Cuspidal Representations Research Articles

Functoriality of Automorphic $\mathrm{L}$-Invariants and Applications

We study the behaviour of automorphic \mathrm{L} -invariants associated to cuspidal representations of \mathrm{GL}(2) of cohomological weight 0 under abelian base change and Jacquet-Langlands lifts to totally definite quaternion algebras. Under a standard non-vanishing hypothesis on automorphic \mathrm{L} -functions and some technical restrictions on the automorphic representation and the base field we get a simple proof of the equality of automorphic and arithmetic \mathrm{L} -invariants. This together with Spieß' results on p -adic \mathrm{L} -functions yields a new proof of the exceptional zero conjecture for modular elliptic curves – at least, up to sign.

On the Distribution of Satake Parameters for Siegel Modular Forms

We prove a harmonically weighted equidistribution result for the p -th Satake parameters of the family of automorphic cuspidal representations of \operatorname{PGSp}(2n) of fixed weight \mathtt{k} and prime-to- p level N\to \infty . The main tool is a new asymptotic Petersson formula for \operatorname{PGSp}(2n) in the level aspect.

Jordan blocks of cuspidal representations of symplectic groups

Let $G$ be a symplectic group over a nonarchimedean local field of characteristic zero and odd residual characteristic. Given an irreducible cuspidal representation of G, we determine its Langlands parameter (equivalently, its Jordan blocks in the language of Moeglin) in terms of the local data from which the representation is explicitly constructed, up to a possible unramified twist in each block of the parameter. We deduce a Ramification Theorem for $G$, giving a bijection between the set of endo-parameters for $G$ and the set of restrictions to wild inertia of discrete Langlands parameters for $G$, compatible with the local Langlands correspondence. The main tool consists in analysing the intertwining Hecke algebra of a good cover, in the sense of Bushnell--Kutzko, for parabolic induction from a cuspidal representation of $G\times\mathrm{GL}_n$, seen as a maximal Levi subgroup of a bigger symplectic group, in order to determine its (ir)reducibility; a criterion of Moeglin then relates this to Langlands parameters.

Symplectic models for unitary groups

In analogy with the study of representations of GL 2 n ⁡ ( F ) \operatorname {GL}_{2n}(F) distinguished by Sp 2 n ⁡ ( F ) \operatorname {Sp}_{2n}(F) , where F F is a local field, we study representations of U 2 n ⁡ ( F ) \operatorname {U}_{2n}(F) distinguished by Sp 2 n ⁡ ( F ) \operatorname {Sp}_{2n}(F) in this paper. (Only quasisplit unitary groups are considered in this paper since they are the only ones which contain Sp 2 n ⁡ ( F ) \operatorname {Sp}_{2n}(F) .) We prove that there are no cuspidal representations of U 2 n ⁡ ( F ) \operatorname {U}_{2n}(F) distinguished by Sp 2 n ⁡ ( F ) \operatorname {Sp}_{2n}(F) for F F a nonarchimedean local field. We also prove the corresponding global theorem that there are no cuspidal automorphic representations of U 2 n ⁡ ( A k ) \operatorname {U}_{2n}(\mathbb {A}_k) with nonzero period integral on Sp 2 n ⁡ ( k ) ∖ Sp 2 n ⁡ ( A k ) \operatorname {Sp}_{2n}(k) \backslash \operatorname {Sp}_{2n}(\mathbb {A}_k) for k k any number field or a function field. We completely classify representations of quasisplit unitary groups in four variables over local and global fields with nontrivial symplectic periods using methods of theta correspondence. We propose a conjectural answer for the classification of all representations of a quasisplit unitary group distinguished by Sp 2 n ⁡ ( F ) \operatorname {Sp}_{2n}(F) .

Rankin–Selberg gamma factors of level zero representations of GL n

Abstract For a p-adic local field F of characteristic 0, with residue field 𝔣 {\mathfrak{f}} , we prove that the Rankin–Selberg gamma factor of a pair of level zero representations of linear general groups over F is equal to a gamma factor of a pair of corresponding cuspidal representations of linear general groups over 𝔣 {\mathfrak{f}} . Our results can be used to prove a variant of Jacquet’s conjecture on the local converse theorem.

N-level densities for twisted cubic Dirichlet L-functions

For any given cuspidal representation π of GLm(AQ), we consider the family of automorphic L-functions L(s,π×χ) twisted by cubic Dirichlet characters. We compute the n-level density for this family and confirm that it agrees with unitary type U.

Twisted Lefschetz number formula and 𝑝-adic trace formula

For a reductive group G / Q G_{/\mathbb {Q}} , we interpolate Arthurs L 2 L^2 -Lefschetz number formula p p -adically and give a geometric expansion of Urbans p p -adic trace formula on overconvergent cuspidal representations. If G / Q G_{/\mathbb {Q}} is anisotropic at infinity and H H is a Weil restriction of G G , we give a twisted version of Arthur’s L 2 L^2 -Lefschetz number formula for H H , and we set up both the spectral side and the geometric side of a twisted p p -adic trace formula for H H .

Transfer to Characteristic Zero of the Jacquet–Mao’s Metaplectic Fundamental Lemma

Abstract Jacquet conjectured that globally a cuspidal automorphic representation of $\mathrm{G}\mathrm{L}_r$ should be in the image of the metaplectic correspondence precisely when it is distinguished with respect to a split orthogonal similitude group. Jacquet–Mao have suggested an approach to solve this problem by using “relative” traces formula. One of the steps of this approach is the Jacquet–Mao’s metaplectic fundamental lemma. The author proved this fundamental lemma for any $r$ in the case of positive characteristic. The aim of this paper is to extend this result to the more general case.

INTERTWINING SEMISIMPLE CHARACTERS FOR -ADIC CLASSICAL GROUPS

Let $G$ be an orthogonal, symplectic or unitary group over a non-archimedean local field of odd residual characteristic. This paper concerns the study of the “wild part” of an irreducible smooth representation of $G$, encoded in its “semisimple character”. We prove two fundamental results concerning them, which are crucial steps toward a complete classification of the cuspidal representations of $G$. First we introduce a geometric combinatorial condition under which we prove an “intertwining implies conjugacy” theorem for semisimple characters, both in $G$ and in the ambient general linear group. Second, we prove a Skolem–Noether theorem for the action of $G$ on its Lie algebra; more precisely, two semisimple elements of the Lie algebra of $G$ which have the same characteristic polynomial must be conjugate under an element of $G$ if there are corresponding semisimple strata which are intertwined by an element of $G$.

Constructing tame supercuspidal representations

A new approach to Jiu-Kang Yu’s construction of tame supercuspidal representations of p p -adic reductive groups is presented. Connections with the theory of cuspidal representations of finite groups of Lie-type and the theory of distinguished representations are also discussed.

On certain degenerate Whittaker Models for cuspidal representations of $${\mathrm{GL}_{k \cdot n}(\mathbb {F}_q)}$$ GL k · n ( F q )

Let $\pi$ be an irreducible cuspidal representation of $\mathrm{GL}_{kn}\left(\mathbb{F}_q\right)$. Assume that $\pi = \pi_{\theta}$, corresponds to a regular character $\theta$ of $\mathbb{F}_{q^{kn}}^{*}$. We consider the twisted Jacquet module of $\pi$ with respect to a non-degenerate character of the unipotent radical corresponding to the partition $(n^k)$ of $kn$. We show that, as a $\mathrm{GL}_{n}\left(\mathbb{F}_q\right)$-representation, this Jacquet module is isomorphic to $\pi_{\theta \upharpoonright_{\mathbb{F}_n^*}} \otimes \mathrm{St}^{k-1}$, where $\mathrm{St}$ is the Steinberg representation of $\mathrm{GL}_{n}\left(\mathbb{F}_q\right)$. This generalizes a theorem of D. Prasad, who considered the case $k=2$. We prove and rely heavily on a formidable identity involving $q$-hypergeometric series and linear algebra.

Cuspidal representations in the cohomology of Deligne–Lusztig varieties for GL(2) over finite rings

We define closed subvarieties of some Deligne–Lusztig varieties for GL(2) over finite rings and study their ´etale cohomology. As a result, we show that cuspidal representations appear in it. Such closed varieties are studied in [Lus2] in a special case. We can do the same things for a Deligne–Lusztig variety associated to a quaternion division algebra over a non-archimedean local field. A product of such varieties can be regarded as an affine bundle over a curve. The base curve appears as an open subscheme of a union of irreducible components of the stable reduction of the Lubin–Tate curve in a special case. Finally, we state some conjecture on a part of the stable reduction using the above varieties. This is an attempt to understand bad reduction of Lubin–Tate curves via Deligne–Lusztig varieties.

Gamma Factors, Root Numbers, and Distinction

AbstractWe study a relation between distinction and special values of local invariants for representations of the general linear group over a quadratic extension of p-adic fields. We show that the local Rankin–Selberg root number of any pair of distinguished representation is trivial, and as a corollary we obtain an analogue for the global root number of any pair of distinguished cuspidal representations. We further study the extent to which the gamma factor at 1/2 is trivial for distinguished representations as well as the converse problem.

Rationality for isobaric automorphic representations: the CM-case

In this note we prove a simultaneous extension of the author’s joint result with M. Harris for critical values of Rankin–Selberg L-functions L(s,Pi times Pi ') (Grobner and Harris in J Inst Math Jussieu 15:711–769, 2016, Thm. 3.9) to (i) general CM-fields F and (ii) cohomological automorphic representations Pi '=Pi _1boxplus cdots boxplus Pi _k which are the isobaric sum of unitary cuspidal automorphic representations Pi _i of general linear groups of arbitrary rank over F. In this sense, the main result of these notes, cf. Theorem 1.9, is a generalization, as well as a complement, of the main results in Raghuram (Forum Math 28:457–489, 2016; Int Math Res Not 2:334–372, 2010. https://doi.org/10.1093/imrn/rnp127), and Mahnkopf (J Inst Math Jussieu 4:553–637, 2005).

Gamma factors and quadratic extension over finite fields

This paper characterizes $${\mathrm {GL}}_n({\mathbb {F}}_q )$$ -distinguished cuspidal representations of $${\mathrm {GL}}_n({\mathbb {F}}_{q^2})$$ in terms of the special values of their twisted gamma factors.

The Spin L-function on GSp6 Via a non-unique model

We give two global integrals that unfold to a non-unique model and represent the partial Spin $L$-function on ${\rm PGSp}_6$. We deduce that for a wide class of cuspidal automorphic representations $\pi$, the partial Spin $L$-function is holomorphic except for a possible simple pole at $s=1$, and that the presence of such a pole indicates that $\pi$ is an exceptional theta lift from ${\rm G}_2$. These results utilize and extend previous work of Gan and Gurevich, who introduced one of the global integrals and proved these facts for a special subclass of these $\pi$ upon which the aforementioned model becomes unique. The other integral can be regarded as a higher rank analogue of the integral of Kohnen-Skoruppa on ${\rm GSp}_4$.

Endoscopic classification of very cuspidal representations of quasi-split unramified unitary groups

Endoscopic classification of very cuspidal representations of quasi-split unramified unitary groups

Reducibility of some generalized principal series of the metaplectic group

We determine reducibility of the representation of the metaplectic group induced from the tensor product of an essentially square integrable representation attached to the Zelevinsky segment and a genuine cuspidal representation of the metaplectic group.

Exponential sums with Fourier coefficients of automorphic forms

Let $$\pi _1,\pi _2, \ldots , \pi _k$$ be cuspidal automorphic representations of $$\mathrm{GL}(2)$$ or $$\mathrm{GL}(3)$$ with trivial conductor and trivial central character, and $$\pi :=\pi _1\boxplus \pi _2\boxplus \cdots \boxplus \pi _k$$ be the isobaric representation associated to them. We establish that there exists a positive constant $$\vartheta _k <1$$ such that for any $$\alpha \in {\mathbb {R}}$$ , any $$x \ge 1$$ and any positive $$\varepsilon $$ , one has $$\begin{aligned} \sum _{n \le x} \lambda _{\pi _1\boxplus \pi _2\boxplus \cdots \boxplus \pi _k}(n)\mathrm{e}(\alpha n) \ll _{\pi _j,\varepsilon } x^{\vartheta _k+\varepsilon }, \end{aligned}$$ where the implied constant does not depend on $$\alpha $$ . We also consider some isobaric representations containing the trivial representation. As an application, we consider averages of shifted convolution sums of the type $$\begin{aligned} \sum _{h=1}^{H}\sum _{n=1}^{X}a(n)\lambda _{\pi _1\boxplus \pi _2\boxplus \cdots \boxplus \pi _k}(n+h), \end{aligned}$$ where a(n) is any arithmetic function.

Rationality results for the exterior and the symmetric square L-function (with an appendix by Nadir Matringe)

Let G=mathrm{GL}_{2n} over a totally real number field F and nge 2. Let Pi be a cuspidal automorphic representation of G(mathbb {A}), which is cohomological and a functorial lift from SO(2n+1). The latter condition can be equivalently reformulated that the exterior square L-function of Pi has a pole at s=1. In this paper, we prove a rationality result for the residue of the exterior square L-function at s=1 and also for the holomorphic value of the symmetric square L-function at s=1 attached to Pi . As an application of the latter, we also obtain a period-free relation between certain quotients of twisted symmetric square L-functions and a product of Gauß sums of Hecke characters.