Shalika Model Research Articles

On a multiplicity formula for spherical varieties

In this paper, we propose a conjectural multiplicity formula for general spherical varieties. For all the cases where a multiplicity formula has been proved, including Whittaker models, Gan–Gross–Prasad models, Ginzburg–Rallis models, Galois models and Shalika models, we show that the multiplicity formulas in our conjecture are the same as the multiplicity formulas that have been proved. We also prove the conjectural multiplicity formula in two new cases.

Annihilator varieties of distinguished modules of reductive Lie algebras

AbstractWe provide a microlocal necessary condition for distinction of admissible representations of real reductive groups in the context of spherical pairs.Let${\mathbf {G}}$be a complex algebraic reductive group and${\mathbf {H}}\subset {\mathbf {G}}$be a spherical algebraic subgroup. Let${\mathfrak {g}},{\mathfrak {h}}$denote the Lie algebras of${\mathbf {G}}$and${\mathbf {H}}$, and let${\mathfrak {h}}^{\bot }$denote the orthogonal complement to${\mathfrak {h}}$in${\mathfrak {g}}^*$. A${\mathfrak {g}}$-module is called${\mathfrak {h}}$-distinguished if it admits a nonzero${\mathfrak {h}}$-invariant functional. We show that the maximal${\mathbf {G}}$-orbit in the annihilator variety of any irreducible${\mathfrak {h}}$-distinguished${\mathfrak {g}}$-module intersects${\mathfrak {h}}^{\bot }$. This generalises a result of Vogan [Vog91].We apply this to Casselman–Wallach representations of real reductive groups to obtain information on branching problems, translation functors and Jacquet modules. Further, we prove in many cases that – as suggested by [Pra19, Question 1] – whenHis a symmetric subgroup of a real reductive groupG, the existence of a temperedH-distinguished representation ofGimplies the existence of a genericH-distinguished representation ofG.Many of the models studied in the theory of automorphic forms involve an additive character on the unipotent radical of the subgroup$\bf H$, and we have devised a twisted version of our theorem that yields necessary conditions for the existence of those mixed models. Our method of proof here is inspired by the theory of modules overW-algebras. As an application of our theorem we derive necessary conditions for the existence of Rankin–Selberg, Bessel, Klyachko and Shalika models. Our results are compatible with the recent Gan–Gross–Prasad conjectures for nongeneric representations [GGP20].Finally, we provide more general results that ease the sphericity assumption on the subgroups, and apply them to local theta correspondence in type II and to degenerate Whittaker models.

L-functions of GL2n: p-adic properties and non-vanishing of twists

The principal aim of this article is to attach and study $p$-adic $L$-functions to cohomological cuspidal automorphic representations $\Pi$ of $\operatorname {GL}_{2n}$ over a totally real field $F$ admitting a Shalika model. We use a modular symbol approach, along the global lines of the work of Ash and Ginzburg, but our results are more definitive because we draw heavily upon the methods used in the recent and separate works of all three authors. By construction, our $p$-adic $L$-functions are distributions on the Galois group of the maximal abelian extension of $F$ unramified outside $p\infty$. Moreover, we work under a weaker Panchishkine-type condition on $\Pi _p$ rather than the full ordinariness condition. Finally, we prove the so-called Manin relations between the $p$-adic $L$-functions at all critical points. This has the striking consequence that, given a unitary $\Pi$ whose standard $L$-function admits at least two critical points, and given a prime $p$ such that $\Pi _p$ is ordinary, the central critical value $L(\frac {1}{2}, \Pi \otimes \chi )$ is non-zero for all except finitely many Dirichlet characters $\chi$ of $p$-power conductor.

Archimedean non-vanishing, cohomological test vectors, and standard L-functions of GL2n: Complex case

The purpose of this paper is to study the local zeta integrals of Friedberg-Jacquet at complex place and to establish similar results to the recent work [4] joint with C. Chen and D. Jiang. In this paper, we will (1) give a necessary and sufficient condition on an irreducible essentially tempered cohomological representation π of GL2n(C) with a non-zero Shalika model; (2) construct a new twisted linear period Λs,χ and give a different expression of the linear model for π; (3) give a necessary and sufficient condition on the character χ such that there exists a uniform cohomological test vector v∈Vπ (which we construct explicitly) for Λs,χ. As a consequence, we obtain the non-vanishing of local Friedberg-Jacquet integral at complex place. All of the above are essential preparations for attacking a global period relation problem in the forthcoming paper ([11]).

Local Rankin–Selberg integrals for Speh representations

We construct analogues of Rankin–Selberg integrals for Speh representations of the general linear group over a$p$-adic field. The integrals are in terms of the (extended) Shalika model and are expected to be the local counterparts of (suitably regularized) global integrals involving square-integrable automorphic forms and Eisenstein series on the general linear group over a global field. We relate the local integrals to the classical ones studied by Jacquet, Piatetski-Shapiro and Shalika. We also introduce a unitary structure for Speh representation on the Shalika model, as well as various other models including Zelevinsky’s degenerate Whittaker model.

Archimedean non-vanishing, cohomological test vectors, and standard L-functions of $${\mathrm {GL}}_{2n}$$: real case

The standard L-functions of $${\mathrm {GL}}_{2n}$$ expressed in terms of the Friedberg-Jacquet global zeta integrals have better structure for arithmetic applications, due to the relation of the linear periods with the modular symbols. The most technical obstacles towards such arithmetic applications are (1) non-vanishing of modular symbols at infinity and (2) the existence or construction of uniform cohomological test vectors. Problem (1) is also called the non-vanishing hypothesis at infinity, which was proved by Sun [Duke Math J 168(1):85–126, (2019), Theorem 5.1], by establishing the existence of certain cohomological test vectors. In this paper, we explicitly construct an archimedean local integral that produces a new type of a twisted linear functional $$\Lambda _{s,\chi }$$ , which, when evaluated with our explicitly constructed cohomological vector, is equal to the local twisted standard L-function $$L(s,\pi \otimes \chi )$$ for all complex values s. With the relations between linear models and Shalika models, we establish (1) with an explicitly constructed cohomological vector using classical invariant theory, and hence proves the non-vanishing results of Sun [24, Theorem 5.1] via a completely different method.

Uniqueness of twisted linear periods and twisted Shalika periods

Let $\rk$ be a local field of characteristic zero. Let $\pi$ be an irreducible admissible smooth representation of $\GL_{2n}(\rk)$. We prove that for all but countably many characters $\chi$ of $\GL_n(\rk)\times \GL_n(\rk)$, the space of $\chi$-equivariant (continuous in the archimedean case) linear functionals on $\pi$ is at most one dimensional. Using this, we prove the uniqueness of twisted Shalika models.

A Godement–Jacquet type integral and the metaplectic Shalika model

We present a novel integral representation for a quotient of global automorphic L-functions, the symmetric square over the exterior square. The pole of this integral characterizes a period of a residual representation of an Eisenstein series. As such, the integral itself constitutes a period, of an arithmetic nature. The construction involves the study of local and global aspects of a new model for double covers of general linear groups, the metaplectic Shalika model. In particular, we prove uniqueness results over p-adic and Archimedean fields, and a new Casselman-Shalika type formula.

A local trace formula for the generalized Shalika model

We study local multiplicities associated to the so-called generalized Shalika models. By establishing a local trace formula for these kind of models, we are able to prove a multiplicity formula for discrete series. As a result, we can show that these multiplicities are constant over every discrete Vogan $L$-packet and that they are related to local exterior square $L$-functions.

On Shalika models and p-adic L-functions

We use modular symbols to construct p-adic L-functions for cohomological cuspidal automorphic representations on GL(2n), which admit a Shalika model. Our construction differs from former ones in that it systematically makes use of the representation theory of p-adic groups.

A Casselman–Shalika formula for the generalized Shalika model of SO4n

A Casselman–Shalika formula for the generalized Shalika model of SO4n

The characterization of theta-distinguished representations of GL(n)

Let θ and θ’ be a pair of exceptional representations in the sense of Kazhdan and Patterson [KP84], of a metaplectic double cover of GL n . The tensor θ ⊗ θ’ 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 contragredients. 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”.

On the local Bump–Friedberg L-function II

Let F be a p-adic field with residue field of cardinality q. To each irreducible representation of GL(n, F), we attach a local Euler factor $$L^{BF}(q^{-s},q^{-t},\pi )$$ via the Rankin–Selberg method, and show that it is equal to the expected factor $$L(s+t+1/2,\phi _\pi )L(2s,\Lambda ^2\circ \phi _\pi )$$ of the Langlands’ parameter $$\phi _\pi $$ of $$\pi $$ . The corresponding local integrals were introduced in Bump and Friedberg (The exterior square automorphic L-functions on $$\mathrm{GL}(n)$$ 47–65, 1990), and studied in Matringe (J Reine Angew Math doi: 10.1515/crelle-2013-0083 ). This work is in fact the continuation of Matringe (J Reine Angew Math doi: 10.1515/crelle-2013-0083 ). The result is a consequence of the fact that if $$\delta $$ is a discrete series representation of GL(2m, F), and $$\chi $$ is a character of Levi subgroup $$L=GL(m,F)\times GL(m,F)$$ which is trivial on GL(m, F) embedded diagonally, then $$\delta $$ is $$(L,\chi )$$ -distinguished if an only if it admits a Shalika model. This result was only established for $$\chi =\mathbf {1}$$ before.

On the arithmetic of Shalika models and the critical values of L -functions for GL 2 n

Let $\Pi$ be a cohomological cuspidal automorphic representation of ${\rm GL}_{2n}(\Bbb{A})$ over a totally real number field $F$. Suppose that $\Pi$ has a Shalika model. We define a rational structure on the Shalika model of $\Pi_f$. Comparing it with a rational structure on a realization of $\Pi_f$ in cuspidal cohomology in top-degree, we define certain periods $\omega^{\epsilon}(\Pi_f)$. We describe the behavior of such top-degree periods upon twisting $\Pi$ by algebraic Hecke characters $\chi$ of $F$. Then we prove an algebraicity result for all the critical values of the standard $L$-functions $L(s,\Pi\otimes\chi)$; here we use the recent work of B. Sun on the non-vanishing of a certain quantity attached to $\Pi_\infty$. As applications, we obtain algebraicity results in the following cases: Firstly, for the symmetric cube $L$-functions attached to holomorphic Hilbert modular cusp forms; we also discuss the situation for higher symmetric powers. Secondly, for certain (self-dual of symplectic type) Rankin-Selberg $L$-functions for ${\rm GL}_3\times{\rm GL}_2$; assuming Langlands Functoriality, this generalizes to certain Rankin-Selberg $L$-functions of ${\rm GL}_n\times{\rm GL}_{n-1}$. Thirdly, for the degree four $L$-functions attached to Siegel modular forms of genus $2$ and full level. Moreover, we compare our top-degree periods with periods defined by other authors. We also show that our main theorem is compatible with conjectures of Deligne and Gross.

Linear and Shalika local periods for the mirabolic group, and some consequences

Using linear periods on the mirabolic subgroup of GL(n,F), for F a nonarchimedean local field, we give a list of the maximal Levi subgroups of GL(n,F) which can distinguish discrete series and generic representations. We also obtain the functional equation of the local exterior-square L-function of a generic representation of GL(n,F) when n=2m is even. Then we discuss the relation between Shalika models and local models for representations of GL(2m,F). Finally we give a necessary and sufficient condition on the cuspidal representation ρ for a generalised Steinberg representation Stk(ρ) of GL(2m,F) to have a local model.

Generalized Shalika models of p-adic SO4n and functoriality

Let F denote a p-adic local field of characteristic zero. In this paper, we investigate the structures of irreducible admissible representations of SO4n (F) having nonzero generalized Shalika models and find relations between the generalized Shalika models and the local Arthur parameters, which support our conjectures on the local Arthur parametrization and the local Langlands functoriality in terms of the dual group associated to the spherical variety, which is attached to the generalized Shalika models.

Uniqueness of Shalika Models

Abstract Let 𝔽q be a finite field of q elements, 𝓕 a p-adic field, and D a quaternion division algebra over 𝓕. This paper proves uniqueness of Shalika models for GL2n(𝔽q) and GL2n(D), and re-obtains uniqueness of Shalika models for GL2n(𝔽q) and GL2n(D), and re-obtains uniqueness of Shalika models for GL2n (𝓕) for any n ∈ ℕ.

Local uniqueness of generalized Shalika models for [formula omitted

This paper shows the uniqueness of generalized Shalika model on SO ( 4 n , F ) , n ⩾ 1 , where F is a p-adic field or a finite field with characteristic p, p ≠ 2 .

Local Shalika models and functoriality

We prove, over a p-adic local field F, that an irreducible supercuspidal representation of GL2n(F) is a local Langlands functorial transfer from SO2n+1(F) if and only if it has a nonzero Shalika model (Corollary 5.2, Proposition 5.4 and Theorem 5.5). Based on this, we verify (Sect. 6) in our cases a conjecture of Jacquet and Martin, a conjecture of Kim, and a conjecture of Speh in the theory of automorphic forms.

On the Restriction to 𝒟* × 𝒟* of Representations of p-adic GL2(𝒟)

AbstractLet be a division algebra over a nonarchimedean local field. Given an irreducible representation π of GL2(), we describe its restriction to the diagonal subgroup × . The description is in terms of the structure of the twisted Jacquet module of the representation π. The proof involves Kirillov theory that we have developed earlier in joint work with Dipendra Prasad. The main result on restriction also shows that π is × -distinguished if and only if π admits a Shalika model. We further prove that if is a quaternion division algebra then the twisted Jacquetmodule is multiplicity-free by proving an appropriate theorem on invariant distributions; this then proves a multiplicity-one theorem on the restriction to × in the quaternionic case.