Let $F$ be a non-archimedean locally compact field of residue characteristic $p\neq2$, let $G=\mathrm{GL}_{n}(F)$ and let $H$ be an orthogonal subgroup of $G$. For $π$ a complex smooth supercuspidal representation of $G$, we give a full characterization for the distinguished space $\mathrm{Hom}_{H}(π,1)$ being non-zero and we further study its dimension as a complex vector space, which generalizes a similar result of Hakim for tame supercuspidal representations. As a corollary, the embeddings of $π$ in the space of smooth functions on the set of symmetric matrices in $G$, as a complex vector space, is non-zero and of dimension four, if and only if the central character of $π$ evaluating at $-1$ is $1$.
For a $p$-adic field $F$, the embeddings of a tame supercuspidal representation of $G= {\rm GL}_n (F)$ in the space of smooth functions on the set of symmetric matrices in $G$ are determined.
This paper develops the theory of distinguished regular supercuspidal representations, and it highlights how the correspondence between regular characters and regular supercuspidal representations resembles induction in certain ways.
Based on recent work of Kaletha, we apply Hakim--Murnaghan's result to study distinguished regular supercuspidal representations of tamely ramified reductive $p$-adic groups. Assuming $p$ is sufficiently large, we obtain a necessary and sufficient condition for regular supercuspidal representations to be distinguished. We also investigate the relation between the distinction problem and the Langlands functoriality, and confirm a conjecture of Lapid for regular depth-zero or epipelagic supercuspidal representations.
Let $F/F_0$ be a quadratic extension of non-Archimedean locally compact fields of residual characteristic $p\neq2$, and let $σ$ denote its non-trivial automorphism. Let $R$ be an algebraically closed field of characteristic different from $p$. To any cuspidal representation $π$ of ${\rm GL}_n(F)$, with coefficients in $R$, such that $π^σ\simeqπ^{\vee}$ (such a representation is said to be $σ$-selfdual) we associate a quadratic extension $D/D_0$, where $D$ is a tamely ramified extension of $F$ and $D_0$ is a tamely ramified extension of $F_0$, together with a quadratic character of $D_0^{\times…
This paper studies the behavior of Jiu-Kang Yu's tame supercuspidal representations relative to involutions of reductive p-adic groups. Symmetric space methods are used to illuminate various aspects of Yu's construction. Necessary conditions for a tame supercuspidal representation of G to be distinguished by (the fixed points of) an involution of G are expressed in terms of properties of the G-orbit of the associated G-datum. When these conditions are satisfied, the question of whether a tame supercuspidal representation is distinguished reduces to the question of whether certain cuspidal repr…
The author's work with Murnaghan on distinguished tame supercuspidal representations is re-examined using a simplified treatment of Jiu-Kang Yu's construction of tame supercuspidal representations of $p$-adic reductive groups. This leads to a unification of aspects of the theories of distinguished cuspidal representations over $p$-adic and finite fields.
We further develop and simplify the general theory of distinguished tame supercuspidal representations of reductive $p$-adic groups due to Hakim and Murnaghan, as well as the analogous theory for finite reductive groups due to Lusztig. We apply our results to study the representations of ${\rm GL}_n(F)$, with $n$ odd and $F$ a nonarchimedean local field, that are distinguished with respect to an orthogonal group in $n$ variables. In particular, we determine precisely when a supercuspidal representation is distinguished with respect to an orthogonal group and, if so, that the space of distingui…
This article has a twofold purpose. First, by recent works of Kaletha and Loke-Ma, we give an explicit description of the local theta correspondence between regular supercuspidal representations in the equal rank symplectic-orthogonal case. Second, based on this description, we show that the local theta correspondence preserves distinction with respect to unramified Galois involutions.
Distinguished involutions in the affine Weyl groups, defined by G. Lusztig, play an essential role in the Kazhdan-Lusztig combinatorics of these groups. A distinguished involution is called canonical if it is the shortest element in its double coset with respect to the finite Weyl group. Each two-sided cell in the affine Weyl group contains precisely one canonical distinguished involution. In this note we calculate the canonical distinguished involutions in the affine Weyl groups of rank <8. We also prove some partial results relating canonical distinguished involutions and Dynkin's diagrams o…
Let $G$ be a quasi-split tamely ramified connected reductive group defined over a $p$-adic field $F$. We show that if $-1$ is in the $F$-points of the absolute Weyl group of $G$, then self-dual supercuspidal representations of $G(F)$ exist. Now assume further that $G$ is unramified and that the center of $G$ is connected. Let $π$ be a generic self-dual depth-zero regular supercuspidal representation of $G(F)$. We show that the Frobenius-Schur indicator of $π$ is given by the sign by which a certain distinguished element of the center of $G(F)$ of order two acts on $π$.
Let $Π_0$ be a representation of a group $H$. We say that a representation $τ$ is $(H,Π_0)$-distinguished, if it is a quotient of $Π_0$. It is natural to ask whether this notion "inflates" to larger groups, in the sense that a representation $\mathrm{I}(τ)$ induced from $τ$ and $H$ to a group $G$, is $(G,Π)$-distinguished. We study representations distinguished by theta representations: $H=GL_n$, $Π_0$ is a pair of the exceptional representations of Kazhdan and Patterson, $G=GSpin_{2n+1}$ and $Π$ is a pair of the small representations of Bump, Friedberg and Ginzburg. We prove a Rodier-type her…
We exhibit a basis for the space of spherical characters of a distinguished supercuspidal representation $π$ of a connected reductive $p$-adic group, subject to the assumption that $π$ is obtained via induction from a representation of an open compact mod centre subgroup. We derive an integral formula for each spherical character belonging to the basis. This formula involves integration of a particular kind of matrix coefficient of $π$. We also obtain a similar formula for the function realizing the spherical character. In addition, we determine, subject to some conditions, which of these sphe…
Let $F$ be a $p$-adic field, and $K$ a quadratic extension of $F$. Using Tadic's classification of the unitary dual of $GL(n,K)$, we give the list of irreducible unitary representations of this group distinguished by $GL(n,F)$, in terms of distinguished discrete series. As it is known that a generalised Steinberg representation $St(k,ρ)$ is distinguished if and only if the cuspidal representation $ρ$ is $η^{k-1}$-distinguished, for $η$ the character of $F^*$ with kernel the norms of $K^*$, this actually gives a classification of distinguished unitary representations in terms of distinguished c…
Let $\rm E/\rm F$ be an unramified quadratic extension of local non archimedean fields of characteristic 0. Let $\underline{H}$ be an algebraic reductive group, defined and split over $\rm F$. We assume that the split connected component of the center of $\underline{H}$ is trivial. Let $(τ,V)$ be a $\underline{H}(\rm F)$-distinguished supercuspidal representation of $\underline{H}(\rm E)$. Using the recent results of C. Zhang, and the geometric side of a local relative trace formula obtained by P. Delorme, P. Harinck and S. Souaifi, we describe spherical characters associated to $\underline{H}…
Let E/F be a quadratic extension of non-archimedean local fields of characteristic 0. In this paper, we investigate two approaches which attempt to describe the smooth irreducible representations of GL(n,E) that are distinguished by its subgroup GL(n,F). One relates this class to representations which come as base change lifts from a quasi-split unitary group F, while another deals with a certain symmetry condition. By characterizing the union of images of the base change maps we show that these two approaches are closely related. Using this observation, we are able to prove a statement relati…
Let $F$ be a finite extension of ${\mathbb{Q}} \_p$. Any dihedral supercuspidal representation of $GL \_2 (K)$ arises from an admissible multiplicative character $ω$ of a quadratic extension $L$ of $K$. We show that such a representation is distinguished for $GL \_2 (F)$ if and only if $L$ biquadratic over $F$ and $ω$ restricted to invertibles of one of the two other quadratic extensions of $F$ in $L$ is trivial. We then observe a similar statement for the principal series and we study all dihedral representations.
In this article, we study Prasad's conjecture for regular supercuspidal representations based on the machinery developed by Hakim and Murnaghan to study distinguished representations, and the fundamental work of Kaletha on parameterization of regular supercuspidal representations. For regular supercuspidal representations, we give some new interpretations of the numerical quantities appearing in Prasad's formula, and reduce the proof to the case of tori. The proof of Prasad's conjecture then reduces to a comparison of various quadratic characters appearing naturally in the above process. We al…
Let $\mathrm{G} = \mathrm{Gl}_{n}(K)$, and $\mathrm{H} = \mathrm{G}^σ$ for $σ$ an involution of the form $g\rightarrow aga^{-1}$, It is known that for $K =\mathbb{Q}_q$ any irreducible representation of $\mathrm{G}$ with an $\mathrm{H}$ invariant functional, is self dual, we prove an analogous result for $\mathbb{F}_{q}$.
Let $E/F$ be a quadratic extension of fields, and $G$ a connected quasi-split reductive group over $F$. Let $G^{op}$ be the opposition group obtained by twisting $G$ by the duality involution considered by Prasad. Assume that the field $F$ is finite. Let $π$ be an irreducible generic representation of $G(E)$. When $π$ is a Shintani base change lift of some representation of $G^{op}(F)$, we give an explicit nonzero $G(F)$-invariant vector in terms of the Whittaker vector of $π$. This shows particularly that $π$ is $G(F)$-distinguished. When the field $F$ is $p$-adic, the paper also proves that …
We describe a special case of base change of certain supercuspidal representations from a ramified unitary group to a general linear group, both defined over a p-adic field of odd residual characteristic. Roughly speaking, we require the underlying stratum of a given supercuspidal representation to be skew maximal simple, and the field datum of this stratum to be of maximal degree, tamely ramified over the base field, and quadratic ramified over its subfield fixed by the Galois involution that defines the unitary group. The base change of this supercuspidal representation is described by a can…
Let $G$ be a complex reductive group and $H=G^θ$ be its fixed point subgroup under a Galois involution $θ$. We show that any $H$-distinguished representation $π$ (i.e $\mathrm{dim}_{\mathbb{C}}\left(π^{*}\right)^{H}\neq0$) satisfies: 1) $π^θ\simeq\tildeπ$, where $\tildeπ$ is the contragredient representation and $π^θ$ is the twist of $π$ under $θ$. 2) $\mathrm{dim}_{\mathbb{C}}\left(π^{*}\right)^{H}\leq\left|B\backslash G/H\right|$, where $B$ is a Borel subgroup of $G$. By proving Statement 1), we give a partial answer to a conjecture by Lapid.
For a symmetric space (G,H), one is interested in understanding the vector space of H-invariant linear forms on a representation πof G. In particular an important question is whether or not the dimension of this space is bounded by one. We cover the known results for the pair (G=R_{E/F}GL(n),H=GL(n)), and then discuss the corresponding SL(n) case. In this paper, we show that (G=R_{E/F}SL(n),H=SL(n)) is a Gelfand pair when n is odd. When $n$ is even, the space of H-invariant forms on πcan have dimension more than one even when πis supercuspidal. The latter work is joint with Dipendra Prasad.
Let K/F be a quadratic extension of non archimedean local fields. Let V be an irreducible level zero discrete series representation of the group G = GL2(R), where R is a division algebra of center K and of index r. One assumes that V is not a supercuspidal representation. Let D be a division algebra of center F and index 2r. In the following work, we try to answer this question : Is the representation V D^{*}-distinguished if and only if its image by the Jacquet-Langlands correspondence is GL2(F)-distinguished. To answer this question, we use the coefficients system on the Bruhat-Tits building…
Let $(W,S)$ be any Coxeter system and let $w \mapsto w^*$ be an involution of $W$ which preserves the set of simple generators $S$. Lusztig and Vogan have shown that the corresponding set of twisted involutions (i.e., elements $w \in W$ with $w^{-1} = w^*$) naturally generates a module of the Hecke algebra of $(W,S)$ with two distinguished bases. The transition matrix between these bases defines a family of polynomials $P^σ_{y,w}$ which one can view as a "twisted" analogue of the much-studied family of Kazhdan-Lusztig polynomials of $(W,S)$. The polynomials $P^σ_{y,w}$ can have negative coeffi…
Let $F$ be a $p$-adic field and $\mathbf{U}$ be a unipotent group defined over $F$, and set $U=\mathbf{U}(F)$. Let $σ$ be an involution of $\mathbf{U}$ defined over $F$. Adapting the arguments of Yves Benoist in the real case, we prove the following result: an irreducible representation $π$ of $U$ is $U^σ$-distinguished if and only if it is $σ$-self-dual and in this case $\mathrm{Hom}_{U^σ}(π,\mathbb{C})$ has dimension one. When $σ$ is a Galois involution these results imply a bijective correspondence between the set $\mathrm{Irr}(U^σ)$ of isomorphism classes of irreducible representations of …
Let $(W,S)$ be a Coxeter system and let $w \mapsto w^*$ be an involution of $W$ which preserves the set of simple generators $S$. Lusztig and Vogan have recently shown that the set of twisted involutions (i.e., elements $w \in W$ with $w^{-1} = w^*$) naturally generates a module of the Hecke algebra of $(W,S)$ with two distinguished bases. The transition matrix between these bases defines a family of polynomials $P^σ_{y,w}$ which one can view as "twisted" analogues of the much-studied Kazhdan-Lusztig polynomials of $(W,S)$. The polynomials $P^σ_{y,w}$ can have negative coefficients, but displa…
We consider the parabolically induced representations of the symmetric space $SO_4\backslash G_2$ over a p-adic field using the geometric lemma when the inducing parabolic is $P_β$. Using an explicit description of the embedding of $G_2$ in $GL_8$, we characterize precisely the induced representations which are $(SO_4, χ)$-distinguished, given a certain type of involutions is chosen.
An irreducible smooth representation of a $p$-adic group $G$ is said to be distinguished with respect to a subgroup $H$ if it admits a non-trivial $H$-invariant linear form. When $H$ is the fixed group of an involution on $G$ it is suggested by the works of Hervé Jacquet from the nineties that distinction can be characterized in terms of the principle of functoriality. If the involution is the Galois involution then a recent conjecture of Dipendra Prasad predicts a formula for the dimension of the space of invariant linear forms which once again involves base change. We will describe the proof…
Let n be a positive integer, F be a non-Archimedean locally compact field of odd residue characteristic p and G be an inner form of GL(2n,F). This is a group of the form GL(r,D) for a positive integer r and division F-algebra D of reduced degree d such that rd=2n. Let K be a quadratic extension of F in the algebra of matrices of size r with coefficients in D, and H be its centralizer in G. We study selfdual cuspidal representations of G and their distinction by H from the point of view of type theory. Given such a representation pi of G, we compute the value at 1/2 of the epsilon factor of the…
The purpose of this article is to shed new light on the combinatorial structure of Kazhdan-Lusztig cells in infinite Coxeter groups $W$. Our main focus is the set $\D$ of distinguished involutions in $W$, which was introduced by Lusztig in one of his first papers on cells in affine Weyl groups. We conjecture that the set $\D$ has a simple recursive structure and can be enumerated algorithmically starting from the distinguished involutions of finite Coxeter groups. Moreover, to each element of $\D$ we assign an explicitly defined set of equivalence relations on $W$ that altogether conjecturally…
In this paper we study the question of determining when an irreducible admissible representation of ${\rm GL}_n(D)$ admits a symplectic model, that is when such a representation has a linear functional invariant under ${\rm Sp}_n(D)$, where $D$ is a quaternion division algebra over a non-Archimedian local field $k$ and ${\rm Sp}_{n}(D)$ is the unique non-split inner form of the symplectic group ${\rm Sp}_{2n}(k)$. We show that if a representation has a symplectic model it is necessarily unique. For ${\rm GL}_2(D)$ we completely classify those representations which have a symplectic model. Glob…
Let $E/F$ be a quadratic extension of non-archimedean local fields of characteristic different from $2$. Let $A$ be an $F$-central simple algebra of even dimension so that it contains $E$ as a subfield, set $G=A^\times$ and $H$ for the centralizer of $E^\times$ in $G$. Using a Galois descent argument, we prove that all double cosets $H g H\subset G$ are stable under the anti-involution $g\mapsto g^{-1}$, reducing to Guo's result for $F$-split $G$ which we extend to fields of positive characteristic different from $2$. We then show, combining global and local results, that $H$-distinguished irr…
Let θbe an involution of the semisimple Lie algebra g and g=k+p be the associated Cartan decomposition. The nilpotent commuting variety of (g,θ) consists in pairs of nilpotent elements (x,y) of p such that [x,y]=0. It is conjectured that this variety is equidimensional and that its irreducible components are indexed by the orbits of p-distinguished elements. This conjecture was established by A. Premet in the case (g \times g, θ) where θ(x,y)=(y,x). In this work we prove the conjecture in a significant number of other cases.
Let $F$ be a $p$-adic field, $E$ be a quadratic extension of $F$, $D$ be an $F$-central division algebra of odd index and let $θ$ be the Galois involution attached to $E/F$. Set $H=GL(m,D)$, $G=GL(m,D\otimes_F E)$, and let $P=MU$ be a standard parabolic subgroup of $G$. Let $w$ be a Weyl involution stabilizing $M$ and $M^{θ_w}$ be the subgroup of $M$ fixed by the involution $θ_w:m\mapsto θ(wmw)$. We denote by $X(M)^{w,-}$ the complex torus of $w$-anti-invariant unramified characters of $M$. Following the global methods of Jacquet, Lapid and Rogawski, we associate to a finite length representat…
