Let $E/F$ be a quadratic unramified extension of non-archimedean local fields and $\mathbb H$ a simply connected semisimple algebraic group defined and split over F. We establish general results (multiplicities, test vectors) on ${\mathbb H} (F)$-distinguished Iwahori-spherical representations of ${\mathbb H} (E)$. For discrete series Iwahori-spherical representations of ${\mathbb H} (E)$, we prove a numerical criterion of ${\mathbb H} (F)$-distinction. As an application, we classify the ${\mathbb H} (F)$-distinguished discrete series representations of ${\mathbb H} (E)$ corresponding to degree $1$ characters of the Iwahori-Hecke algebra.

We study harmonic analysis on the symmetric space $\text{GL}_n \times \text{GL}_n \backslash \text{GL}_{2n}$ . We prove several standard results, e.g. Shalika germ expansion of orbital integrals, representability of the Fourier transform of orbital integrals and representability of spherical characters. These properties are not expected to hold for symmetric spaces in general.

We study basic geometric properties of Kottwitz–Viehmann varieties, which are certain generalizations of affine Springer fibers that encode orbital integrals of spherical Hecke functions. Based on the previous work of A. Bouthier and the author, we show that these varieties are equidimensional and give a precise formula for their dimension. Also we give a conjectural description of their number of irreducible components in terms of certain weight multiplicities of the Langlands dual group and we prove the conjecture in the case of unramified conjugacy class.

Consider the action of $\operatorname{GL}(n,\mathbb{Q}_{p})$ on the $p$-adic unit sphere ${\mathcal{S}}_{n}$ arising from the linear action on $\mathbb{Q}_{p}^{n}\setminus \{0\}$. We show that for the action of a semigroup $\mathfrak{S}$ of $\operatorname{GL}(n,\mathbb{Q}_{p})$ on ${\mathcal{S}}_{n}$, the following are equivalent: (1) $\mathfrak{S}$ acts distally on ${\mathcal{S}}_{n}$; (2) the closure of the image of $\mathfrak{S}$ in $\operatorname{PGL}(n,\mathbb{Q}_{p})$ is a compact group. On ${\mathcal{S}}_{n}$, we consider the ‘affine’ maps $\overline{T}_{a}$ corresponding to $T$ in $\operatorname{GL}(n,\mathbb{Q}_{p})$ and a nonzero $a$ in $\mathbb{Q}_{p}^{n}$ satisfying $\Vert T^{-1}(a)\Vert _{p}<1$. We show that there exists a compact open subgroup $V$, which depends on $T$, such that $\overline{T}_{a}$ is distal for every nonzero $a\in V$ if and only if $T$ acts distally on ${\mathcal{S}}_{n}$. The dynamics of ‘affine’ maps on $p$-adic unit spheres is quite different from that on the real unit spheres.

The Peter–Weyl idempotent $e_{\mathscr{P}}$ of a parahoric subgroup $\mathscr{P}$ is the sum of the idempotents of irreducible representations of $\mathscr{P}$ that have a nonzero Iwahori fixed vector. The convolution algebra associated with $e_{\mathscr{P}}$ is called a Peter–Weyl Iwahori algebra. We show that any Peter–Weyl Iwahori algebra is Morita equivalent to the Iwahori–Hecke algebra. Both the Iwahori–Hecke algebra and a Peter–Weyl Iwahori algebra have a natural conjugate linear anti-involution $\star$, and the Morita equivalence preserves irreducible hermitian and unitary modules. Both algebras have another anti-involution, denoted by $\bullet$, and the Morita equivalence preserves irreducible and unitary modules for $\bullet$.

We provide an explicit construction of representations in the discrete spectrum of two $p$-adic symmetric spaces. We consider $\text{G}{{\text{L}}_{n}}\left( F \right)\,\times \,\text{G}{{\text{L}}_{n}}\left( F \right)\backslash \text{G}{{\text{L}}_{2n}}\left( F \right)$ and $\text{G}{{\text{L}}_{n}}\left( F \right)\,\backslash \text{G}{{\text{L}}_{n}}\left( E \right)$, where $E$ is a quadratic Galois extension of a nonarchimedean local field $F$ of characteristic zero and odd residual characteristic. The proof of the main result involves an application of a symmetric space version of Casselman’s Criterion for square integrability due to Kato and Takano.

In this paper, we prove the spherical fundamental lemma for metaplectic group $M{{p}_{2n}}$ based on the formalism of endoscopy theory by J. Adams, D. Renard, and W.-W. Li.

Suppose that $\widetilde{G}$ is a connected reductive group defined over a field $k$, and $\Gamma$ is a finite group acting via $k$-automorphisms of $\widetilde{G}$ satisfying a certain quasi-semisimplicity condition. Then the identity component of the group of $\Gamma$-fixed points in $\widetilde{G}$ is reductive. We axiomatize the main features of the relationship between this fixed-point group and the pair $\left( \tilde{G},\Gamma \right)$, and consider any group $G$ satisfying the axioms. If both $\widetilde{G}$ and $G$ are $k$-quasisplit, then we can consider their duals $\widetilde{{{G}^{*}}}$ and ${{G}^{*}}$. We show the existence of and give an explicit formula for a natural map from the set of semisimple stable conjugacy classes in ${{G}^{*}}\,(k)$ to the analogous set for $\widetilde{{{G}^{*}}}\,(k)$. If $k$ is finite, then our groups are automatically quasisplit, and our result specializes to give a map of semisimple conjugacy classes. Since such classes parametrize packets of irreducible representations of $G(k)$ and $\widetilde{G}\,(k)$, one obtains a mapping of such packets.

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 $\Pi _{i=1}^{r}\,\text{S}{{\text{L}}_{ni}}\,\subseteq \,M\,\subseteq \,\Pi _{i=1}^{r}\,\text{G}{{\text{L}}_{ni}}$ for positive integers $r$ and ${{n}_{i}}$. We prove that the Plancherel measure for any unitary supercuspidal representation of $M\left( F \right)$ 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 $\text{S}{{\text{O}}_{4n}}$ and $\text{S}{{\text{p}}_{4n}}$ 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 ${{D}_{n}}$.

For V a two-dimensional p-adic representation of Gℚp, we denote by B(V ) the admissible unitary representation of GL2(ℚp) attached to V under the p-adic local Langlands correspondence of GL2(ℚp) initiated by Breuil. In this paper, building on the works of Berger–Breuil and Colmez, we determine the locally analytic vectors B(V )an of B (V ) when V is irreducible, crystabelian and Frobenius semisimple with distinct Hodge–Tate weights; this proves a conjecture of Breuil. Using this result, we verify Emerton’s conjecture that dim Ref η⊗ψ (V )=dim Exp η∣⋅∣⊗xψ (B (V )an ⊗(x∣⋅∣∘det )) for those V which are irreducible, crystabelian and Frobenius semisimple.

When $F$ is a $p$-adic field, and $G\,=\,\mathbb{G}\left( F \right)$ is the group of $F$-rational points of a connected algebraic $F$-group, the complex vector space $\mathcal{H}\left( G \right)$ 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, $\mathcal{H}\left( G \right)$ has drawbacks such as the lack of an identity element, and the process $G\,\mapsto \,\mathcal{H}\left( G \right)$ is not a functor. Bernstein introduced an enlargement ${{\mathcal{H}}^{\hat{\ }}}\left( G \right)$ of $\mathcal{H}\left( G \right)$. The algebra ${{\mathcal{H}}^{\hat{\ }}}\left( G \right)$ consists of the distributions that are left essentially compact. We show that the process $G\,\mapsto \,{{\mathcal{H}}^{\hat{\ }}}\left( G \right)$ is a functor. If $\tau \,:\,G\,\to \,H$ is a morphism of $p$-adic groups, let $F\left( \tau \right):\,{{\mathcal{H}}^{\hat{\ }}}\left( G \right)\,\to \,{{\mathcal{H}}^{\hat{\ }}}\left( H \right)$ be the morphism of $\mathbb{C}$-algebras. We identify the kernel of $F\left( \tau \right)$ in terms of $\text{Ker}\left( \tau \right)$. In the setting of $p$-adic Lie algebras, with $\mathfrak{g}$ a reductive Lie algebra, $\mathfrak{m}$ a Levi, and $\tau \,:\,\mathfrak{g}\,\to \,\mathfrak{m}$ the natural projection, we show that $F\left( \tau \right)$ maps $G$-invariant distributions on $\mathcal{G}$ to ${{N}_{G}}\left( \mathfrak{m} \right)$-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 $SL\left( 2 \right)$ show how certain members of the family can be moved to the group.

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.

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.

Colmez has given a recipe to associate a smooth modular representation Ω(W) of the Borel subgroup of GL2(Qp) to a -representation W of by using Fontaine’s theory of (φ,Γ)-modules. We compute Ω(W) explicitly and we prove that if W is irreducible and dim (W)=2, then Ω(W) is the restriction to the Borel subgroup of GL2(Qp) of the supersingular representation associated to W by Breuil’s correspondence.

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.

Let $\mathcal{D}$ be a division algebra over a nonarchimedean local field. Given an irreducible representation $\pi $ of $G{{L}_{2}}(\mathcal{D})$, 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.

In this paper we determine the structure of the full lift for the Howe correspondence of $\left( Sp\left( n \right),O\left( V \right) \right)$ for rank-one reducibilities.

Soit $F$ un corps local non archimédien, et $G$ le groupe des $F$-points d’un groupe réductif connexe quasi-déployé défini sur $F$. Dans cet article, on s’intéresse aux distributions sur $G$ invariantes par conjugaison, et à l’espace de leurs restrictions à l’algèbre de Hecke $\mathcal{H}$ des fonctions sur $G$ à support compact biinvariantes par un sous-groupe d’Iwahori $I$ donné. On montre tout d’abord que les valeurs d’une telle distribution sur $\mathcal{H}$ sont entièrement déterminées par sa restriction au sous-espace de dimension finie des éléments de $\mathcal{H}$ à support dans la réunion des sous-groupes parahoriques de $G$ contenant $I$. On utilise ensuite cette propriété pour montrer, moyennant certaines conditions sur $G$, que cet espace est engendré d’une part par certaines intégrales orbitales semi-simples, d’autre part par les intégrales orbitales unipotentes, en montrant tout d’abord des résultats analogues sur les groupes finis.

We study reducibility of representations parabolically induced from discrete series representations of $S{{U}_{n}}(F)$ for $F$ a $p$-adic field of characteristic zero. We use the approach of studying the relation between $R$-groups when a reductive subgroup of a quasi-split group and the full group have the same derived group. We use restriction to show the quotient of $R$-groups is in natural bijection with a group of characters. Applying this to $S{{U}_{n}}(F)\,\subset \,{{U}_{n}}(F)$ we show the $R$ group for $S{{U}_{n}}$ is the semidirect product of an $R$-group for ${{U}_{n}}(F)$ and this group of characters. We derive results on nonabelian $R$-groups and generic elliptic representations as well.