Soit $G$ le groupe $\mathrm{GL}(N,F)$, où $F$ est un corps localement compact et non archimédien. En utilisant la théorie des types simples de Bushnell et Kutzko, ainsi qu'une idée originale d'Henniart, nous construisons des pseudo-coefficients explicites pour les représentations de la série discrète de $G$. Comme application, nous en déduisons des formules inédites pour la valeur du charactère d'Harish-Chandra de certaines telles représentations en certains éléments elliptiques réguliers.

Let $F$ be a $p$-adic field of characteristic $0$, and let $M$ be an $F$-Levi subgroup of a connected reductive $F$-split group such that $\Pi_{i=1}^{r} SL_{n_i} \subseteq M \subseteq \Pi_{i=1}^{r} GL_{n_i}$ for positive integers $r$ and $n_i$. We prove that the Plancherel measure for any unitary supercuspidal representation of $M(F)$ is identically transferred under the local Jacquet-Langlands type correspondence between $M$ and its $F$-inner forms, assuming a working hypothesis that Plancherel measures are invariant on a certain set. This work extends the result of Muić and Savin (2000) for Siegel Levi subgroups of the groups $SO_{4n}$ and $Sp_{4n}$ under the local Jacquet-Langlands correspondence. It can be applied to a simply connected simple $F$-group of type $E_6$ or $E_7$, and a connected reductive $F$-group of type $A_{n}$, $B_{n}$, $C_n$ or $D_n$.

Dans cet article, on énonce une variante du lemme fondamental pondéré d'Arthur pour le groupe métaplectique de Weil, qui sera un ingrédient indispensable de la stabilisation de la formule des traces. Pour un corps de caractéristique résiduelle suffisamment grande, on en donne une démonstration à l'aide de la méthode de descente, qui est conditionnelle: on admet le lemme fondamental pondéré non standard sur les algèbres de Lie. Vu les travaux de Chaudouard et Laumon, on s'attend à ce que cette condition soit ultérieurement vérifiée.

W. Casselman defined a basis $f_u$ of Iwahori fixed vectors of a spherical representation $(\pi, V)$ of a split semisimple $p$-adic group $G$ over a nonarchimedean local field $F$ by the condition that it be dual to the intertwining operators, indexed by elements $u$ of the Weyl group $W$. On the other hand, there is a natural basis $\psi_u$, and one seeks to find the transition matrices between the two bases. Thus, let $f_u = \sum_v \tilde{m} (u, v) \psi_v$ and $\psi_u = \sum_v m (u, v) f_v$. Using the Iwahori-Hecke algebra we prove that if a combinatorial condition is satisfied, then $m (u, v) = \prod_{\alpha} \frac{1 - q^{- 1} \mathbf{z}^{\alpha}}{1 -\mathbf{z}^{\alpha}}$, where $\mathbf z$ are the Langlands parameters for the representation and $\alpha$ runs through the set $S (u, v)$ of positive coroots $\alpha \in \hat{\Phi}$ (the dual root system of $G$) such that $u \leqslant v r_{\alpha} < v$ with $r_{\alpha}$ the reflection corresponding to $\alpha$. The condition is conjecturally always satisfied if $G$ is simply-laced and the Kazhdan-Lusztig polynomial $P_{w_0 v, w_0 u} = 1$ with $w_0$ the long Weyl group element. There is a similar formula for $\tilde{m}$ conjecturally satisfied if $P_{u, v} = 1$. This leads to various combinatorial conjectures.

When $F$ is a $p$-adic field, and $G={\mathbb G}(F)$ is the group of $F$-rational points of a connected algebraic $F$-group, the complex vector space ${\mathcal H}(G)$ of compactly supported locally constant distributions on $G$ has a natural convolution product that makes it into a ${\mathbb C}$-algebra (without an identity) called the Hecke algebra. The Hecke algebra is a partial analogue for $p$-adic groups of the enveloping algebra of a Lie group. However, $\mathcal{H}(G)$ has drawbacks such as the lack of an identity element, and the process $G \mapsto \mathcal{H}(G)$ is not a functor. Bernstein introduced an enlargement $\mathcal{H}\,\hat{\,}(G)$ of $\mathcal{H}(G)$. The algebra $\mathcal{H}\,\hat{\,} (G)$ consists of the distributions that are left essentially compact. We show that the process $G \mapsto \mathcal{H}\,\hat{\,} (G)$ is a functor. If $\tau \colon G \rightarrow H$ is a morphism of $p$-adic groups, let $F(\tau) \colon \mathcal{H}\,\hat{\,} (G) \rightarrow \mathcal{H}\,\hat{\,} (H)$ be the morphism of $\mathbb{C}$-algebras. We identify the kernel of $F(\tau)$ in terms of $\textrm{Ker}(\tau)$. In the setting of $p$-adic Lie algebras, with $\mathfrak{g}$ a reductive Lie algebra, $\mathfrak{m}$ a Levi, and $\tau \colon \mathfrak{g} \to \mathfrak{m}$ the natural projection, we show that $F(\tau)$ maps $G$-invariant distributions on $\mathcal{G}$ to $N_G (\mathfrak{m})$-invariant distributions on $\mathfrak{m}$. Finally, we exhibit a natural family of $G$-invariant essentially compact distributions on $\mathfrak{g}$ associated with a $G$-invariant non-degenerate symmetric bilinear form on ${\mathfrak g}$ and in the case of $SL(2)$ show how certain members of the family can be moved to the group.

Let $G$ be the $F$-rational points of the symplectic group $Sp_{2n}$, where $F$ is a non-Archimedean local field of characteristic $0$. Cogdell, Kim, Piatetski-Shapiro, and Shahidi constructed local Langlands functorial lifting from irreducible generic representations of $G$ to irreducible representations of $GL_{2n+1}(F)$. Jiang and Soudry constructed the descent map from irreducible supercuspidal representations of $GL_{2n+1}(F)$ to those of $G$, showing that the local Langlands functorial lifting from the irreducible supercuspidal generic representations is surjective. In this paper, based on above results, using the same descent method of studying $SO_{2n+1}$ as Jiang and Soudry, we will show the rest of local Langlands functorial lifting is also surjective, and for any local Langlands parameter $\phi \in \Phi(G)$, we construct a representation $\sigma$ such that $\phi$ and $\sigma$ have the same twisted local factors. As one application, we prove the $G$-case of a conjecture of Gross-Prasad and Rallis, that is, a local Langlands parameter $\phi \in \Phi(G)$ is generic, i.e., the representation attached to $\phi$ is generic, if and only if the adjoint $L$-function of $\phi$ is holomorphic at $s=1$. As another application, we prove for each Arthur parameter $\psi$, and the corresponding local Langlands parameter $\phi_{\psi}$, the representation attached to $\phi_{\psi}$ is generic if and only if $\phi_{\psi}$ is tempered.

We calculate reducibility for the representations of metaplectic groups induced from cuspidal representations of maximal parabolic subgroups via theta correspondence, in terms of the analogous representations of the odd orthogonal groups. We also describe the lifts of all relevant subquotients.

In this paper we give a classification of discrete series for $SO(2n,F)$, $F$ $p$-adic, similar to that of Mœglin--Tadić for the other classical groups. This is obtained by taking the Mœglin--Tadić classification for $O(2n,F)$ and studying how the representations restrict to $SO(2n,F)$. We then extend this to an analysis of how admissible representations of $O(2n,F)$ restrict.

In this paper we prove holomorphy for certain intertwining operators arising from the theory of Eisenstein series. To do that we need to normalize using the Langlands--Shahidi's normalization arising from the twisted endoscopy and the associated representations of the general linear group.

In this paper we construct an analogue of Iwahori--Hecke algebras of $\operatorname{SL}_2$ over $2$-dimensional local fields. After considering coset decompositions of double cosets of a Iwahori subgroup, we define a convolution product on the space of certain functions on $\operatorname{SL}_2$, and prove that the product is well-defined, obtaining a Hecke algebra. Then we investigate the structure of the Hecke algebra. We determine the center of the Hecke algebra and consider Iwahori--Matsumoto type relations.

Let $G_n=\mathrm{Sp}_n(F)$ be the rank $n$ symplectic group with entries in a nondyadic $p$-adic field $F$. We further let $\widetilde{G}_n$ be the metaplectic extension of $G_n$ by $\mathbb{C}^{1}=\{z\in\mathbb{C}^{\times} \mid |z|=1\}$ defined using the Leray cocycle. In this paper, we aim to demonstrate the complete list of reducibility points of the genuine principal series of ${\widetilde{G}_2}$. In most cases, we will use some techniques developed by Tadić that analyze the Jacquet modules with respect to all of the parabolics containing a fixed Borel. The exceptional cases involve representations induced from unitary characters $\chi$ with $\chi^2=1$. Because such representations $\pi$ are unitary, to show the irreducibility of $\pi$, it suffices to show that $\dim_{\mathbb{C}}\mathrm{Hom}_{{\widetilde{G}}}(\pi,\pi)=1$. We will accomplish this by examining the poles of certain intertwining operators associated to simple roots. Then some results of Shahidi and Ban decompose arbitrary intertwining operators into a composition of operators corresponding to the simple roots of ${\widetilde{G}_2}$. We will then be able to show that all such operators have poles at principal series representations induced from quadratic characters and therefore such operators do not extend to operators in $\mathrm{Hom}_{{\widetilde{G}_2}}(\pi,\pi)$ for the $\pi$ in question.

Let $G$ be a connected semisimple split group over a $p$-adic field. We establish the explicit link between principal nilpotent orbits and the irreducible constituents of principal series in terms of $L$-group objects.

Let $\BF_q$ be a finite field of $q$ elements, $\CF$ a $p$-adic field, and $D$ a quaternion division algebra over $\CF$. This paper proves uniqueness of Shalika models for $\GL_{2n}(\BF_q) $ and $\GL_{2n}(D)$, and re-obtains uniqueness of Shalika models for $\GL_{2n}(\CF)$ for any $n\in \BN$.

Suppose that $P=MN$ is a maximal parabolic subgroup of a quasisplit, connected, reductive classical group $G$ defined over a non-Archimedean field and $A$ is the standard intertwining operator attached to a tempered representation of $G$ induced from $M$. In this paper we determine all the cases in which $\Lie(N)$ is prehomogeneous under $\Ad(m)$ when $N$ is non-abelian, and give necessary and sufficient conditions for $A$ to have a pole at $0$.

The aim of this paper is to prove two general results on parabolic induction of classical $p$-adic groups (actually, one of them holds also in the archimedean case), and to obtain from them some consequences about irreducible unitarizable representations. One of these consequences is a reduction of the unitarizability problem for these groups. This reduction is similar to the reduction of the unitarizability problem to the case of real infinitesimal character for real reductive groups.

This paper shows the existence and uniqueness of Klyachko models for irreducible unitary representations of $\GL_5(\CF)$, where $\CF$ is a $p$-adic field. It is an extension of the work of Heumos and Rallis on $\GL_4(\CF)$.

This paper is the continuation of our previous work on the explicit determination of the structure of theta lifts for dual pairs $(\Sp_{2n}, O(V))$ over a non-archimedean field $F$ of characteristic different than $2$, where $n$ is the split rank of $\Sp_{2n}$ and the dimension of the space $V$ (over $F$) is even. We determine the structure of theta lifts of tempered representations in terms of theta lifts of representations in discrete series.

Let $F$ be a non-archimedean local field of residue characteristic neither 2 nor 3 equipped with a galois involution with fixed field $F_0$, and let $G$ be a symplectic group over $F$ or an unramified unitary group over $F_0$. Following the methods of Bushnell--Kutzko for $\GL(N,F)$, we define an analogue of a simple type attached to a certain skew simple stratum, and realize a type in $G$. In particular, we obtain an irreducible supercuspidal representation of $G$ like $\GL(N,F)$.

Soit $F_0$ un corps local non archim\'edien de caract\'eristique nulle et de ca\-rac\-t\'eristique r\'esiduelle impaire. J. Rogawski a montr\'e l'existence du changement de base entre le groupe unitaire en trois variables $U(2,1)(F_{0})$, d\'efini relativement \`a une extension quadratique $F$ de $F_{0}$, et le groupe lin\'eaire $GL(3,F)$. Par ailleurs, nous avons d\'ecrit les repr\'esentations supercuspidales irr\'eductibles de $U(2,1)(F_{0})$ comme induites \`a partir d'un sous-groupe compact ouvert de $U(2,1)(F_{0})$, description analogue \`a celle des repr\'esentations admissibles irr\'eductibles de $GL(3,F)$ obtenue par C. Bushnell et P. Kutzko. A partir de ces descriptions, nous construisons explicitement le changement de base des repr\'esentations tr\`es cuspidales de $U(2,1)(F_{0})$.

On calcule les restrictions {\`a} l'alg{\`e}bre de Hecke sph{\'e}rique des traces tordues compactes d'un ensemble de repr{\'e}sentations explicitement construites du groupe $\GL(N, F)$, o{\`u} $F$ est un corps $p$-adique. Ces calculs r\'esolve en particulier une question pos{\'e}e dans un article pr\'ec\'edent du m\^eme auteur.

Let $\mathcal{D}$ be a division algebra over a nonarchimedean local field. Given an irreducible representation $\pi$ of $\GL_2(\mathcal{D})$, we describe its restriction to the diagonal subgroup $\mathcal{D}^* \times \mathcal{D}^*$. The description is in terms of the structure of the twisted Jacquet module of the representation $\pi$. The proof involves Kirillov theory that we have developed earlier in joint work with Dipendra Prasad. The main result on restriction also shows that $\pi$ is $\mathcal{D}^* \times \mathcal{D}^*$-distinguished if and only if $\pi$ admits a Shalika model. We further prove that if $\mathcal{D}$ is a quaternion division algebra then the twisted Jacquet module is multiplicity-free by proving an appropriate theorem on invariant distributions; this then proves a multiplicity-one theorem on the restriction to $\mathcal{D}^* \times \mathcal{D}^*$ in the quaternionic case.

In this paper we prove the unitarity of duals of tempered representations supported on minimal parabolic subgroups for split classical $p$-adic groups. We also construct a family of unitary spherical representations for real and complex classical groups

Soient $F$ un corps commutatif localement compact non archim\'edien, $G=\GL (n,F)$ pour un entier $n\geq 2$, et $\kappa$ un caract\`ere de $F^\times$ trivial sur $(F^\times)^n$. On prouve une formule pour les $\kappa$-int\'egrales orbitales r\'eguli\`eres sur $G$ permettant, si $F$ est de caract\'eristique $>0$, de les relever \`a la caract\'eristique nulle. On en d\'eduit deux r\'esultats nouveaux en caract\'eristique $>0$\,: le ``lemme fondamental'' pour l'induction automorphe, et une version simple de la formule des traces tordue locale d'Arthur reliant $\kappa$-int\'egrales orbitales elliptiques et caract\`eres $\kappa$-tordus. Cette formule donne en particulier, pour une s\'erie $\kappa$-discr\`ete de $G$, les $\kappa$-int\'egrales orbitales elliptiques d'un pseudo-coefficient comme valeurs du caract\`ere $\kappa$-tordu.

Dans un travail ant\'erieur, nous avions montr\'e que l'induite parabolique (normalis\'ee) d'une repr\'esentation irr\'eductible cuspidale $\sigma $ d'un sous-groupe de Levi $M$ d'un groupe $p$-adique contient un sous-quotient de carr\'e int\'egrable, si et seulement si la fonction $\mu $ de Harish-Chandra a un p\^ole en $\sigma $ d'ordre \'egal au rang parabolique de $M$. L'objet de cet article est d'interpr\'eter ce r\'esultat en termes de fonctorialit\'e de Langlands.

The Casselman--Shalika method is a way to compute explicit formulas for periods of irreducible unramified representations of $p$-adic groups that are associated to unique models (i.e., multiplicity-free induced representations). We apply this method to the case of the Shalika model of $GL_n$, which is known to distinguish lifts from odd orthogonal groups. In the course of our proof, we further develop a variant of the method, that was introduced by Y. Hironaka, and in effect reduce many such problems to straightforward calculations on the group.