Let $θ$ and $θ'$ be a pair of exceptional representations in the sense of Kazhdan and Patterson [KP], of a metaplectic double cover of $GL_n$. The tensor $θ\otimesθ'$ is a (very large) representation of $GL_n$. We characterize its irreducible generic quotients. In the square-integrable case, these are precisely the representations whose symmetric square L-function has a pole at s=0. Our proof of this case involves a new globalization result. In the general case these are the representations induced from distinguished data or pairs of representations and their contragradients. The combinatorial analysis is based on a complete determination of the twisted Jacquet modules of $θ$. As a corollary, $θ$ is shown to admit a new "metaplectic Shalika model".
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…
For Brylinski-Deligne covering groups of an arbitrary split reductive group, we consider theta representations attached to certain exceptional genuine characters. The goal of the paper is to determine when a theta representation has exactly a one-dimensional space of Whittaker functionals, in which case it is called distinguished. For this purpose, we first give effective lower and upper bounds for the dimension of Whittaker functionals for general theta representations. As a consequence, the dimension in many cases can be reduced to simple combinatorial computations, e.g., the Kazhdan-Patters…
We continue our work on understanding Howe correspondences by using theta representations from p-adic groups to compact groups. We prove some results for unitary theta representations of compact groups with respect to the induction and restriction functors.
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.
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.
We study representations of GL(n) appearing as quotients of a tensor of exceptional representations, in the sense of Kazhdan and Patterson. Such representations are called distinguished. We characterize distinguished principal series representations in terms of their inducing data. In particular, we complete the proof of a conjecture of Savin, relating distinguished spherical representations to the image of the tautological lift from a suitable classical group.
In this paper we give the description of generic representations of metaplectic groups over p-adic fields in terms of their Langlands parameters and calculate their theta lifts on all levels for any tower of odd orthogonal groups. We also describe precisely all the occurrences of the failure of the standard module conjecture for metaplectic groups.
In this paper we study the local theta correspondences between epipelagic supercupsidal representations of a type I classical dual pair $(G,G')$ over $p$-adic fields. We show that, besides an exceptional case, an epipelagic supercupsidal representation $π$ of $\widetilde{G}$ lifts to an epipelagic supercupsidal representation $π'$ of $\widetilde{G}'$ if and only if the epipelagic data of $π$ and $π'$ are related by the moment maps.
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…
We determine the occurrence and explicitly describe the theta lifts on all levels of all the irreducible generic representations for the dual pair of groups $(\mathrm{Sp}_{2n}, \mathrm{O}(V))$ defined over a local nonarchimedean field $\mathbb{F}$ of characteristic $0$. As a direct application of our results, we are able to produce a series of non-generic unitarizable representations of these groups.
Let F be a p-adic field with p odd. Quadratic base change and theta-lifting are shown to be compatible for supercuspidal representations of SL(2,F). The argument involves the theory of types and the lattice model of the Weil representation.
By the works of Yu, Kim and Hakim-Murnaghan, we have a parameterization and construction of all supercuspidal representations of a reductive $p$-adic group in terms of supercuspidal data, when $p$ is sufficiently large. In this paper, we will define a correspondence of supercuspidal data via moment maps and theta correspondences over finite fields. Then we will show that local theta correspondences between supercuspidal representations are completely described by this notion. In Appendix B, we give a short proof of a result of Pan on "depth preservation".
We study the algebraic framework in which one can define, in the manner of the theta correspondence, a correspondence between representations of two locally profinite groups $H_1$, $H_2$. In particular, we examine when and how such a correspondence can be extended to bigger groups $G_1$, $G_2$ containing $H_1$, $H_2$ respectively as normal subgroups. As an application, we discuss the theta correspondence for a reductive dual pair of the similitude groups in the non-archimedean case.
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.
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 $V$ be a $GL_n(\mathbb{R})$-distinguished, irreducible, admissible representation of $GL_n(\mathbb{C})$. We prove that any continuous linear functional on $V$, which is invariant under the action of the real mirabolic subgroup, is automatically $GL_n(\mathbb{R})$-invariant.
In this paper we consider the problem of decomposing tensor products of certain singular unitary representations of a semisimple Lie group G. Using explicit models for these representations (constructed earlier by one of us) we show that the decomposition is controlled by a reductive homogeneous space G'/H'. Our procedure establishes a correspondence between certain unitary representations of G and those of G'. This extends the usual theta--correspondence for dual reductive pairs. As a special case we obtain a correspondence between certain representations of real forms of E_7 and F_4.
We study ${\rm Sp}_{2n}(F)$-distinction for representations of the quasi-split unitary group $U_{2n}(E/F)$ in $2n$ variables with respect to a quadratic extension $E/F$ of $p$-adic fields. A conjecture of Dijols and Prasad predicts that no tempered representation is distinguished. We verify this for a large family of representations in terms of the Moeglin-Tadic classification of the discrete series. We further study distinction for some families of non-tempered representations. In particular, we exhibit $L$-packets with no distinguished members that transfer under stable base change to ${\rm …
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.
Let $(π,V)$ be a $GL_n(\mathbb{R})$-distinguished, irreducible, admissible representation of $GL_n(\mathbb{C})$, let $π'$ be an irreducible, admissible, $GL_m(\mathbb{R})$-distinguished representation of $GL_m(\mathbb{C})$, and let $ψ$ be a non-trival character of $\mathbb{C}$ which is trivial on $\mathbb{R}$. We prove that Rankin-Selberg gamma factor at $s=\frac{1}{2}$ is $γ(\frac{1}{2},π\times π'; ψ) = 1$. The result follows as a simple consequence from the characterisation of $GL_n(\mathbb{R})$-distinguished representations in terms of their Langlands data.
In the following article, we give a description of the distingushed irreducible principal series representations of the general linear group over a p-adic field in terms of inducing datum. This provides a counter-example to a conjecture of Jacquet about distinction (Conjecture 1 in U.K Anandavardhanan, "Distinguished non-Archimedean representations ", Proc. Hyderabad Conference on Algebra and Number Theory, 2005, 183-192).
Let $ρ$ is a cuspidal representation of $GL(n,F)$, with $F$ a non archimedean local field, and $H$ a maximal Levi subgroup of $GL(n,F)$. We show that if $ρ$ is $H$-distinguished, then $n$ is even, and $H\simeq GL(n/2,F)\times GL(n/2,F)$.
In this article, we proove that it is equivalent for a generic irreducible representation of GL(n,K), with K a p-adic field, to be distinguished, and for its Rankin-Selberg Asai L-function to have an exceptional pole at zero. This extends a criterion of Kable claiming that a discrete series representation is distinguished if and only if its Asai L-function has a pole at zero. As an application we compute by local methods Asai L-functions of ordinary representations of GL(2,K), in particular we give a formula for Asai L-functions of principal series representations of GL(2,K).
Let $F$ be a non archimedean local field of characteristic zero, we give a classification of generic representations of $GL(n,F)$ distinguished by a maximal Levi subgroup, in terms of inducing discrete series.
In his article "Transcending Classical Invariant Theory" (J.A.M.S., 1989, Vol 2), Roger Howe established a correspondence between representations of a dual pair of reductive groups. This correspondence is known as Howe's correspondence or local theta correspondence. In my paper "Unitary Representations and Theta Correspondence for Classical Group of Type I"(to appear in J. Func. Anal.), we proved that theta correspondence over the real field preserves unitarity under certain restrictions, generalizing some results of Jian-Shu Li (Invent. Math., Vol 97, 1989). In this paper, we study the compos…
S.-Y. Pan decomposes the uniform projection of the Weil representation of a finite symplectic-odd orthogonal dual pair, in terms of Deligne-Lusztig virtual characters, assuming that the order of the finite field is large enough. In this paper we use Pan's decomposition to study the theta correspondence for this kind of dual pairs, following the approach of Adams-Moy and Aubert-Michel-Rouquier. Our results give the theta correspondence between unipotent representations and certain quadratic unipotent representations.
In this article, we consider a dual pair $(G, G')$ in the symplectic group $Sp(W)$ with $G$ compact and let $(\tilde{G}, \tilde{G}')$ be the preimages of $G$ and $G'$ in the metaplectic group $\widetilde{Sp(W)}$. For every irreducible representation $Π$ of $\tilde{G}$ appearing in Howe correspondence, we compute explicitly the restriction of the character $Θ_{Π'}$ of the associated representation $Π'$ of $\tilde{G}'$ on the set of regular points on the compact Cartan subgroup $\tilde{H}'$ of $\tilde{G}'$.
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.
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 $G$ be a split simply laced group defined over a $p$-adic field $F$. In this paper we study the restriction of the minimal representation of $G$ to various dual pairs in $G$. For example, the restriction of the minimal representation of $E_7$ to the dual pair $G_2 \times{}$Sp(6) gives the non-endoscopic Langlands lift of irreducible representations of $G_2$ to Sp(6).
We study modular invariance of normalized supercharacters of tame integrable modules over an affine Lie superalgebra, associated to an arbitrary basic Lie superalgebra $ \mathfrak{g}. $ For this we develop a several step modification process of multivariable mock theta functions, where at each step a Zwegers' type "modifier" is used. We show that the span of the resulting modified normalized supercharacters is $ SL_2(\mathbb{Z}) $-invariant, with the transformation matrix equal, in the case the Killing form on $\mathfrak{g}$ is non-degenerate, to that for the subalgebra $ \mathfrak{g}^! $ of $…
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.
We study the theta lifting for real unitary groups and completely determine the theta lifts of discrete series representations. In particular, we show that these theta lifts can be expressed as cohomologically induced representations in the weakly fair range. This extends a result of J.-S. Li in the case of discrete series representations with sufficiently regular infinitesimal character, whose theta lifts can be expressed as cohomologically induced representations in the good range.
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…
We study the theta lifting for real unitary groups and completely determine the theta lifts of tempered representations. In particular, we show that the theta lifts of (limits of) discrete series representations can be expressed as cohomologically induced representations in the weakly fair range. This extends a result of J.-S. Li in the case of discrete series representations with sufficiently regular infinitesimal character, whose theta lifts can be expressed as cohomologically induced representations in the good range.
We study model transition for representations occurring in the local theta correspondence between split even special orthogonal groups and symplectic groups, over a non-archimedean local field of characteristic zero.
Theta representations appear globally as the residues of Eisenstein series on covers of groups; their unramified local constituents may be characterized as subquotients of certain principal series. A cuspidal theta representation is one which is equal to the local twisted theta representation at almost all places. Cuspidal theta representations are known to exist but only for covers of $GL_j$, $j\leq 3$. In this paper we establish necessary conditions for the existence of cuspidal theta representations on the $r$-fold metaplectic cover of the general linear group of arbitrary rank.
In this paper we study representations of the indefinite orthogonal group O(n,m) which are local theta lifts of one dimensional characters or unitary lowest weight modules of the double covers of the symplectic groups. We apply the transfer of K-types on these representations of O(n,m), and we study their effects on the dual pair correspondences. These results provide examples that the theta lifting is compatible with the transfer of K-types. Finally we will use these results to study subquotients of some cohomologically induced modules.
In this paper, we give an explicit determination of the theta lifting for symplectic-orthogonal and unitary dual pairs over a nonarchimedean field $F$ of characteristic $0$. We determine when theta lifts of tempered representations are nonzero, and determine the theta lifts in terms of the local Langlands correspondence.
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 non-Archimedean locally compact field, $q$ be the cardinality of its residue field, and $\R$ be an algebraically closed field of characteristic $\ell$ not dividing $q$.We classify all irredu\-cible smooth $\R$-representations of $\GL\_n(\F)$ having a nonzero $\GL\_{n-1}(\F)$-inva\-riant linear form, when $q$ is not congruent to $1$ mod $\ell$.Partial results in the case when $q$ is $1$ mod $\ell$ show that, unlike the complex case, the space of $\GL\_{n-1}(\F)$-invariant linear forms has dimension $2$ for certain irreducible representations.
We study the generalized theta lifting between the double covers of split special orthogonal groups, which uses the non-minimal theta representations constructed by Bump, Friedberg and Ginzburg. We focus on the theta liftings of non-generic representations and make a conjecture that gives an upper bound of the first non-zero occurrence of the liftings, depending only on the unipotent orbit. We prove both global and local results that support the conjecture.
In his paper 'Theta lifting for representations with non zero cohomlogy', Jian-Shu Li proved that a certain kind of cohomological representations of $U(a,b)$ is automorphic. In this paper, this result is generalized to a more general class of cohomological representations of this group. It comes from the fact that these cohomological representations are the image by the theta correspondance of some representations obtained by cohomological induction. In proving this theorem, we also prove some cases of Adams conjecture.
The Weil representation discovered by Andre Weil plays an important role in the study of the tranformation properties of theta series. In this paper, we define the Weil-Schroedinger representation of the Jacobi group and prove that the theta series associated with the Weil-Schroedinger representation is a Jacobi form with respect to a suitable arithmetic subgroup of the Jacobi modular group.
In this paper, we give an explicit determination of the non-vanishing of the theta liftings for unitary dual pairs (U(p,q), U(r,s)). Assuming the local Gan-Gross-Prasad conjecture, we determine when theta lifts of tempered representations are nonzero in terms of the local Langlands correspondence.
In this manuscript, we develope the theory of harmonic analysis on the Heisenberg group G of high dimension. We investigate the theta functions and the Weil representation related to this Heisenberg group and describe the connection among them.
