [ABV 92, Def. 11.13 p. 130]. Finally, we apply [ABV 92, Th. 11.14 (a) p. 131]. □

**Remark 7.4.** We place ourselves in the case where $G = SL(3, \mathbb{R})$ and $\lambda$ is the element of $\mathfrak{g}^{ssInc}_G$ which is the image of $\begin{pmatrix} 1 \\ \lambda_+ \\ -2 \end{pmatrix}$ by the isomorphism of $G$-modules that maps $A \in \mathfrak{g}$ to $\text{tr}(A.) \in \mathfrak{g}^*$. The group $MA$ which is here equal to $G(\lambda)$, is the image of $GL(2, \mathbb{R})$ by the embedding of Lie groups that maps $x \in GL(2, \mathbb{R})$ to $\begin{pmatrix} x \\ (\det x)^{-1} \end{pmatrix} \in G$. But the representation $\text{Ind}_{M.A.N}^G(T_{\tilde{\mu},\tau_M}^M \otimes e^{i\nu\circ\text{Log}} \otimes \mathbf{1}_N)$ is not given by “nondegenerate data” in the sense of [KZ 82, p. 473], because the Weyl group of a fundamental Cartan subalgebra of $\mathfrak{gl}(2, \mathbb{R})$ is represented in $GL(2, \mathbb{R})$. ■
**Definition 7.5.** Define $T_{\tilde{\lambda}, \mathfrak{a}^{*+}, \tau_{+}}^{G} = \text{Ind}_{M.A.N}^{G}(T_{\tilde{\mu}, \tau_{M}}^{M} \otimes e^{i\nu\circ\text{Log}} \otimes 1_{N}) \in \hat{G}$, independently of the choice of $N$ as in the lemma above.

Thus, when $M$ is connected, $X_M^{irr}(\tilde{\mu})$ has as its only element $\chi_{\tilde{\mu}}^M$, and the representation classes $T_{\tilde{\mu}, \chi_M^M}$ and $T_{\tilde{\mu}, \{0\}, \chi_M^M}$ are both equal to $T_{\tilde{\mu}}^{M_0}$.

## 8. The representations $T_{\tilde{\lambda}, \mathfrak{a}^{*+}, \tau_{+}}^G$

Given $(\tilde{\lambda}, \mathfrak{a}^{*+}, \tau_+)$ as on page 161, we will apply the construction of the connected case to $M'_0 := G(\nu_+)_0$. A detour through homology will, thanks to a result by Vogan, allow us to normalize some intertwining operators in order to extend $T_{\tilde{\lambda}}^{M'_0}$ into a representation of a certain group $M'$.

Let $M'$ denote $G(\lambda_+)G(\nu_+)_0$ and $\mathfrak{m}' = \mathfrak{g}(\nu_+)$ its Lie algebra (thus $\tilde{\lambda} \in \mathfrak{m}'_{\text{fund},M'}$ by 8.2 (a), $\mathfrak{m}'(\lambda)(i\rho_{\mathcal{F}^+}) = \mathfrak{h}$ and $X_{M'}^{irr}(\tilde{\lambda}) = X_{M'}^{\text{final},+}(\tilde{\lambda}, \mathfrak{a}^*)$),
$R^+(\mathfrak{m}'_C, \mathfrak{h}_C) = R(\mathfrak{m}'_C, \mathfrak{h}_C) \cap R^+(\mathfrak{g}_C, \mathfrak{h}_C)$ (see 1.3 (b) relative to $\mathfrak{m}'$ and $(\tilde{\lambda}, \mathfrak{a}^*)$),
$\lambda_{+,m'} = \lambda_{m',\tilde{\lambda},\mathfrak{a}^*,1}$ the element $\mu_+ + \nu$ of $\mathfrak{h}^*$ where $\mu_+$ still denotes $\mu_{\mathfrak{g},\tilde{\lambda},\mathfrak{a}^{*+}}$ (thus $B_{\lambda_{+,m'}}$ is a restriction of $B_{\lambda_+}$, $M'(\tilde{\lambda}) = G(\lambda_+)$ and $\rho_{\tilde{\lambda}}^{m'/\mathfrak{h}} = \rho_{\lambda_{+,m'}}^{m'/\mathfrak{h}}$),
$\mathfrak{n}_{M'} = \sum_{\alpha \in R^+(\mathfrak{m}'_C, \mathfrak{h}_C)} \mathfrak{g}_C^\alpha$ and $\mathfrak{b}_{M'} = \mathfrak{h}_C \oplus \mathfrak{n}_{M'}$,
$q' = |\{\alpha \in R^+(\mathfrak{m}'_C, \mathfrak{h}_C) | \alpha \text{ compact}\}| + \frac{1}{2} |\{\alpha \in R^+(\mathfrak{m}'_C, \mathfrak{h}_C) | \alpha \text{ complex}\}|$,
$M_{\nu_+}$ the intersection of the kernels of the positive real characters of $C_{M'_0}(\mathfrak{a})$,
$T_{\tilde{\mu}}^{M_{\nu_+}} = T_{\tilde{\mu}, \chi_M^M}$ (see 7.1 (b) and $M_{\nu_+}(\tilde{\mu}) = \exp \mathfrak{t}$),
$T_{\tilde{\lambda}}^{M'_0} = T_{\tilde{\lambda}, \mathfrak{a}^*, \chi_M^{M'}}$ (see $M'_0(\tilde{\lambda}) = \exp \mathfrak{h}$),
$\mathfrak{u} = (\sum_{\alpha \in R(\mathfrak{g}_C, \mathfrak{h}_C) \text{ and } \nu_+(H_\alpha) > 0} \mathfrak{g}_C^\alpha) \cap \mathfrak{g}$, $\mathfrak{U} = \exp \mathfrak{u}$ (hence $\text{int } M' \cdot \mathfrak{U} \subseteq \mathfrak{U}$ and $M' \cap \mathfrak{U} = \{1\}$).

**Proposition 8.1.** *Denote by $\mathcal{H}$ “the space” of $T_{\tilde{\lambda}}^{M'_0}$ and by $\mathcal{H}^{\infty}$ the set of its $C^{\infty}$ vectors.*

(a) The eigenspace of weight $-(i\lambda + \rho_{m',\mathfrak{h}})$ under the action of $\mathfrak{h}_C$ on $H_{q'}(\mathfrak{n}_{M'}, \mathcal{H}^{\infty})^*$ resulting from the action of $\mathfrak{h}_C$ on $\bigwedge \mathfrak{n}_{M'} \otimes_C \mathcal{H}^{\infty}$, is one-dimensional. Denote it by $(H_{q'}(\mathfrak{n}_{M'}, \mathcal{H}^{\infty})^*)_{-(i\lambda + \rho_{m',\mathfrak{h}})}$.

(b) There exists a unique continuous unitary representation $S$ of $M'(\tilde{\lambda})^{m'/\mathfrak{h}}$ in $\mathcal{H}$ satisfying the following conditions (i) and (ii):

(i) $S(\hat{x}) T_{\tilde{\lambda}}^{M'_0}(y) S(\hat{x})^{-1} = T_{\tilde{\lambda}}^{M'_0}(xyx^{-1})$ for $\hat{x} \in M'(\tilde{\lambda})^{m'/\mathfrak{h}}$ lying over $x \in M'(\tilde{\lambda})$ and $y \in M'_0$;

(ii) the action of $M'(\tilde{\lambda})^{m'/\mathfrak{h}}$ on $(H_{q'}(\mathfrak{n}_{M'}, \mathcal{H}^{\infty})^*)_{-(i\lambda + \rho_{m',\mathfrak{h}})}$ resulting from the action of $M'(\tilde{\lambda})^{m'/\mathfrak{h}}$ on $\bigwedge \mathfrak{n}_{M'} \otimes_C \mathcal{H}^{\infty}$ obtained from $S$, is $(\rho_{\lambda_{+,m'}}^{m'/\mathfrak{h}})^{-1}$ id.

**Proof of the proposition.** (a) Assume $G$ is connected and use the notation from Section 7.

Fix a maximal compact subgroup $K$ of $G$ whose Cartan involution normalizes $\mathfrak{h}$. Therefore $K_{M'_0} := K \cap M'_0$ and $K_{M_{\nu_+}} := K \cap M_{\nu_+}$ are maximal compact subgroups of $M'_0$ and $M_{\nu_+}$. By Remark 7.2, the $(\mathfrak{m}_C, K_{M_{\nu_+}})$-module associated with $T_{\tilde{\mu}}^{M_{\nu_+}}$ is isomorphic to $\mathcal{R}_{M_{\nu_+}}^q((\rho_{\mu_{+,m}}^{m/t})^{-1}\chi_{\tilde{\mu}}^M)$, where $\mathcal{R}_{M_{\nu_+}}^q$ is the cohomological induction functor relative to $\mathfrak{b}_M$.
Given Definition 4.2 (d), and Lemma 5.2 (b) with its notation, for every $e \in C_M(\mathfrak{m})T_0$ (see 2 (a)) we have

$$ \frac{(\rho_{\lambda_{+,m'}}^{\mathfrak{m}/\mathfrak{h}} \delta_{\lambda_{+,m'}}^{\mathfrak{m}/\mathfrak{h}})(e)}{(\rho_{\mu_{+,m}}^{\mathfrak{m}/\mathfrak{t}} \delta_{\mu_{+,m}}^{\mathfrak{m}/\mathfrak{t}})(e)} = \det(\mathrm{Ad} e^{\mathcal{C}})_{\mathfrak{n}_{M'}/\mathfrak{n}_M} \times \frac{((\rho_{\lambda_{+,m'}}^{\mathfrak{m}/\mathfrak{h}})^{-1} \delta_{\lambda_{+,m'}}^{\mathfrak{m}/\mathfrak{h}})(e)}{((\rho_{\mu_{+,m}}^{\mathfrak{m}/\mathfrak{t}})^{-1} \delta_{\mu_{+,m}}^{\mathfrak{m}/\mathfrak{t}})(e)} = u_1 \cdots u_b $$

where $\{\beta_1, -\beta_1\}, \dots, \{\beta_b, -\beta_b\}$ are the elements of $\tilde{R}_\mathfrak{C}^+(\mathfrak{m}'_\mathfrak{C}, \mathfrak{h}_\mathfrak{C})$ and $u_1, \dots, u_b$ are the ratios of the homotheties $(\mathrm{Ad} e^{\mathcal{C}})_{\mathfrak{g}_\mathfrak{C}^1}, \dots, (\mathrm{Ad} e^{\mathcal{C}})_{\mathfrak{g}_\mathfrak{C}^b}$.

We apply [KV 95, Th. 11.225 p.759] (where (11.222) need to change “$z_1z_2$” to “$\xi_{\delta_c(u)}(t)^{-1}z_1z_2$” on line 19 p.753) and [KV 95, Cor. 8.28 p.566] to the group $M'_0$ and to the parabolic subalgebra $\mathfrak{b}_{M'}$ of $\mathfrak{m}'_\mathfrak{C}$. The $(\mathfrak{m}'_\mathfrak{C}, K_{M'_0})$-module $\mathcal{H}^f$ associated with $T_{\tilde{\lambda}}^{M'_0}$ is therefore irreducible and isomorphic to $\mathcal{R}_{M'_0}^{q'}((\rho_{\tilde{\lambda}}^{\mathfrak{m}/\mathfrak{h}})^{-1}\chi_{\tilde{\lambda}}^{M'})$, where $\mathcal{R}_{M'_0}^{q'}$ is the cohomological induction functor relative to $\mathfrak{b}_{M'}$. We obtain

$$ \dim \mathrm{Hom}_{\mathfrak{h}_\mathfrak{C}, T_0} (H_{q'}(\mathfrak{n}_{M'}, \mathcal{H}^f), \rho_{\tilde{\lambda}}^{\mathfrak{m}/\mathfrak{h}} \chi_{\tilde{\lambda}}^{M'}) = 1 $$

by taking $X = \mathcal{R}_{M'_0}^{q'}(Z)$ with $Z = (\rho_{\tilde{\lambda}}^{\mathfrak{m}/\mathfrak{h}})^{-1}\chi_{\tilde{\lambda}}^{M'}$ in [KV 95, Prop. 8.11 p. 555], and taking into account Schur's Lemma (see [Wal 88, Lem. 3.3.2 p.80]). Since the canonical homomorphism of $(\mathfrak{h}_\mathfrak{C}, T_0)$-modules from $H_{q'}(\mathfrak{n}_{M'}, \mathcal{H}^f)$ to $H_{q'}(\mathfrak{n}_{M'}, \mathcal{H}^\infty)$ is bijective according to [Duf 82a, Lem. 4 p. 165], this yields the result.

(b) By 9.6 (a), we have $\int x \cdot T_{\tilde{\lambda}}^{M'_0} = T_{\tilde{\lambda}}^{M'_0}$ for all $x \in M'(\tilde{\lambda})$. We then repeat the proof of [Duf 82a, Lem. 6 p. 169], almost verbatim. (The uniqueness of $S$ follows, of course, from 7.3 (b) and Schur's Lemma.) $\square$

**Lemma 8.2.** We retain the notation from the previous proposition.

(a) There exists a unique $\tau_{M'} \in X_{M'}^{irr}(\tilde{\lambda})$ such that $\delta_{\lambda_{+,m'}}^{\mathfrak{m}/\mathfrak{h}} \tau_{M'} = \delta_{\lambda_+}^{\mathfrak{g}/\mathfrak{h}} \tau_+$. Denote by $T_{\tilde{\lambda}, \tau_{M'}}^{M'} = \tau_{M'} \otimes S T_{\tilde{\lambda}}^{M'_0}$ the representation of $M'$ defined (unambiguously) by $(\tau_{M'} \otimes S T_{\tilde{\lambda}}^{M'_0})(xy) = \tau_{M'}(\hat{x}) \otimes S(\hat{x}) T_{\tilde{\lambda}}^{M'_0}(y)$ for $\hat{x} \in M'(\tilde{\lambda})^{\mathfrak{m}/\mathfrak{h}}$ lying over $x \in M'(\tilde{\lambda})$ and $y \in M'_0$.

(b) When $G$ is connected, we have: $T_{\tilde{\lambda}, \mathfrak{a}^{*+}, \tau_+}^G = \mathrm{Ind}_{M'U}^G((\tau_{M'} \otimes S T_{\tilde{\lambda}}^{M'_0}) \otimes \mathbb{1}_U)$.

**Proof of the lemma.** (a) Lemma 5.2 (b) allows us to define $\tau_{M'}$ using the equality $\frac{\rho_{\lambda_{+,m'}}^{\mathfrak{m}/\mathfrak{h}}}{|\rho_{\lambda_{+,m'}}^{\mathfrak{m}/\mathfrak{h}}|} (\delta_{\lambda_{+,m'}}^{\mathfrak{m}/\mathfrak{h}})^{-1} = \frac{\rho_{\lambda_+}^{\mathfrak{g}/\mathfrak{h}}}{|\rho_{\lambda_+}^{\mathfrak{g}/\mathfrak{h}}|} (\delta_{\lambda_+}^{\mathfrak{g}/\mathfrak{h}})^{-1}$, where $\rho_{\lambda_{+,m'}}^{\mathfrak{m}/\mathfrak{h}}$ and $\rho_{\lambda_+}^{\mathfrak{g}/\mathfrak{h}}$ are Lie group homomorphisms. Furthermore, we have $S(\exp X) = e^{-i\lambda(X)} T_{\tilde{\lambda}}^{M'_0}(\exp X)$ for $X \in \mathfrak{h}$.

(b) Assume $G$ is connected and use the notation from Section 7. We therefore have $M' = C_M(\mathfrak{m})M'_0$, $M'(\tilde{\lambda}) = C_M(\mathfrak{m}) \exp \mathfrak{h}$ and $\delta_{\lambda_{+,m'}}^{\mathfrak{m}/\mathfrak{h}}$ is a unitary character of $M'(\tilde{\lambda})^{\mathfrak{m}/\mathfrak{h}}$.

Choose a connected component $\mathfrak{a}_0^{*+}$ of the set of elements of $\mathfrak{a}^*$ that do not vanish on any $H_\alpha$ with nonimaginary $\alpha \in R(\mathfrak{g}_\mathfrak{C}, \mathfrak{h}_\mathfrak{C})$, whose closure contains $\nu_+$. Set $\mathfrak{n} = (\sum_{\alpha \in R(\mathfrak{g}_\mathfrak{C}, \mathfrak{h}_\mathfrak{C})} \mathfrak{g}_\mathfrak{C}^\alpha) \cap \mathfrak{g}$, $N = \exp \mathfrak{n}$ and $N_{\nu_+} = N \cap M'_0$.

The group $N$ (resp. $N_{\nu_+}$) is the unipotent radical of a parabolic subgroup of $G$ (resp. $M'_0$) with split component $A$. Furthermore, the product of $G$ restricts to a diffeomorphism from $N_{\nu_+} \times U$ onto $N$.

Denote by $\mathcal{V}$ and $\mathcal{W}$ “the spaces” of $\tau_{M'}$ and $T_{\tilde{\mu}}^{M_0}$. The space of $T_{\tilde{\lambda}}^{M'_0}$ is
realized in the form $\mathcal{H} := \mathrm{Ind}_{M_0.A.N_{\nu_+}}^{M'_0}(\mathcal{W} \otimes e^{i\nu\circ\mathrm{Log}} \otimes \mathbb{1}_{N_{\nu_+}})$. Identify $\mathcal{V} \otimes_{\mathbb{C}} \mathcal{H}$ with a set of functions on $M'_0$ with values in $\mathcal{V} \otimes_{\mathbb{C}} \mathcal{W}$. The map $\Phi \mapsto \Phi|_{M'_0}$ from the space of $\mathrm{Ind}_{C_M(m)}^{C_M(m)} M'_0$ $(\delta_{\lambda_{+,m'}}^{m/\mathfrak{h}} \tau_{M'} \otimes T_{\tilde{\mu}}^{M_0} \otimes e^{i\nu\circ\mathrm{Log}} \otimes \mathbb{1}_{N_{\nu_+}})$ to $\mathcal{V} \otimes_{\mathbb{C}} \mathcal{H}$ equipped with $1 \otimes T_{\tilde{\lambda}}^{M'_0}$ is a unitary isomorphism of $M'_0$-modules. The action of a $x \in C_M(m)$ on $\mathcal{V} \otimes_{\mathbb{C}} \mathcal{H}$ that follows from this is given by $\tau_{M'}(\hat{x}) \otimes S_0(\hat{x})$, where $\hat{x} \in M'(\tilde{\lambda})^{m/\mathfrak{h}}$ lies over $x$ and $S_0(\hat{x}) \cdot \varphi = \delta_{\lambda_{+,m'}}^{m/\mathfrak{h}}(\hat{x}) \times \varphi(x^{-1} \cdot x)$ for $\varphi \in \mathcal{H}$.

Consider a closed subgroup $\Gamma$ of $C_M(m)$. Set $M_\Gamma = \Gamma M_{\nu_+}$ and $M'_\Gamma = \Gamma M'_0$. Fix an irreducible component $\tau_{M',\Gamma}$ (in $X_{M'_\Gamma}^{irr}(\tilde{\lambda})$) of the restriction of $\tau_{M'}$ to $M'_\Gamma(\tilde{\lambda})^{m/\mathfrak{h}}$. As above, we obtain a unitary isomorphism of $M'_0$-modules, that allows us to transport the action of $\Gamma$, replacing $C_M(m)$ by $\Gamma$, $\tau_{M'}$ by $\tau_{M',\Gamma}$ and $\mathcal{V}$ by the space $\mathcal{V}_\Gamma$ of $\tau_{M',\Gamma}$. Determine an element $\tau_{M,\Gamma}$ of $X_{M_\Gamma}^{irr}(\tilde{\mu})$ by the equality $\delta_{\mu_{+,m}}^{m/t} \tau_{M,\Gamma} = (\delta_{\lambda_{+,m'}}^{m/\mathfrak{h}} \tau_{M',\Gamma})|_{\Gamma T_0}$ (see . 2 (a)).

We will see that $S_0 = S$ (see 8.1 (b)). The asserted equality will then follow from the induction in stages theorem and the fact that $\mathrm{Ind}_{H_1}^{G_1}(\pi \circ p|_{H_1}) \simeq (\mathrm{Ind}_{H_2}^{G_2} \pi) \circ p$ when $p: G_1 \to G_2$ is a surjective homomorphism of real Lie groups, $H_1$ is the inverse image by $p$ of a closed subgroup $H_2$ of $G_2$, and $\pi$ is a continuous unitary representation of $H_2$.

Resume the arguments and notation from the proof of 8.1 (a). The groups $K_{M'_\Gamma} := K \cap M'_\Gamma$ and $K_{M_\Gamma} := K \cap M_\Gamma$ are maximal compact subgroups of $M'_\Gamma$ and $M_\Gamma$. Note that the $(m_\mathbb{C}, K_{M_\Gamma})$-module associated to the representation $\mathrm{Ind}_{\Gamma.M_0}^{M_\Gamma}(\delta_{\mu_{+,m}}^{m/t} \tau_{M,\Gamma} \otimes T_{\tilde{\mu}}^{M_0})$ of the group $M_\Gamma$ is isomorphic to $\mathcal{R}_{M_\Gamma}^q((\rho_{\mu_{+,m}}^{m/t})^{-1}\tau_{M,\Gamma})$, where $\mathcal{R}_{M_\Gamma}^q$ is the cohomological induction functor relative to $b_M$. Next, using the induction in stages theorem, we see that the $(m'_\mathbb{C}, K_{M'_\Gamma})$-module associated to $\mathrm{Ind}_{\Gamma.M_0.A.N_{\nu_+}}^{M'_\Gamma}(\delta_{\lambda_{+,m'}}^{m/\mathfrak{h}} \tau_{M'} \otimes T_{\tilde{\mu}}^{M_0} \otimes e^{i\nu\circ\mathrm{Log}} \otimes \mathbb{1}_{N_{\nu_+}})$ is isomorphic to $\mathcal{R}_{M'_\Gamma}^{q'}((\rho_{\tilde{\lambda}}^{m/\mathfrak{h}})^{-1}\tau_{M',\Gamma})$, where $\mathcal{R}_{M'_\Gamma}^{q'}$ is the cohomological induction functor relative to $b_{M'}$. Finally, we find that

$$ \dim \operatorname{Hom}_{\mathfrak{b}_{\mathbb{C}}, \Gamma T_0} (\mathcal{V}_{\Gamma} \otimes_{\mathbb{C}} H_{q'}(\mathfrak{n}_{M'}, \mathcal{H}^f), \rho_{\tilde{\lambda}}^{m/\mathfrak{h}} \tau_{M', \Gamma}) = 1, $$

where each $y \in \Gamma$ projection of a $\hat{y} \in M'(\tilde{\lambda})^{m/\mathfrak{h}}$ acts on $\mathcal{V}_{\Gamma} \otimes_{\mathbb{C}} H_{q'}(\mathfrak{n}_{M'}, \mathcal{H}^f)$ by the tensor product of $\tau_{M',\Gamma}(\hat{y})$ and the endomorphism obtained from $S_0(\hat{y})$.

Consider $\hat{x} \in M'(\tilde{\lambda})^{m/\mathfrak{h}}$ lying over some $x \in C_M(m)$. We choose for $\Gamma$ the closed subgroup of $C_M(m)$ generated by $x$. Therefore $\dim \mathcal{V}_\Gamma = 1$ and the preceding calculation proves that the action of $\hat{x}$ on $(H_{q'}(\mathfrak{n}_{M'}, \mathcal{H}^\infty)^*)_{-(i\lambda+\rho_{M',\mathfrak{h}})}$ arising from $S_0(\hat{x})$ is indeed the homothety with ratio $\rho_{\tilde{\lambda}}^{m/\mathfrak{h}}(\hat{x})^{-1}$. $\square$

**Definition 8.3.** We retain the notation from Proposition 8.1 and Lemma 8.2. Define $T_{\tilde{\lambda},a^{*+},\tau_+}^G = \mathrm{Ind}_{M'.U}^G((\tau_{M'} \otimes S T_{\tilde{\lambda}}^{M'_0}) \otimes \mathbb{1}_U)$.

To simplify the notation $T_{\tilde{\lambda},a^{*+},\tau_+}^G$, I will omit $a^{*+}$ when $a^{*+} = a^*$ (compatible with 9.2) and $\tau_+$ when $X_G^{irr,+}(\tilde{\lambda}, a^{*+})$ is a singleton, and I will replace $\tilde{\lambda}$ with $\lambda$ when $\tilde{\lambda} = (\lambda, \mathfrak{h}_{(\mathbb{R})}^*)$ (conventions applied for $T_{\tilde{\mu}}^{M_0}$, $T_{\tilde{\mu},\sigma}^M$, $T_{\tilde{\mu}}^{M_{\nu_+}}$ and $T_{\tilde{\lambda}}^{M'_0}$).

**Proposition 8.4.** (a) For each semisimple $l \in \mathfrak{g}_\mathbb{C}^*$ denote by $\chi_l^{\mathfrak{g}_\mathbb{C}}$ the character of $Z(U\mathfrak{g}_\mathbb{C})$ canonically associated with the orbit of $l$ under the action of int $\mathfrak{g}_\mathbb{C}$ (see
[KV 95, (4.114) p. 297]). The set $(U\mathfrak{g}_C)^G$ of $G$-invariant elements of $U\mathfrak{g}_C$ acts on the space of $C^\infty$ vectors of $T_{\tilde{\lambda}, \mathfrak{a}^{*+}, \tau_+}^G$ via restriction of the character $\chi_{i\tilde{\lambda}}^{U\mathfrak{g}_C}$ of $Z(U\mathfrak{g}_C)$.

(b) The group homomorphism $z \mapsto (z, 1)$ from $Z(G)$ to $G(\lambda_+)^{\mathfrak{g}/\mathfrak{h}}$ is injective. The central character of $T_{\tilde{\lambda}, \mathfrak{a}^{*+}, \tau_+}^G$ is “the restriction” of that of $\tau_+$.

**Proof of the proposition.** (a) Corollary 5.25 (b) of [KV 95, p.344] allows us to calculate the infinitesimal character of $\tau_{M'} \otimes S T_{\tilde{\lambda}}^{M'_0}$ (equal to that of $T_{\tilde{\lambda}}^{M'_0}$). To then move on to $G$, the proof of [KV 95, Prop. 11.43 p. 665] can be immediately adapted.

(b) Let $z \in Z(G)$. The equality $S(z, 1) = \text{id}$ yields the result. $\square$

**9. The injection $G \cdot (\tilde{\lambda}, \tau) \mapsto T_{\tilde{\lambda}, \tau}^G$ from $G \setminus X_G^{\text{Ind}}$ to $\hat{G}$**

Theorem 9.6 will generalize a large part of the one written by Duflo in [Duf 82a, Lem. 8 p. 173]. My proof is similar to his, except for one unexpected result: Lemma 9.3 (b). Recall that the equivalence class of an irreducible unitary representation of trace class of a real Lie group is determined by its character (see [Cow 88, p. 64]).

Given $(\tilde{\lambda}, \mathfrak{a}^{*+}, \tau_+)$ as on page 161, construct $\tau$ from $\tau_+$ as in Definition 5.5 (c).

**Lemma 9.1.** *The parameters $\tilde{\lambda}$ and $\tau$ determine $T_{\tilde{\lambda}, \mathfrak{a}^{*+}, \tau_+}^G$.*

**Proof of the lemma.** This result will follow from Theorem 10.2, by first removing all references to $T_{\tilde{\lambda}, \tau}^G$ (for consistency of reasoning). $\square$

**Definition 9.2.** (a) Define $T_{\tilde{\lambda}, \tau}^G = T_{\tilde{\lambda}, \mathfrak{a}^{*+}, \tau_+}^G$. In the notation $T_{\tilde{\lambda}, \tau}^G$, I will omit $\tau$ when $X_G^{\text{Ind}}(\tilde{\lambda})$ is a singleton, and I will replace $\tilde{\lambda}$ with $\lambda$ when $\tilde{\lambda} = (\lambda, \mathfrak{h}_{(\mathbb{R})}^*)$.

(b) Let $a$ be an automorphism of the Lie group $G$. It induces canonical isomorphisms, still denoted by $a$, of vector spaces from $\mathfrak{g}/\mathfrak{g}(\lambda)(i\rho_{\mathcal{F}+})$ to $\mathfrak{g}/\mathfrak{g}(a\lambda)(i\rho_{a\mathcal{F}+})$, of Lie algebras from $\mathfrak{sl}(\mathfrak{g}/\mathfrak{g}(\lambda)(i\rho_{\mathcal{F}+}))$ to $\mathfrak{sl}(\mathfrak{g}/\mathfrak{g}(a\lambda)(i\rho_{a\mathcal{F}+}))$ by conjugacy, of Lie groups from $\mathfrak{DL}(\mathfrak{g}/\mathfrak{g}(\lambda)(i\rho_{\mathcal{F}+}))$ to $\mathfrak{DL}(\mathfrak{g}/\mathfrak{g}(a\lambda)(i\rho_{a\mathcal{F}+}))$ by integration, and finally of Lie groups from $G(\tilde{\lambda})^{\mathfrak{g}/\mathfrak{g}(\lambda)(i\rho_{\mathcal{F}+})}$ to $G(a\tilde{\lambda})^{\mathfrak{g}/\mathfrak{g}(a\lambda)(i\rho_{a\mathcal{F}+})}$.

**Lemma 9.3.** Let $\tau_0 \in (G_0(\lambda_+)^{\mathfrak{g}/\mathfrak{h}})^\wedge$ occurring in $\tau_+|_{G_0(\lambda_+)^{\mathfrak{g}/\mathfrak{h}}}$.

(a) We have: $\tau_0 \in X_{G_0}^{\text{final},+}(\tilde{\lambda}, \mathfrak{a}^{*+})$ and $G_0(\lambda_+) = C_{G_0}(\mathfrak{h}) = N_{G_0(\tilde{\lambda})}(\mathfrak{a}^{*+})$.

(b) Define: $\dot{\tau}_0 = \frac{\rho_{\tilde{\lambda}, \mathfrak{a}^{*+}}^{\mathfrak{g}/\mathfrak{h}}}{|\rho_{\tilde{\lambda}, \mathfrak{a}^{*+}}^{\mathfrak{g}/\mathfrak{h}}|} \tau_0$ (see 5.5 (c)). The action of the group $G_0(\tilde{\lambda})$ on the representation class $\dot{\tau}_0$ of $G_0(\lambda_+)$ is trivial.

**Proof of the lemma.** (a) Definition 5.5 (b) ensures that $\tau_0$ is final. The equalities $G_0(\lambda_+) = C_{G_0}(\mathfrak{h}) = N_{G_0(\tilde{\lambda})}(\mathfrak{a}^{*+})$ follows from Lemma 1.4.

(b) We prove that $\text{tr} \, \dot{\tau}_0(x\gamma x^{-1}) = \text{tr} \, \dot{\tau}_0(\gamma)$ for $x \in G_0(\tilde{\lambda})$ and $\gamma \in G_0(\lambda_+)$.

Let $G_1$ be the subgroup of $G_0$ consisting of the elements that fix $\mu$ and $i\rho_{\mathcal{F}+}$. Denote by $\mathfrak{g}_1$ its Lie algebra. Since the linear forms $\mu$ and $i\rho_{\mathcal{F}+}$ are infinitesimally elliptic, $\mathfrak{g}_1$ is reductive, $\mathfrak{h} \in \text{Car } \mathfrak{g}_1$, and $G_1$ is connected. The
root system $R(\mathfrak{g}_1\mathbb{C}, \mathfrak{h}_\mathbb{C})$ has no imaginary root. Denote by $W(\mathfrak{g}_1(\nu), \mathfrak{a})$ the Weyl group of the root system formed by the restricted roots of $(\mathfrak{g}_1(\nu), \mathfrak{a})$. By [Kna 86, Th. 5.17 p. 125, (5.5) p. 126 and Lem. 5.16 p. 124] applied to $G_1$ on the one hand, and [Kna 86, Prop. 4.12 p. 81] on the other hand, the canonical map from $G_0(\tilde{\lambda})/G_0(\lambda_+)^\beta$ to $W(\mathfrak{g}_1(\nu), \mathfrak{a})$ is bijective. Consider $\beta \in R(\mathfrak{g}_1(\nu)\mathbb{C}, \mathfrak{h}_\mathbb{C})$, $X \in \mathfrak{g}_1(\nu)^{\beta|_\mathfrak{a}}$ and $Y \in \mathfrak{g}_1(\nu)^{-\beta|_\mathfrak{a}}$ such that $[X, Y]$ is the coroot of $\beta|_\mathfrak{a}$ in $\mathfrak{a}$. Set $g = \exp(\frac{\pi}{2}(X - Y))$. Therefore $g \in G_0(\tilde{\lambda})$ projects into $W(\mathfrak{g}_1(\nu), \mathfrak{a})$ on the root reflection $s_{(\beta|_\mathfrak{a})}$ given the proof of [Kna 86, Prop. 5.15 (c) p. 123]. It now suffices to show the equality $\text{tr} \, \dot{\tau}_0(g\gamma g^{-1}) = \text{tr} \, \dot{\tau}_0(\gamma)$ for $\gamma \in G_0(\lambda_+)$.

A calculation in $SL(2, \mathbb{C})$ shows that for every real root $\alpha \in R(\mathfrak{g}_\mathbb{C}, \mathfrak{h}_\mathbb{C})$, we have $\text{Ad} \, \gamma_\alpha = \exp(i\pi \text{ ad } H_\alpha)$ and $\text{Ad} \, \gamma_\alpha^2 = \text{id}$, with $\gamma_\alpha$ from 5.5 (a). Therefore $\gamma_\alpha^2 \in Z(G_0)$. By [Kna 86, Lem. 12.30 (c) p. 469] and (a), every element $\gamma$ of $G_0(\lambda_+)$ can be written as $\gamma = h z \gamma_{\alpha_1} \dots \gamma_{\alpha_k}$ with $h \in \exp \mathfrak{h}$, $z \in Z(G_0)$ and real roots $\alpha_1, \dots, \alpha_k \in R(\mathfrak{g}_\mathbb{C}, \mathfrak{h}_\mathbb{C})$. Given Proposition 4.3 (b), the differential at 1 of $\frac{\rho_{\tilde{\lambda}, \mathfrak{a}^*+}^{g/\mathfrak{h}}}{|\rho_{\tilde{\lambda}, \mathfrak{a}^*+}^{g/\mathfrak{h}}|}$ is the half sum of $\alpha \in R(\mathfrak{g}(\nu)\mathbb{C}, \mathfrak{h}_\mathbb{C})$ such that: $i\mu(H_\alpha) > 0$ or, $i\mu(H_\alpha) = 0$ and $\rho_{\mathcal{F}^+}(H_\alpha) > 0$. Therefore $G_0(\tilde{\lambda})$ leaves invariant the unitary characters by which $\exp \mathfrak{h}$ and $Z(G_0)$ act in the space of $\dot{\tau}_0$. In the rest of this proof, we will therefore take $\gamma$ of the form $\gamma = \gamma_{\alpha_1} \cdots \gamma_{\alpha_k}$ with real roots $\alpha_1, \dots, \alpha_k \in R(\mathfrak{g}_\mathbb{C}, \mathfrak{h}_\mathbb{C})$.

For every real root $\alpha \in R(\mathfrak{g}_\mathbb{C}, \mathfrak{h}_\mathbb{C})$, the element $\gamma_\alpha g \gamma_\alpha^{-1}$ (resp. $\gamma_\alpha g^{-1} \gamma_\alpha^{-1}$) is equal to $g$ (resp. $g^{-1}$) if $\beta(H_\alpha)$ is even and to $g^{-1}$ (resp. $g$) otherwise. Consequently, we have:

$$ g(\gamma g^{-1} \gamma^{-1}) = \begin{cases} 1 & \text{if } \beta(H_{\alpha_1} + \cdots + H_{\alpha_k}) \text{ is even} \\ g^2 & \text{otherwise.} \end{cases} $$

First, consider the case where $\beta$ is real and $g\gamma g^{-1}\gamma^{-1} \neq 1$. We make the choices of $g$ and $\gamma_\beta$ compatible by choosing $X=X_\beta$ and $Y=X_{-\beta}$, so $\gamma_\beta = g^2$ (see 5.5 (a)). Denote by $\zeta$ the ratio of the homothety $\dot{\tau}_0(\gamma_\beta)$. We have $\gamma_\beta\gamma\gamma_\beta^{-1} = \gamma_\beta^2\gamma$ and $\gamma_\beta(g\gamma g^{-1})\gamma_\beta^{-1} = \gamma_\beta^2(g\gamma g^{-1})$. Therefore $\text{tr} \, \dot{\tau}_0(\gamma) = 0 = \text{tr} \, \dot{\tau}_0(g\gamma g^{-1})$ when $\zeta \neq 1$. We will reason differently when $\zeta = 1$, using $\dot{\tau}_0(g\gamma g^{-1})\dot{\tau}_0(\gamma)^{-1} = \dot{\tau}_0(\gamma_\beta)$. The element $\gamma_\beta + \gamma_\beta^{-1}$ of the group algebra of $G_0(\lambda_+)$ is central. Therefore $\dot{\tau}_0(\gamma_\beta)$ is an homothety when $\zeta \neq -1$, equal to the $(\dot{\tau}_0(\gamma_\beta) + \dot{\tau}_0(\gamma_\beta^{-1}))(\dot{\tau}_0(\gamma_\beta^2)^{-1} + \text{id})^{-1}$. In particular, since $\tau_0$ is final, we have $(\delta_{\lambda_+}^{g/\mathfrak{h}}\tau_0)(\gamma_\beta) = (-1)^{n_\beta+1} \text{id}$ when $\zeta = 1$. Let $F$ denote the subgroup of $G$ generated by the $\gamma_\alpha$ with $\alpha \in R(\mathfrak{g}_\mathbb{C}, \mathfrak{h}_\mathbb{C})$ real. According to [DV 88, (42) p. 335], to be corrected, there exists a $G(\tilde{\lambda})$-invariant character $\chi_1$ of $F$ that sends $\gamma_\alpha$ to $(-1)^{n_\alpha+1}$ for every real root $\alpha \in R(\mathfrak{g}_\mathbb{C}, \mathfrak{h}_\mathbb{C})$. Furthermore, the restriction $\chi_2$ of $\frac{\rho_{\tilde{\lambda}, \mathfrak{a}^*+}^{g/\mathfrak{h}}}{|\rho_{\tilde{\lambda}, \mathfrak{a}^*+}^{g/\mathfrak{h}}|} (\delta_{\lambda_+}^{g/\mathfrak{h}})^{-1}$ to $G_0(\lambda_+)$ is $G(\tilde{\lambda})$-invariant (see 5.4 (a)). Thus, $\chi := \chi_1\chi_2|_F$ is a $G(\tilde{\lambda})$-invariant unitary character of $F$, such that when $\zeta = 1$ we have: $\dot{\tau}_0(\gamma_\beta) = \chi(\gamma_\beta)$ id then $\dot{\tau}_0(\gamma_\beta) = \chi(g\gamma g^{-1})\chi(\gamma)^{-1}$ id = id.

Finally, assume that $\beta$ is nonreal and that $g\gamma g^{-1}\gamma^{-1} \neq 1$. Denote by $R_\beta$ the system of roots $(\mathbb{R}\beta \oplus \mathbb{R}\bar{\beta}) \cap R(\mathfrak{g}_1(\nu)\mathbb{C}, \mathfrak{h}_\mathbb{C})$ whose roots are neither imaginary nor real. Examining the Satake diagram of the semisimple Lie subalgebra of $\mathfrak{g}_1(\nu)$ of complexification $(\mathbb{C}H_\beta \oplus \mathbb{C}H_{\bar{\beta}}) \oplus \sum_{\alpha \in R_\beta} \mathfrak{g}_\mathbb{C}^\alpha$, we see that there is an injective Lie algebra homomorphism $\eta: \mathfrak{g}_0 \to \mathfrak{g}_1(\nu)$, with $\mathfrak{g}_0 = \text{sl}(2, \mathbb{C})|_{\mathbb{R}}$ or $\mathfrak{g}_0 = \text{su}(1, 2)$, which sends, for every $(x, y) \in \mathbb{R}^2$ and according to the value of $\mathfrak{g}_0$, the matrix $\begin{pmatrix} x+iy & 0 \\ 0 & -(x+iy) \end{pmatrix}$ or the matrix $\begin{pmatrix} iy & x & 0 \\ x & iy & 0 \\ 0 & 0 & -2iy \end{pmatrix}$ to $x(H_\beta+H_{\bar{\beta}})+iy(H_\beta-H_{\bar{\beta}})$.
Choose $X \in (\mathfrak{g}_C^\beta \oplus \mathfrak{g}_C^{\bar{\beta}}) \cap \mathfrak{g}$ and $Y \in (\mathfrak{g}_C^{-\beta} \oplus \mathfrak{g}_C^{-\bar{\beta}}) \cap \mathfrak{g}$, equals according to $\mathfrak{g}_0$ to:

$$X = \eta((\begin{smallmatrix} 0 & 1 \\ 0 & 0 \end{smallmatrix})) \quad \text{and} \quad Y = \eta((\begin{smallmatrix} 0 & 0 & 1 \\ 0 & 0 & 0 \\ 1 & -1 & 0 \end{smallmatrix})), \text{ or, } X = \eta((\begin{smallmatrix} 0 & 0 & 1 \\ 0 & 0 & 0 \\ 1 & 1 & 0 \end{smallmatrix})) \quad \text{and} \quad Y = \eta((\begin{smallmatrix} 0 & 0 & 1 \\ 0 & 0 & -1 \\ 1 & 1 & 0 \end{smallmatrix})).$$

A calculation in one of the simply connected groups $SU(2)$ or $\{1\} \times SU(2)$ then shows that $g^2$ belongs to $\exp(\mathfrak{h} \cap \mathrm{D}(\mathfrak{g}_1(\nu)))$. The definition of $\mathfrak{g}_1$ ensures that $W(\mathfrak{g}_1(\nu)_C, \mathfrak{h}_C)$ fixes $d_1\dot{\tau}_0$. Therefore $\dot{\tau}_0$ is trivial on $\exp(\mathfrak{h} \cap \mathrm{D}(\mathfrak{g}_1(\nu)))$. Thus, we obtain: $\dot{\tau}_0(g\gamma g^{-1}\gamma^{-1}) = \dot{\tau}_0(g^2) = \mathrm{id}$. $\square$

In the following proposition, I state the results of Mackey's little group theory that will be useful in proving the next theorem. I will not review the concepts of "type I group" and "Hilbert integral" (see [Dix 69]). For this proposition, we consider a (Hausdorff) locally compact group $A$, whose topology have a countable base, and a normal closed subgroup $B$ of $A$ which is of type $I$. The group $A$ acts canonically on $\hat{B}$.

A few words about terminology. A measurable unitary cocycle of $A$ is a measurable map $c$ from $A \times A$ to the set of complex numbers of absolute value 1 that satisfies $c(1, 1) = 1$ and $c(xy, z)c(x, y) = c(x, yz)c(y, z)$ for $x, y, z \in A$. In this case, given a separable complex Hilbert space $V$, we will call $c$-projective representation of $A$ in $V$ a map $\tilde{\pi}$ from $A$ to the unitary group of $V$, for which each of the functions $x \in A \mapsto \langle \tilde{\pi}(x) \cdot v, w \rangle \in \mathbb{C}$ with $v, w \in V$ is measurable, and such that $\tilde{\pi}(1) = \mathrm{id}$ and $\tilde{\pi}(xy) = c(x, y) \times \tilde{\pi}(x)\tilde{\pi}(y)$ for $x, y \in A$.

**Proposition 9.4. (Mackey)** (a) Let $\pi \in \hat{B}$. Denote by $A_\pi$ the stabilizer of $\pi$ in $A$ and by $\mathrm{pr}$ the canonical projection from $A_\pi$ onto $A_\pi/B$. The group $A_\pi$ is closed in $A$. There exists a measurable unitary cocycle $c$ of $A_\pi/B$ and an isomorphism class $\tilde{\pi}$ of $c \circ (\mathrm{pr} \times \mathrm{pr})$-projective representation of $A_\pi$ that extends $\pi$ in the same Hilbert space. Furthermore, there exists a nonzero $\sigma$-finite measure $m$ on $\hat{B}$, unique up to equivalence, such that the orbit of $\pi$ under $A$ in $\hat{B}$ has an $m$-negligible complement and $a \cdot m$ is equivalent to $m$ pour for all $a \in A$.

(b) Retain the notation from (a). We can define the mapping $\tilde{\eta} \mapsto \mathrm{Ind}_{A_\pi}^A (\tilde{\eta} \circ \mathrm{pr} \otimes \tilde{\pi})$ from the set of isomorphism classes of irreducible $c^{-1}$-projective representations of $A_\pi/B$, to the set of isomorphism classes of unitary representations of $A$. It is injective and its image consists of the elements of $\hat{A}$ whose restriction to $B$ has a common multiple with $\int_{\hat{B}} \rho \, dm(\rho)$.

(c) When the measurable quotient space $A\backslash\hat{B}$ is countably separated (i.e., we can find a sequence $(E_n)_{n \in \mathbb{N}}$ of measurable subsets of $E = A\backslash\hat{B}$ such that for $x \neq y$ in $E$, there exists $m \in \mathbb{N}$ satisfying $x \in E_m$ and $y \notin E_m$), every element of $\hat{A}$ is obtained as in (b) for a unique orbit of $A$ in $\hat{B}$.

**Proof of the proposition.** (a) Mackey's "metrically smooth of type I" hypothesis translates to "of type I" by [Dix 64, Prop. 4.6.1 p.95 and Th. 9.1 p.168]. The group $A_\pi$ is closed in $A$ and $m$ exists by [Mac 58, Th. 7.5 p. 295]. The existence of $c$ and of $\tilde{\pi}$ is given by [Mac 58, Th. 8.2 p. 298].

(b) This result is cited in [Mac 58, Th. 8.1 p. 297 and 8.3 p. 300]. It refers to the representation class of $B$ associated with $m$, constructed at the end of its Section 7 p. 296, which refers to lines 4–6 p. 273.

(c) Follows from [Mac 58, Th. 9.1 p. 302 and end of Section 7 p. 296]. $\square$
The representations classified a little further on are the “tempered” irreducible unitary representations of $G$. This notion deserves a definition.

**Definition 9.5.** (a) Let $\pi_0 \in \hat{G}_0$. We say that $\pi_0$ is tempered if it is equivalent to a subrepresentation of a representation $\text{Ind}_{\tilde{P}}^{G_0}(\tilde{\sigma} \otimes \tilde{\eta} \otimes \mathbb{1}_{\tilde{N}})$, where $\tilde{P}$ is a parabolic subgroup of $G_0$ of Langlands decomposition $\tilde{P} = \tilde{M}\tilde{A}\tilde{N}$, $\tilde{\sigma}$ is a limit of discrete series representation of $\tilde{M}$, and $\tilde{\eta}$ is a unitary character of $\tilde{A}$. It can be shown, as in [Kna 86, Th. 8.53 p. 260 and Th. 12.23 p. 456] that this is equivalent to the following condition: $|D_{G_0}|^{1/2} \text{tr } \pi_0$ is bounded on $(G_0)_{\text{ss reg}}$.

(b) Let $\pi \in \hat{G}$. By 9.4 end of (b) (c), there exist $n \in \mathbb{N}$ and $\pi_1, \dots, \pi_n \in \hat{G}_0$ such that $\pi|_{G_0} = \bigoplus_{1 \le k \le n} \pi_k$ (see [Dix 69, Prop. 2.1 p. 425], [Dix 64, Prop. 4.6.1 p. 95 and Th. 9.1 p. 168], and 2.1.21 of Zimmer’s book “Ergodic theory and semi-simple groups” for $(G/G_0)\backslash\hat{G_0}$). We say that $\pi$ is tempered if $\pi_1, \dots, \pi_n$ are tempered.

**Theorem 9.6.** (a) We have: $a \cdot T_{\tilde{\lambda}, \tau}^G = T_{a \cdot \tilde{\lambda}, a \cdot \tau}^G$ for every automorphism $a$ of the Lie group $G$.

(b) The set $G \setminus X_G^{\text{Ind}}$ (resp. $G \setminus X_G^{\text{final,+}}$) injects into $\hat{G}$ by the map that sends the orbit of $(\tilde{\lambda}', \tau')$ (resp. $(\tilde{\lambda}', \mathfrak{a}^{*+}, \tau_+')$) to $T_{\tilde{\lambda}', \tau'}^G$ (resp. $T_{\tilde{\lambda}', \mathfrak{a}^{*+}, \tau_+'}^G$).

The image of this map consists of the representation classes in $\hat{G}$ that are tempered. (By Lemma 6.1, the $\tilde{\lambda}' \in \mathfrak{g}_{\text{reg}}^*$ involved here belong to $\mathfrak{g}_{\text{reg},G}^*$).

**Proof of the theorem.** We will use the notation from the beginning of Part III. and Sections 7. and 8.

(a) Given a property of induction cited in the proof of Lemma 8.2 (b), we are reduced to proving that $a \cdot T_{\tilde{\mu}}^{M_0} = T_{a \cdot \tilde{\mu}}^{a \cdot M_0}$ when $G$ is connected. This equality comes from some formulas for the characters of $a \cdot T_{\tilde{\mu}}^{M_0}$ and $T_{a \cdot \tilde{\mu}}^{a \cdot M_0}$ obtained (without Theorem 9.6) in the last part of the proof of Theorem 10.2.

(b) First assume that $G$ is connected. To begin with, the limits of discrete series representations of $G$ are tempered according to [Wal 88, Th. 6.8.1 p. 202, Prop. p. 142, Prop. p. 139] (see also [Kna 86, Cor. 12.27 p. 461] for a study of the character property given in Definition 9.5 (a)). It follows that the representation class $T_{\tilde{\lambda}, \mathfrak{a}^{*+}, \tau_+}^G$ of Definition 7.5 is tempered by adapting the proof of the implication (3) ⇒ (1) of [BW 80, Prop. IV 3.7], and applying the implication (1) ⇒ (3) of the same proposition.

Consider a tempered irreducible unitary representation of $G$ in a Hilbert space $V$. Let $K$ be a maximal compact subgroup of $G$. By Theorem 11.14 (b) of [ABV 92, p. 131] with $z = 1$, there exists a “final limit character $\Lambda_V$” such that the $(\mathfrak{g}_C, K)$-module underlying $V$ is isomorphic to that of the representation $\pi(\Lambda_V)$ which is defined in [ABV 92, p. 122] modulo [KV 95, 5] p. 742]. I use the dictionary proposed in the proof of Lemma 7.3 (b). The equality (11.196) of [KV 95, p. 740] shows the representation $\pi(\Lambda_V)$ as a quotient of a representation $\text{Ind}_{\widetilde{M} \cdot \tilde{A} \cdot \tilde{N}}^{G}(T_{\tilde{\lambda}', \mathfrak{a}^{*+}, \tau_+'}^{\widetilde{M}} \otimes \tilde{\eta} \otimes \mathbb{1}_{\tilde{N}})$ induced from a parabolic subgroup $\widetilde{M}\tilde{A}\tilde{N}$ of $G$, for some $(\tilde{\lambda}', \mathfrak{a}^{*+}, \tau_+') \in X_{\widetilde{M}}^{\text{final,+}}$ and some real character $\tilde{\eta}$ of $\tilde{A}$. We therefore have $\widetilde{M}\tilde{A}\tilde{N} = G$ according to [BW 80, Lem. IV 4.9] and [KV 95, (11.197) p. 741], and the group $G$ acts in $V$ by a representation whose isomorphism class is $T_{\tilde{\lambda}', \mathfrak{a}^{*+}, \tau_+'}^G$.
Consider the case where $(\tilde{\lambda}', \mathfrak{a}'^{*+}, \tau_+')$ satisfies $T_{\tilde{\lambda}', \mathfrak{a}'^{*+}, \tau_+'}^G = T_{\tilde{\lambda}, \mathfrak{a}^{*+}, \tau_+}^G$. Introduce the ordered triples $(\Lambda^{can}, R_{i\mathbb{R}}^+, R_{\mathbb{R}}^+)$ and $(\Lambda'^{can}, R'_{i\mathbb{R}}^+, R'_{\mathbb{R}}^+)$ attached to $\tau_+$ and $\tau_+'$ as in the proof of 7.3 (b), and set $\mathfrak{h}' = \mathfrak{g}(\tilde{\lambda}')$. The differentials of the central characters of $\Lambda^{can}$ and $\Lambda'^{can}$ are equal to $i\lambda|_{\mathfrak{h}}$ and $i\lambda'|_{\mathfrak{h}'}$. By [ABV 92, Th. 11.14 (c) p. 131 and Def. 11.6 p. 124], there exists $g \in G$ such that, denoting by $\zeta$ the map that sends $x \in G(\lambda_+)$ to the determinant of the restriction of $\mathrm{Ad} x^\mathbb{C}$ to the sum of $\mathfrak{g}^\alpha_\mathbb{C}$ with $\alpha \in R_\mathbb{R}^+$ and $g\alpha \notin R'_{\mathbb{R}}^+$, we have: $\tilde{\lambda}' = g\tilde{\lambda}$ and $\Lambda'^{can} = g \cdot (\Lambda^{can} \otimes \frac{\zeta}{|\zeta|})$. Using Lemma 5.4 (a) and a calculation made in the proof of 7.3 (b), we obtain: $\frac{\rho_{\tilde{\lambda}, \mathfrak{a}'^{*+}}^{g/\mathfrak{h}}}{|\rho_{\tilde{\lambda}, \mathfrak{a}'^{*+}}^{g/\mathfrak{h}}|} \tau_+' = g \cdot \left( \frac{\rho_{\tilde{\lambda}, \mathfrak{a}^{*+}}^{g/\mathfrak{h}}}{|\rho_{\tilde{\lambda}, \mathfrak{a}^{*+}}^{g/\mathfrak{h}}|} \tau_+ \right)$. Since the chambers of $(\mathfrak{g}(\lambda)(i\rho_{\mathbb{F}^+}), \mathfrak{a})$ are conjugate under $N_{G(\lambda)(i\rho_{\mathbb{F}^+})_0}(\mathfrak{a})$, there exists $u \in G(\tilde{\lambda})$ such that $(\tilde{\lambda}', \mathfrak{a}'^{*+}) = gu \cdot (\tilde{\lambda}, \mathfrak{a}^{*+})$. A fortiori, we have $\tau_+' = gu \cdot \tau_+$ by Lemma 9.3 (b).

We no longer assume $G$ connected. We adapt Duflo's proof in [Duf 82a, p. 176 to p. 179]. Note that the closed normal subgroups $B_1 = G_0(\lambda_+)^{g/\mathfrak{h}}$ and $B_2 = G_0$, respectively of $A_1 = G(\lambda_+)^{g/\mathfrak{h}}$ and $A_2 = G$, are of type I by [Dix 69, Prop. 2.1 p. 425]. In particular, the measurable spaces $\widehat{B}_1$ and $\widehat{B}_2$ are standard by [Dix 64, Prop. 4.6.1 p. 95 and Th. 9.1 p. 168]. Their quotients under the left actions of the finite groups respectively equal to $A_1/B_1$ and $A_2/B_2$ are therefore countably separated.

We will introduce below a certain $\tau_0 \in (G_0(\lambda_+)^{g/\mathfrak{h}})^\wedge$. Let $G(\lambda_+)^{g/\mathfrak{h}}_{\tau_0}$ denote the stabilizer of $\tau_0$ in $G(\lambda_+)^{g/\mathfrak{h}}$, let $G(\lambda_+)^{g/\mathfrak{h}}_{\tau_0}$ denote the image of $G(\lambda_+)^{g/\mathfrak{h}}_{\tau_0}$ in $G(\lambda_+)$ and let $M'(\tilde{\lambda})^{m/\mathfrak{h}}_{\tau_0}$ denote the inverse image of $G(\lambda_+)^{m/\mathfrak{h}}$ in $M'(\tilde{\lambda})^{m/\mathfrak{h}}$. We will use the same notation pr to denote the canonical projections from $G(\lambda_+)^{g/\mathfrak{h}}_{\tau_0}$, $G(\lambda_+)^{m/\mathfrak{h}}_{\tau_0}$ and $M'(\tilde{\lambda})^{m/\mathfrak{h}}_{\tau_0}$ onto $G(\lambda_+)^{m/\mathfrak{h}}_{\tau_0}/G_0(\lambda_+)$. By Proposition 9.4 (a) and (c), there exists an irreducible subrepresentation $\tau_0$ of $\tau_+|_{G_0(\lambda_+)^{g/\mathfrak{h}}}$, a measurable unitary cocycle $c$ of $G(\lambda_+)^{m/\mathfrak{h}}_{\tau_0}/G_0(\lambda_+)$, an isomorphism class $\tilde{\tau}_0$ of $c(\mathrm{pr} \times \mathrm{pr})$-projective representation of $G(\lambda_+)^{g/\mathfrak{h}}_{\tau_0}$ that extends $\tau_0$ in its space, and an isomorphism class $\tilde{\eta}_0$ of irreducible $c^{-1}$-projective representations of $G(\lambda_+)^{m/\mathfrak{h}}_{\tau_0}/G_0(\lambda_+)$, such that:

$$ \tau_+ = \mathrm{Ind}_{G(\lambda_+)^{g/\mathfrak{h}}_{\tau_0}}^{G(\lambda_+)^{g/\mathfrak{h}}}(\tilde{\eta}_0 \circ \mathrm{pr} \otimes \tilde{\tau}_0). $$

We now construct three isomorphisms of unitary representations.

The first, of $M'(\tilde{\lambda})^{m/\mathfrak{h}}$-modules, goes from the space of $\mathrm{Ind}_{G(\lambda_+)^{g/\mathfrak{h}}_{\tau_0}}^{G(\lambda_+)^{g/\mathfrak{h}}}(\tilde{\eta}_0 \circ \mathrm{pr} \otimes \tilde{\tau}_0)$ equipped with $\tau_{M'}$ (see Lemma 8.2 (a)) to that of $\mathrm{Ind}_{M'(\tilde{\lambda})^{m/\mathfrak{h}}_{\tau_0}}^{M'(\tilde{\lambda})^{m/\mathfrak{h}}}(\tilde{\eta}_0 \circ \mathrm{pr} \otimes (\tilde{\tau}_0)_{M'})$, where $(\tilde{\tau}_0)_{M'}$ is the $c(\mathrm{pr} \times \mathrm{pr})$-projective representation of $M'(\tilde{\lambda})^{m/\mathfrak{h}}_{\tau_0}$ such that $\delta_{\lambda_+, m'}^{m/\mathfrak{h}}(\tilde{\tau}_0)_{M'} = \delta_{\lambda_+}^{g/\mathfrak{h}}\tilde{\tau}_0$ on $G(\lambda_+)^{g/\mathfrak{h}}_{\tau_0}$. It is written $\iota_1: \varphi \mapsto \varphi_{M'}$ with: $(\delta_{\lambda_+, m'}^{m/\mathfrak{h}})^{-1} \varphi_{M'} = (\delta_{\lambda_+}^{g/\mathfrak{h}})^{-1} \varphi$ (see the proof of 8.2 (a)).

The second, for $M'$, goes from the space of $(\mathrm{Ind}_{M'(\tilde{\lambda})^{m/\mathfrak{h}}_{\tau_0}}^{M'(\tilde{\lambda})^{m/\mathfrak{h}}}(\tilde{\eta}_0 \circ \mathrm{pr} \otimes (\tilde{\tau}_0)_{M'}) \otimes ST_{\tilde{\lambda}}^{M'_0}$ to that of $\mathrm{Ind}_{G(\lambda_+)^{m/\mathfrak{h}}_{\tau_0}}^{M'(\tilde{\lambda})^{m/\mathfrak{h}}}(\tilde{\eta}_0 \circ \mathrm{pr} \otimes (\tilde{\tau}_0)_{M'}) \otimes ST_{\tilde{\lambda}}^{M'_0}$, where the representations in tensor products of $M' = M'(\tilde{\lambda})M'_0$ and $G(\lambda_+)^{m/\mathfrak{h}}_{\tau_0}M'_0$ referred to are constructed as $T_{\tilde{\lambda}, \tau_{M'}}^{M'}$ in 8.2 (a). It sends an element of the form $\varphi \otimes v$ to the element $\psi$ satisfying

$$ \psi(xy) = \varphi(\hat{x}) \otimes T_{\tilde{\lambda}}^{M'_0}(y)^{-1} S(\hat{x})^{-1} \cdot v $$
for $\hat{x} \in M'(\tilde{\lambda})^{m/\mathfrak{h}}$ lying over $x \in G(\lambda_+)$ and $y \in M'_0$.
Denote by $(M'_{G_0}, (\tau_0)_{M'})$ the “pair $(M', \tau_{M'})$” obtained from $(G_0, \tilde{\lambda}, \mathfrak{a}^{*+}, \tau_0)$ instead of $(G, \tilde{\lambda}, \mathfrak{a}^{*+}, \tau_+)$. There exists a unique $c \circ (\text{pr} \times \text{pr})$-projective representation $\tilde{T}_0$ of $G(\lambda_+)_{\tau_0} G_0$ that extends $T_{\tilde{\lambda}, \mathfrak{a}^{*+}, \tau_0}^{G_0}$ so that its action on an element $\phi$ of the space of $\text{Ind}_{M'_{G_0}, U}^{G_0}((\tau_0)_{M'} \otimes ST_{\tilde{\lambda}}^{M'_0}) \otimes \mathbb{1}_U)$ satisfies

$$ (\tilde{T}_0(xg) \cdot \phi)(h) = |\det(\text{Ad } x)_U|^{1/2} \times ((\tilde{\tau}_0)_{M'}(\hat{x}) \otimes S(\hat{x})) \cdot \phi(g^{-1}x^{-1}hx) \\ \text{for } \hat{x} \in M'(\tilde{\lambda})_{\tau_0}^{m'/h} \text{ lying over } x \in G(\lambda_+)_{\tau_0} \text{ and } g, h \in G_0. $$

The third isomorphism, of $G(\lambda_+)_{\tau_0} G_0$-modules, is provided by the restriction map to $G_0$, from the space of $\text{Ind}_{(G(\lambda_+)_{\tau_0} M'_0)_U}^{G(\lambda_+)_{\tau_0} G_0}((\tilde{\eta}_0 \circ \text{pr} \otimes (\tilde{\tau}_0)_{M'}) \otimes ST_{\tilde{\lambda}}^{M'_0}) \otimes \mathbb{1}_U)$ to that of $\tilde{\eta}_0 \circ \text{pr} \otimes \tilde{T}_0$ (identified with a space of functions on $G_0$).

To summarize, we find that:

$$ T_{\tilde{\lambda}, \mathfrak{a}^{*+}, \tau_+}^{G} = \text{Ind}_{G(\lambda_+)_{\tau_0} G_0}^{G}(\tilde{\eta}_0 \circ \text{pr} \otimes \tilde{T}_0). $$

It remains to apply Mackey's theory. According to the connected case study above, we have $T_{\tilde{\lambda}, \mathfrak{a}^{*+}, \tau_0}^{G_0} \in \tilde{G}_0$ and the stabilizer of $T_{\tilde{\lambda}, \mathfrak{a}^{*+}, \tau_0}^{G_0}$ in $G$ is equal to $G(\lambda_+)_{\tau_0} G_0$. Proposition 9.4 (b) provides the bijection $\tilde{\sigma} \mapsto \text{Ind}_{G(\lambda_+)_{\tau_0} G_0}^{G}(\tilde{\sigma} \circ \text{pr} \otimes \tilde{T}_0)$ from the set of isomorphism classes of irreducible $c^{-1}$-projective representations of $G(\lambda_+)_{\tau_0}/G_0(\lambda_+)$ onto the set of elements of $\tilde{G}$ whose restriction to $G_0$ is equal to the Hilbert direct sum of the $g \cdot T_{\tilde{\lambda}, \mathfrak{a}^{*+}, \tau_0}^{G_0}$ with $\dot{g} \in G/G(\lambda_+)_{\tau_0} G_0$. Thus, given the connected case, we have: the representation class $T_{\tilde{\lambda}, \mathfrak{a}^{*+}, \tau_+}^{G}$ is tempered irreducible, every $\pi \in \tilde{G}$ which is tempered is obtained by Proposition 9.4 (c), and by an easy calculation, any $(\tilde{\lambda}', \mathfrak{a}'^{*+}, \tau_+') \in X_G^{\text{final}, +}$ such that $T_{\tilde{\lambda}', \mathfrak{a}'^{*+}, \tau_+'}^{G} = T_{\tilde{\lambda}, \mathfrak{a}^{*+}, \tau_+}^{G}$ is conjugate to $(\tilde{\lambda}, \mathfrak{a}^{*+}, \tau_+)$ under $G$. A fortiori for such a triple $(\tilde{\lambda}', \mathfrak{a}'^{*+}, \tau_+')$ we have $(\tilde{\lambda}', \tau_') \in G \cdot (\tilde{\lambda}, \tau)$, where $\tau_+$ is deduced from $\tau_+$ as in Definition 5.5 (c). $\square$

**Remark 9.7.** In this remark we assume that $\tilde{\lambda} \in \tilde{g}_I^*$ and set $\tau = \tau_+$. We will explicitly find the orbit of $(\tilde{\lambda}, \tau)$ under $G$ starting from $\text{tr} T_{\tilde{\lambda}, \tau}^G$.

First, I retrieve $G \cdot \tilde{\lambda}$. Theorem 10.2 provides the equality

$$ (\text{tr} T_{\tilde{\lambda}, \tau}^G)_1 = \dim \tau \times \beta_{G \cdot \tilde{\lambda}}|_{V_1} $$

where $V_1$ is an open neighborhood of 0 in $g$, which therefore intersects every connected component of $g_{ss reg}$. By Proposition 2.5 (c) and the end of Remark 2.4 (1), we deduce that $T_{\tilde{\lambda}, \tau}^G$ determines $\beta_{R_G(G\tilde{\lambda})}$. The tempered Radon measures $\beta_{\Omega}$ on $g^*$ with $\Omega \in G \setminus g_{reg}^*$ are linearly independent, because each $\beta_{\Omega}$ is nonzero and concentrated on $\Omega$. Given Proposition 2.2 (a), the representation class $T_{\tilde{\lambda}, \tau}^G$ determines the orbit $G \cdot \tilde{\lambda}$.

I will now reconstruct $\text{tr} \tau$ from $T_{\tilde{\lambda}, \tau}^G$ and $\tilde{\lambda}$. We already know that $\tau(\exp X) = e^{i\lambda(X)}$ id for $X \in \mathfrak{h}$. Let $\hat{e} \in G(\lambda_+)^{g/\mathfrak{h}}$ lie over an elliptic $e \in G(\lambda_+)$. Associate it to a $e_0 \in e \exp t(e)$ as in Lemma 3.2 (b). Theorem 10.2 expresses $(\text{tr} T_{\tilde{\lambda}, \tau}^G)_{e_0}$ as a linear combination of the locally integrable functions $\beta_{G(e_0) \cdot \tilde{\lambda}'_0}|_{V_{e_0}}$ with $\tilde{\lambda}'_0 \in G(e_0) \setminus \tilde{g}(e_0)^* \setminus \tilde{g}(e_0)^*$ that are linearly independent. The coefficient of $(\text{tr} T_{\tilde{\lambda}, \tau}^G)_{e_0}$ following $\beta_{G(e_0) \cdot \tilde{\lambda}[e_0]}|_{V_{e_0}}$ provides $\text{tr} \tau(\hat{e})$. $\blacksquare$
## IV. Characters of representations

Again, consider an element $\tilde{\lambda} = (\lambda, \mathcal{F}^+)$ of $\tilde{\mathfrak{g}}_{reg}^*$ and set $\mathfrak{h} = \mathfrak{g}(\tilde{\lambda})$. Denote by $\mu$ and $\nu$ (resp. $t$ and $a$) the infinitesimally elliptic and hyperbolic components of $\lambda$ (resp. $\mathfrak{h}$). Let $a^{*+}$ be a chamber of $(\mathfrak{g}(\lambda)(i\rho_{\mathcal{F}^+}), a)$. I will use the other notations from Part III. (including $M$, $\mathfrak{m}$, $\tau_M$ for Section 7., and, $M'$, $\mathfrak{m}'$, $U$, $\tau_{M'}$ for Section 8.) without recalling what they represent.

In this section, we consider an elliptic element $e$ of $G$ (not to be confused with the base $e$ of natural logarithms).

## 10. Character restriction formula

The formula I will now write generalizes that of Bouaziz in [Bou 87, Th. 5.5.3 p. 52] (see also [DHV 84, Th. (7) p. 106]), while reducing to it, and that of Rossmann [Ros 80, p. 64] relating to the case $e = 1$.

**Definition 10.1.** (a) Denote by $D_G$ the function on $G$ whose restriction to each connected component $G^+$ of $G$ sends $x \in G^+$ to the coefficient of $T^{r^+}$ in $\det(T \text{id} + \text{id} - \text{Ad} x)$, where $r^+$ is the common rank of the reductive Lie algebras $\mathfrak{g}(x^+)$ as $x^+$ ranges over the set of semisimple elements of $G^+$ (see [Bou 87, Lem. 1.4.1 p. 6]). It is analytic and invariant under the action of the group $\text{int } G := \{\text{int } g; g \in G\}$, where $\text{int } g$ denote the map $x \in G \mapsto gxg^{-1} \in G$.

(b) Denote by $G_{ss reg}$ the set of semisimple $x \in G$ such that $\mathfrak{g}(x)$ is abelian. Therefore $G_{ss reg}$ is the dense open set of $G$ with negligible complement, formed by the points where $D_G$ does not vanish (see [Bou 87, 1.3 p. 5]).

(c) Write $d_e = \frac{1}{2} \dim(1 - \text{Ad} e)(\mathfrak{g}/i)$ and $D_e = \det(1 - \text{Ad} e)_{(1-\text{Ad} e)(\mathfrak{g}/i)} > 0$, independently of the choice of a Cartan subalgebra $j_e$ of $\mathfrak{g}(e)$ to which we associate the Cartan subalgebra $j := C_{\mathfrak{g}}(j_e)$ of $\text{Car } \mathfrak{g}$.

(d) Set

$$ \mathcal{V}_e = \{ X \in \mathfrak{g}(e) \mid |\text{Im } z| < \varepsilon_e \text{ for any eigenvalue } z \text{ of } \text{ad}_{\mathfrak{g}} X \} $$

$$ \text{and } k_e(X) = \left( \det \left( \frac{\text{e}^{\text{ad } X/2} - \text{e}^{-\text{ad } X/2}}{\text{ad } X} \right)_{\mathfrak{g}(e)} \frac{\det(1 - \text{Ad}(e \exp X))_{\mathfrak{g}/\mathfrak{g}(e)}}{\det(1 - \text{Ad } e)_{\mathfrak{g}/\mathfrak{g}(e)}} \right)^{1/2} > 0 $$

$$ \text{for } X \in \mathcal{V}_e, \text{ where } \varepsilon_e = \inf\{\theta \in ]0, 2\pi] \mid \text{e}^{i\theta} \text{ is an eigenvalue of } \text{Ad}_G e\} \le 2\pi. $$

(e) To every $\text{int } G$-invariant generalized function $\Theta$ on $G$, we associate (see [DHV 84, (6) p. 98]) the $\text{Ad } G(e)$-invariant generalized function $\Theta_e$ on $\mathcal{V}_e$, determined by the equality $\Theta_e(X) = k_e(X) \times \Theta|_{e \exp \mathcal{V}_e}(e \exp X)$ of generalized functions in $X \in \mathcal{V}_e$.

$$ \text{The equality } G = \bigcup_{\substack{e_0 \in G \\ e_0 \text{ elliptique}}} \text{int } G \cdot (e_0 \exp \mathcal{V}_{e_0}) \text{ of [Bou 87, Lem. 8.1.1 p. 72]} $$

(see [DV 93, Lem. 40 p. 41]) shows that in the situation of (e), the set of $\Theta_{e_0}$ with $e_0 \in G$ elliptic determine $\Theta$.

According to [Bou 87, 3.1 top of p. 21], any topologically irreducible unitary representation $\pi$ of $G$ is trace class, ant its character is a locally integrable function $\text{tr } \pi$ on $G$ invariant under $\text{int } G$ whose restriction to $G_{ss reg}$ is analytic.
**Theorem 10.2.** Given $\tau \in X_G^{\text{Ind}}(\tilde{\lambda})$ and $\tau_+ \in X_G^{\text{final},+}(\tilde{\lambda}, \mathfrak{a}^{*+})$ associated as in 5.5 (c), we have

$$ (\operatorname{tr} T_{\tilde{\lambda}, \tau})_e = D_e^{-\frac{1}{2}} \sum_{\tilde{\lambda}'_e \in G(e) \setminus \tilde{\mathfrak{g}}(e)^*_{\text{reg}}} |\{\mathfrak{a}'^{*+}_e\}|^{-1} \left( \sum_{\tilde{\lambda}' \in G \setminus \tilde{\lambda} \cap \tilde{\mathfrak{g}}^*_{\text{reg}}(e)} \pm_{\tilde{e}'} i^{-d_{e', \tilde{\lambda}}} \operatorname{tr} \tau(\tilde{e}') \right) \left. \hat{\beta}_{G(e) \cdot \tilde{\lambda}'_e} \right|_{V_e} \\ \text{such that } \tilde{\lambda}'[e] \text{ exists} \\ \text{and } \tilde{\lambda}'[e] = \tilde{\lambda}'_e \\ = (\operatorname{tr} T_{\tilde{\lambda}, \mathfrak{a}^{*+}, \tau_+})_e = i^{-d_e} D_e^{-\frac{1}{2}} \sum_{\tilde{\lambda}'_e \in G(e) \setminus \tilde{\mathfrak{g}}(e)^*_{\text{reg}}} |\{\mathfrak{a}'^{*+}_e\}|^{-1} \left( \sum_{\lambda'_+ \in G \setminus \lambda_+ \cap \mathfrak{g}(e)^*} \pm_{\tilde{e}'} \operatorname{tr} \tau_+(\tilde{e}'_+) \right) \left. \hat{\beta}_{G(e) \cdot \tilde{\lambda}'_e} \right|_{V_e} \\ \text{(hence } \tilde{\lambda}'[e] \text{ exists, see 1.4)} \\ \text{such that } \tilde{\lambda}'[e] = \tilde{\lambda}'_e $$

where the sum over $\tilde{\lambda}'$ is understood by choosing $g \in G$ such that $\tilde{\lambda}' = g\tilde{\lambda}$ and then fixing $\tilde{e}' \in G(\tilde{\lambda})\mathfrak{g}/\mathfrak{g}(\lambda)(i\rho_{\mathcal{F}+})$ lying over $e' = g^{-1}eg$.

(resp.: the sum over $\lambda'_+$ is understood by choosing $g \in G$ such that $\lambda'_+ = g\lambda_+$ then setting $\tilde{\lambda}' = g\tilde{\lambda}$ and fixing $\tilde{e}'_+ \in G(\lambda_+)\mathfrak{g}/\mathfrak{h}$ above $e' = g^{-1}eg$),

$\pm_{\tilde{e}'}$ is the sign such that $\mathcal{O}(\tilde{e}')_{(1-\text{Ad }e')(\mathfrak{g}/\mathfrak{g}(\lambda)(i\rho_{\mathcal{F}+}))} = \pm_{\tilde{e}'} \mathcal{O}(B_{\lambda_{\text{can}}})_{(1-\text{Ad }e')(\mathfrak{g}/\mathfrak{g}(\lambda)(i\rho_{\mathcal{F}+}))}$

(resp.: $\pm_{\tilde{e}'_+}$ is the sign such that $\mathcal{O}(\tilde{e}'_+)^{(1-\text{Ad }e')(\mathfrak{g}/\mathfrak{h})} = \pm_{\tilde{e}'_+} \mathcal{O}(B_{\lambda_+})_{(1-\text{Ad }e')(\mathfrak{g}/\mathfrak{h})}$), $d_{\tilde{e}'}^{(\lambda)(i\rho_{\mathcal{F}+})}$ is the coefficient $d_{e'}$ relative to $\mathfrak{g}(\lambda)(i\rho_{\mathcal{F}+})$, and $d_{e', \tilde{\lambda}} = d_{\tilde{e}'} - d_{\tilde{e}'}^{(\lambda)(i\rho_{\mathcal{F}+})}$,

$\mathfrak{h}'_e = \mathfrak{g}(e)(\tilde{\lambda}'[e])$ and $\mathfrak{a}'_e$ is the infinitesimally hyperbolic component of $\mathfrak{h}'_e$, $(\lambda', \mathcal{F}'^+) = \tilde{\lambda}'[e]$ and $\{\mathfrak{a}'^{*+}_e\}$ is the set of chambers of $(\mathfrak{g}(e)(\lambda')(i\rho_{\mathcal{F}+})), \mathfrak{a}'_e)$.

The summations relating to $G(e)\setminus \tilde{\mathfrak{g}}(e)^*_{\text{reg}}$ involve a finite number of nonzero terms.

The other summations relate to finite sets. The condition "$\tilde{\lambda}'[e]$ exists" (see 3.2) can be omitted, as it is satisfied when $\operatorname{tr} \tau(\tilde{e}') \neq 0$ (calculation of $\operatorname{tr} \tau$ from 5.5 (c)).

In particular, for almost all $X \in \mathcal{V}_1$ we have

$$ \det \left( \frac{e^{\text{ad }X/2} - e^{-\text{ad }X/2}}{\text{ad }X} \right)^{1/2} \times \operatorname{tr} T_{\tilde{\lambda}, \mathfrak{a}^{*+}, \tau_+} (\exp X) = \dim \tau_+ \times \hat{\beta}_{G \cdot \tilde{\lambda}}(X). $$

**Remark 10.3.** Here are two examples and an important clarification.

(1) First example: $G = GL(2, \mathbb{R})$ and $e = \begin{pmatrix} 0 & -1 \\ 1 & 0 \end{pmatrix}$, $\tilde{\lambda} = (0, \mathcal{F}^+)$ with $\mathcal{F}^+ \subseteq \text{i so}(2)^* \oplus \text{Z}(\mathfrak{g})^*$ and $\tau = \chi_{\tilde{\lambda}}^G$. So $\tilde{\lambda}[e]$ exists in $\tilde{\mathfrak{g}}(e)^*_{\text{reg}}$. The sum over $G \cdot \tilde{\lambda} \cap \tilde{\mathfrak{g}}^*_{\text{reg}}(e)$ here involves two nonzero, opposite complex numbers.

(2) Second example: $G$ is the semidirect product of $\mathbb{Z}/2\mathbb{Z}$ by $SL(2, \mathbb{R})^2$ where $\mathbb{Z} \in \mathbb{Z}/2\mathbb{Z}$ operates on $SL(2, \mathbb{R})^2$ by permutation of coordinates and $e = ((\begin{pmatrix} 0 & -1 \\ 1 & 0 \end{pmatrix}), (\begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix}))$, $\tilde{\lambda} = (0, \mathcal{F}^+)$ with $\mathcal{F}^+ \subseteq (\text{i so}(2)^*)^2$ not stable under $\mathbb{Z}/2\mathbb{Z}$ and $\tau = \chi_{\tilde{\lambda}}^G$. Therefore $\tilde{\lambda}[e]$ exists in $\tilde{\mathfrak{g}}(e)^*_{\text{reg}}$. The two elements $\tilde{\lambda}'$ of $G \cdot \tilde{\lambda} \cap \tilde{\mathfrak{g}}^*_{\text{reg}}(e)$ are conjugate under $G(e)$, and satisfy: $\tilde{\lambda}'[e]$ exists and $\tilde{\lambda}'[e] = \tilde{\lambda}[e]$. The corresponding summed complex numbers are nonzero and, of course, equal.

(3) It may happen that nonlinearly independent measures $\hat{\beta}_{G(e) \cdot \tilde{\lambda}'_e}$ occur with nonzero coefficients in the character formula.

Indeed, consider the following case: $G$ is the canonical semidirect product of the symmetric group $\mathfrak{S}_4$ by $SL(3, \mathbb{R})^4$, $\lambda = 0$, $\mathfrak{h}$ is the product of two copies of a split Cartan subalgebra of $\mathfrak{sl}(3, \mathbb{R})$ with two copies of a fundamental Cartan subalgebra of $\mathfrak{sl}(3, \mathbb{R})$, $\mathfrak{a}^{*+}$ is stable under the permutation of the first two components. Define $e = (12)$ and $g = (13)(24)$ in $\mathfrak{S}_4$. Therefore, we have: $G \cdot \tilde{\lambda} \cap \tilde{\mathfrak{g}}^*_{\text{reg}}(e) =$
$G(e) \cdot \tilde{\lambda} \cup G(e) \cdot g\tilde{\lambda}, \tilde{\lambda}[e]$ and $(g\tilde{\lambda})[e]$ exist in $\tilde{g}(e)_{reg}^*$, $G(e) \cdot \tilde{\lambda}[e] \neq G(e) \cdot (g\tilde{\lambda})[e]$ and $\hat{\beta}_{G(e) \cdot \tilde{\lambda}[e]} = \hat{\beta}_{G(e) \cdot (g\tilde{\lambda})[e]}$ (see 2.4 (3)). Furthermore, there exists $\tau_+ \in X_G^{final,+}(\tilde{\lambda}, a^{*+})$ for which the coefficients of $\hat{\beta}_{G(e) \cdot \tilde{\lambda}[e]}|_{V_e}$ and $\hat{\beta}_{G(e) \cdot (g\tilde{\lambda})[e]}|_{V_e}$ as sums over $\tilde{\lambda}'$ are equal and nonzero. ■

**Remark 10.4.** Assume that $G$ is connected and use the notations of Section 7. Various authors have, in certain situations, used character formulas equivalent to those in the previous theorem, but relating to other parametrizations.

To link these formulas, fix a system of positive roots $R_0^+(\mathfrak{g}_C, \mathfrak{h}_C)$ of $R(\mathfrak{g}_C, \mathfrak{h}_C)$ that contains the conjugates of its nonimaginary roots. Let $\rho_{g,\mathfrak{h},0}$ and $\rho_{m,t,0}$ denote the half sums of positive roots associated with $R_0^+(\mathfrak{g}_C, \mathfrak{h}_C)$ and with $R_0^+(\mathfrak{g}_C, \mathfrak{h}_C) \cap R(\mathfrak{m}_C, \mathfrak{t}_C)$.

(a) The map from $C(\mathfrak{g}(\lambda), \mathfrak{h})$ to the set of chambers of $(\mathfrak{m}_C, \mathfrak{t}_C)$ in $i\mathfrak{t}^*$ whose closure contains $i\mu$, which to an element $\mathcal{F}^{\prime+}$ of $C(\mathfrak{g}(\lambda), \mathfrak{h})$ associates the chamber of $(\mathfrak{m}_C, \mathfrak{t}_C)$ containing $i\mu_{\mathfrak{m},(\lambda,\mathcal{F}^{\prime+}),\{0\}}$, is bijective. Denote by $\mathcal{C}^+$ the image of $\mathcal{F}^{\prime+}$ under this map. Let

$$ \mathrm{sg}_0(\mathcal{F}^+, \mu) = \prod_{\substack{\alpha \in R_0^+(\mathfrak{g}_C, \mathfrak{h}_C) \\ \alpha \text{ compact}}} \mathrm{sg}(\mathcal{C}^+(H_\alpha)) \times \prod_{\substack{\alpha \in R_0^+(\mathfrak{g}_C, \mathfrak{h}_C) \\ \alpha \text{ noncompact imaginary}}} \mathrm{sg}(-\mathcal{C}^+(H_\alpha)). $$

(b) Let $T = C_M(t) = C_M(m) \exp t$. The map from $X_M^{irr}(\tilde{\mu})$ to the set of unitary characters of $T$ of differential $i\mu - \rho_{m,t,0}$ which to $\sigma \in X_M^{irr}(\tilde{\mu})$ associates the unitary character $\eta$ of $T$ such that $\eta(x \exp X) = \sigma(x, 1) e^{(i\mu - \rho_{m,t,0})(X)}$ for $x \in C_M(m)$ and $X \in t$, is bijective.

(c) Let $\sigma \in X_M^{irr}(\tilde{\mu})$. Denote by $\eta$ its image under the map of (b). The function $\Theta_{\eta}^{(C^+)}$ of [Var 77, Th. 23 p. 260] relative to the group $M$ is given by:

$$ \Theta_{\eta}^{(C^+)} = \mathrm{sg}_0(\mathcal{F}^+, \mu) \times \mathrm{tr} T_{\tilde{\mu}, \sigma}^M. $$

In particular, when $M_0$ is semisimple and “acceptable,” the function $\Theta_{i\mu, C^+}$ of [Har 65, p. 305] relative to the group $M_0$ is given by:

$$ \Theta_{i\mu, C^+} = \mathrm{sg}_0(\mathcal{F}^+, \mu) \times \mathrm{tr} T_{\tilde{\mu}}^{M_0}. $$

Furthermore, when $G$ is linear, $\mathrm{tr} T_{\tilde{\mu}, \sigma}^M$ is denoted $\Theta^M(i\mu, C^+, \sigma(\cdot, 1)|_{Z(M)})$ in [KZ 82, p. 397] by identifying $t$ and $t^*$.

(d) Consider the case where $G$ is the neutral component of the group of real points of a complex linear algebraic group defined over $\mathbb{R}$ which is semisimple and simply connected. This gives us a complex character $\xi_\rho$ of $\exp \mathfrak{h}_C$ of differential $\rho_{g,\mathfrak{h},0}$. Fix a unitary character $\eta$ of $T$ as in (b). When $\nu$ is $(\mathfrak{g}, \mathfrak{a})$-regular, the function $\theta(TA, \xi_\rho \eta, \nu)$ of [Her 83, p. 244] is written:

$$ \theta(TA, \xi_\rho \eta, \nu) = |C(\mathfrak{g}(\lambda), \mathfrak{h})|^{-1} \times \sum_{\mathcal{F}^{\prime+} \in C(\mathfrak{g}(\lambda), \mathfrak{h})_{reg}} \mathrm{sg}_0(\mathcal{F}^{\prime+}, \mu) \times \mathrm{tr} T_{\tilde{\lambda}', \tau'}^G $$

where $\tilde{\lambda}' = (\lambda, \mathcal{F}^{\prime+})$, $\lambda'_+$ is associated with $(\tilde{\lambda}', a^*)$ as in 1.3 (b), and $\tau' \in X_G^{irr}(\tilde{\lambda}')$ is determined by the equality $(\delta_{\lambda'_+}^{g/\mathfrak{h}} \tau')|_{Z(M)} = \eta|_{Z(M)}$. ■

The theorem will be proven in three steps. The first two steps follow Bouaziz: from $G$ to $G(\lambda_+)G_0$ (“beginning of the proof”), then from $G(\lambda_+)G_0$ to $M'$ (“continuation of the proof”). In the last step (“end of the proof”), a variant
of Zuckerman’s translation functor will enable me to relate the characters of the
representations of $M'$ that interest me to the characters studied by Bouaziz.

**Beginning of the proof of the theorem.** Let $\widehat{e}'_+ \in G(\lambda_+)^{\mathfrak{g}/\mathfrak{h}}$ and $\widehat{e}' \in G(\tilde{\lambda})^{\mathfrak{g}/\mathfrak{g}(\lambda)(i\rho_{\mathcal{F}+})}$ lie over the same elliptical element $e'$ of $G(\lambda_+)$. The eigenvalues other than $-1$ and $1$ of the restriction of $\mathrm{Ad} \, e'^{\mathcal{C}}$ to $\mathfrak{w}_{\mathcal{C}} := \sum_{\alpha \in R^+(\mathfrak{g}(\lambda)(i\rho_{\mathcal{F}+})_{\mathcal{C}}, \mathfrak{h}_{\mathcal{C}})} \mathfrak{g}_{\mathcal{C}}^{\alpha}$ are nonreal pairwise conjugates of absolute value $1$. Since $\mathcal{L}_{\tilde{\lambda}, \mathfrak{a}^{*+}}/\mathfrak{h}_{\mathcal{C}}$ (see 5.4 (a)) is a direct sum of the $\mathrm{Ad} \, e'^{\mathcal{C}}$-invariants subspaces $\mathcal{L}_{\tilde{\lambda}}/\mathfrak{g}(\lambda)(i\rho_{\mathcal{F}+})_{\mathcal{C}}$ (see 5.4 (c)) and $\mathfrak{w}_{\mathcal{C}}$, we have $q\mathcal{L}_{\tilde{\lambda}, \mathfrak{a}^{*+}}/\mathfrak{h}_{\mathcal{C}}((\mathrm{Ad} \, e')_{\mathfrak{g}/\mathfrak{h}}) = q\mathcal{L}_{\tilde{\lambda}}/\mathfrak{g}(\lambda)(i\rho_{\mathcal{F}+})_{\mathcal{C}}((\mathrm{Ad} \, e')_{\mathfrak{g}/\mathfrak{g}(\lambda)(i\rho_{\mathcal{F}+})})$ relative to $B_{\lambda_+}$, and then by 5.4 and 4.2 (d):

$$
\pm_{\widehat{e}'_+} \rho_{\tilde{\lambda}, \mathfrak{a}^{*+}}^{\mathfrak{g}/\mathfrak{h}}(\widehat{e}') = \pm_{\widehat{e}'} \rho_{\tilde{\lambda}}^{\mathfrak{g}/\mathfrak{g}(\lambda)(i\rho_{\mathcal{F}+})}(\widehat{e}') \times i^{-1/2} \dim(1-\mathrm{Ad} \, e')(\mathfrak{g}(\lambda)(i\rho_{\mathcal{F}+})/\mathfrak{h})
$$

where the signs $\pm \widehat{e}'_+$ and $\pm \widehat{e}'$ are defined as in the theorem.

Let $\tilde{\lambda}'_0 \in G \cdot \tilde{\lambda} \cap \tilde{\mathfrak{g}}_{reg}^*(e)$, $g_0 \in G$ satisfying $\tilde{\lambda}'_0 = g_0 \tilde{\lambda}$, and $\widehat{e}'_0 \in G(\tilde{\lambda})^{\mathfrak{g}/\mathfrak{g}(\lambda)(i\rho_{\mathcal{F}+})}$ lie over $e'_0 = g_0^{-1}eg_0$. By Definition 5.5 (c) and the above, we obtain

$$
\pm_{\widehat{e}_0} i^{-d_{e'_0, \tilde{\lambda}}} \operatorname{tr} \tau(\widehat{e}'_0) = \sum_{\substack{\tilde{x} \in G(\tilde{\lambda})/G(\lambda_+) \\ \text{tel que } x^{-1}e'_0 x \in G(\lambda_+)}} \pm_{\widehat{e}'_{+,x}} i^{-d_e} \operatorname{tr} \tau_+(\widehat{e}'_{+,x})
$$

independently of the choice of $\widehat{e}'_{+,x} \in G(\lambda_+)^{\mathfrak{g}/\mathfrak{h}}$ that lies over $\widehat{e}'_x = x^{-1}e'_0 x$.

Next, for every $\tilde{\lambda}'_e \in \tilde{\mathfrak{g}}(e)_{reg}^*$ we obtain the equality of finite sums

$$
\sum_{\tilde{\lambda}'_0 \in G \cdot \tilde{\lambda} \cap \tilde{\mathfrak{g}}_{reg}^*(e)} \pm_{\widehat{e}_0} i^{-d_{e'_0, \tilde{\lambda}}} \operatorname{tr} \tau(\widehat{e}'_0) = \sum_{\lambda'_+ \in G \cdot \lambda_+ \cap \mathfrak{g}(e)^*} \pm_{\widehat{e}_+} i^{-d_e} \operatorname{tr} \tau_+(\widehat{e}_+). \\
\text{such that } \tilde{\lambda}'_0[e] = \tilde{\lambda}'_e \qquad \text{such that } \tilde{\lambda}'[e] = \tilde{\lambda}'_e
$$

The two expressions of character proposed in the theorem are therefore equal.

Let $\tilde{\lambda}'$ (resp. $\lambda'_+$) be a term associated with a $\tilde{\lambda}'_e$ as in the theorem. We have:
$$ \{ g\tilde{\lambda} \in G \cdot \tilde{\lambda} \cap \tilde{\mathfrak{g}}_{reg}^*(e) \mid (g\tilde{\lambda})[e] = \tilde{\lambda}'_e \text{ and } g\tilde{\lambda} \in G(e) \cdot \tilde{\lambda}' \} = G(e)(\tilde{\lambda}'_e) \cdot \tilde{\lambda}' $$
(resp. $\{ g\lambda_+ \in G \cdot \lambda_+ \cap \mathfrak{g}(e)^* \mid (g\tilde{\lambda})[e] = \tilde{\lambda}'_e \text{ and } g\lambda_+ \in G(e) \cdot \lambda'_+ \} = G(e)(\tilde{\lambda}'_e) \cdot \lambda'_+ $).

The theorem is therefore equivalent to each of the following formulas ($\mathbf{F}$) and ($\mathbf{F}_+$):

$$
(\mathrm{tr} \, T_{\tilde{\lambda}, \tau}^{G})_{e}^{(\mathbf{F})} = D_{e}^{-\frac{1}{2}} \sum_{\substack{\tilde{\lambda}' \in G(e) \setminus G \cdot \tilde{\lambda} \cap \tilde{\mathfrak{g}}_{reg}^{*}(e) \\ \text{such that } \tilde{\lambda}'[e] \in \tilde{\mathfrak{g}}(e)_{reg}^{*}}}
$$

$$
(\mathrm{tr} \, T_{\tilde{\lambda}, \mathfrak{a}^{*+}, \tau_+}^G)_e \stackrel{(\mathbf{F}_+)}{=} i^{-d_e} D_e^{-\frac{1}{2}} \sum_{\substack{\lambda'_+ \in G(e) \setminus G \cdot \lambda_+ \cap \mathfrak{g}(e)^* \\ \text{such that } \tilde{\lambda}'[e] \in \tilde{\mathfrak{g}}(e)_{reg}^*}} \pm_{\widehat{e}_+} \frac{|G(e)(\tilde{\lambda}'[e]) \cdot \lambda'_+|}{|\{\mathfrak{a}_e^{*+}\}|} \mathrm{tr} \, \tau_+(\widehat{e}_+) \hat{\beta}_{G(e) \cdot \tilde{\lambda}'[e]} |_{V_e} .
$$

In particular this makes it possible to specify the results set out in the introduction.

$$
\text{According to Definition 2.3, we have } \hat{\beta}_{\omega} = \sum_{\omega_0 \in G(e)_0 \setminus \omega} \hat{\beta}_{\omega_0} \text{ for } \omega \in G(e) \setminus \tilde{\mathfrak{g}}(e)_{reg}^*.
$$

Thus, the theorem is equivalent to the statement obtained by replacing $G(e)$ with $G(e)_0$, and therefore also to the statement obtained by replacing $G(e)$ with $G(e)_0$ in $(\mathbf{F}_+)$. Let $\lambda'_+$ and $\tilde{\lambda}'$ be as in $(\mathbf{F}_+)$. The group $G(e)_0(\tilde{\lambda}'[e])$ operates transitively on $\{\mathfrak{a}_e^{*+}\}$ by the action of certain elements of $W(\mathfrak{g}(e)_{\mathcal{C}}, \mathfrak{h}'_{e_{\mathcal{C}}})$ (theorem notation). Since $\lambda'_+$ belongs to $\mathfrak{h}'_{e}^* \cap \mathfrak{g}(e)^*_{ss_{reg}}$, the stabilizer of an element of $\{\mathfrak{a}_e^{*+}\}$ is equal to $G(e)_0(\tilde{\lambda}'[e])(\lambda'_+)$ by Lemma 1.4. Therefore $|G(e)_0(\tilde{\lambda}'[e]) \cdot \lambda'_+| = |\{\mathfrak{a}_e^{*+}\}|$.
Consequently, the formulas to be proven are equivalent to:

$$ (\operatorname{tr} T_{\tilde{\lambda}, \mathbf{a}^{*+}, \tau_{+}}^{G})_{e} \stackrel{(\mathbf{F}_{+,0})}{=} i^{-d_e} D_{e}^{-\frac{1}{2}} \sum_{\substack{\lambda'_{+} \in G(e)_{0} \setminus G \cdot \lambda_{+} \cap \mathfrak{g}(e)^{*} \\ \text{such that } (g\tilde{\lambda})[e] \in \tilde{\mathfrak{g}}(e)^{*}_{\text{reg}}}} \frac{\mathcal{O}(\widehat{e'_{+}}^{(1-\mathrm{Ad} \, e')(\mathfrak{g}/\mathfrak{h})})}{\mathcal{O}(B_{\lambda_{+}}^{(1-\mathrm{Ad} \, e')(\mathfrak{g}/\mathfrak{h})})} \operatorname{tr} \tau_{+}(\widehat{e'_{+}}) \hat{\beta}_{G(e)_{0} \cdot (g\tilde{\lambda})[e]} |_{V_{e}} $$

where $g \in G$ satisfies $\lambda'_+ = g\lambda_+$ and $\widehat{e'_+} \in G(\lambda_+)^{\mathfrak{g}/\mathfrak{h}}$ lies over $e' = g^{-1}eg$.

For later use (Section 11.) note that in the previous argument, we can replace $G(e)_0$ by any subgroup $L$ of $G(e)$ that contains $G(e)_0$ and whose adjoint group is included in $\mathrm{int}\,\mathfrak{g}(e)_{\mathbb{C}}$.

Since the orbits over which we sum are regular semisimple orbits, we can reproduce most of the arguments of [Bou 87].

We have $T_{\tilde{\lambda}, \mathbf{a}^{*+}, \tau_{+}}^{G} = \mathrm{Ind}_{G(\lambda_{+}) G_{0}}^{G} T_{\tilde{\lambda}, \mathbf{a}^{*+}, \tau_{+}}^{G(\lambda_{+}) G_{0}}$ by “induction in stages,” and then
$$ (\operatorname{tr} T_{\tilde{\lambda}, \mathbf{a}^{*+}, \tau_{+}}^{G})_{e} = \sum_{\substack{\dot{x} \in G/G(\lambda_{+}) G_{0} \\ \text{such that } x^{-1}ex \in G(\lambda_{+}) G_{0}}} \operatorname{Ad} x \cdot (\operatorname{tr} T_{\tilde{\lambda}, \mathbf{a}^{*+}, \tau_{+}}^{G(\lambda_{+}) G_{0}})_{x^{-1}ex} $$

Since the set $G \cdot \lambda_+ \cap \mathfrak{g}(e)^*$ is the union of the disjoint sets $\mathrm{Ad}^* x \cdot (G(\lambda_+)G_0 \cdot \lambda_+ \cap \mathfrak{g}(x^{-1}ex)^*)$ where $\dot{x} \in G/G(\lambda_+)G_0$ satisfies $x^{-1}ex \in G(\lambda_+)G_0$, we can assume that $G = G(\lambda_+)G_0$. . .

## 11. Transition from $T_{\tilde{\lambda}, \tau_{M'}}^{M'}$ to $T_{\tilde{\lambda}, \mathbf{a}^{*+}, \tau_{+}}^{G}$

In this section, we restrict ourselves to proving the theorem for $T_{\tilde{\lambda}, \tau_{M'}}^{M'}$ repeating Bouaziz's proof for the regular semisimple case verbatim.

The results stated in the following proposition are proved in [Bou 87, Lem. 7.1.3 p. 65 and Lem. 4.2.1 p. 38].

**Proposition 11.1. (Bouaziz)** (a) Let $\pi$ be a topologically irreducible unitary representation of $M'$. Associate to it $\Pi = \mathrm{Ind}_{M'U}^{M'G_0}(\pi \otimes \mathbb{1}_U)$. The unitary representation $\Pi$ of $M'G_0$ is a Hilbertian sum of a finite number of topologically irreducible subrepresentations (see [Dix 64, Prop. 5.4.13 (i) p. 110] or [Wal 88, Prop. p. 25]).

Assume that $e \in M'G_0$. For every $j_e \in \mathrm{Car}\,\mathfrak{g}(e)$ and every $X \in j_e$ such that $e \exp X \in (M'G_0)_{ss\,reg}$, we have

$$ |D_G(e \exp X)|^{1/2} (\mathrm{tr} \, \Pi)(e \exp X) = \sum_{(M' \cap G_0) g \in M' \cap G_0 \setminus \{g_0 \in G_0 | g_0 \in \exp j_e g_0^{-1} \subseteq M'\}} |D_{M'}(ge \exp Xg^{-1})|^{1/2} (\mathrm{tr} \, \pi)(ge \exp Xg^{-1}) $$

where the sum on the right relates to a finite set.

(b) Let $\nu' \in \mathfrak{g}^*$ be a hyperbolic linear form, let $\mathfrak{h}'$ be a fundamental Cartan subalgebra of $\mathfrak{g}(\nu')$, and let $M'_1$ be a subgroup of $G_0(\nu')$ that contains the group $C_{G_0}(\mathfrak{h}')G(\nu')_0$. For all $\lambda' \in \mathfrak{h}'^* \cap \mathfrak{g}_{ss\,reg}^*$, $j \in \mathrm{Car}\,\mathfrak{g}$ and $X \in j \cap \mathfrak{g}_{ss\,reg}$, we have

$$ |\mathcal{D}_{\mathfrak{g}}(X)|^{1/2} \hat{\beta}_{G_0 \cdot \lambda'}(X) = \sum_{M'_1 x \in M'_1 \setminus \{x_0 \in G_0 | x_0 \in \mathfrak{g}(\nu')\}} |\mathcal{D}_{\mathfrak{g}(\nu')}(x X)|^{1/2} \hat{\beta}_{M'_1 \cdot \lambda'}(x X) $$

where the sum on the right relates to a finite set.

**Continuation of the proof of the theorem.** We assume here the equality $G = G(\lambda_+)G_0$, to which we have reduced the problem. We (provisionally) assume that the theorem holds for $T_{\tilde{\lambda}, \tau_{M'}}^{M'}$.