# Representations and Lie Theory Seminar

## Schedule of talks:

 TIME SPEAKER TITLE HOST September 16 Eyal Kaplan The metaplectic Shalika model and a new Godement-Jacquet type integral September 23 Yiannis Sakellaridis Paley-Wiener theorems for p-adic spherical varieties Eyal Kaplan September 30 Fan Zhou Double Dirichlet Series and Voronoi Formula October 14 Roman Holowinsky Sup-norms of Hecke-Maass cusp forms October 21 Thorsten Heidersdorf Invariant theory and interpolating categories for the orthosymplectic supergroup November 4 Vitaly Bergelson Sarnak's conjecture November 18 Eyal Kaplan The metaplectic Shalika model and symmetric square L-function December 2 David Soudry On Rankin-Selberg integrals for classical groups Eyal Kaplan January 27 Eyal Kaplan Covering groups - computations and techniques February 3 Eyal Kaplan Covering groups - computations and techniques - part II February 10 Eyal Kaplan Covering groups - computations and techniques - part III February 24 Eyal Kaplan Covering groups part IV - the case of GL(n) March 2 Eyal Kaplan A new Godement-Jacquet type integral March 9 Eyal Kaplan Doubling global constructions for tensor product L-functions March 30 Mahdi Asgari Local Langlands Correspondence for Small Rank GSpin Groups Eyal Kaplan April 6 Yuanqing Cai Fourier Coefficients for Generalized Theta Representations Eyal Kaplan April 20 Beth Romano Representations of p-adic groups via geometric invariant theory Eyal Kaplan

## Abstracts

September 16: One of the tools frequently used in the study of representations and L-functions is called a model. Generally speaking, if G is a locally compact group, pi is an irreducible representation of G, X is a closed subgroup and Psi is a character of X, an (X,Psi)-equivariant functional on the space of pi gives rise to a model of pi, as a summand of the representation of G induced from X and Psi. In the context of covering groups, one naturally looks for subgroups X that are split under the cover. We describe a novel notion of a metaplectic Shalika model, which is the analog of the Shalika model of GL(2n) introduced by Jacquet and Shalika. We study uniqueness and prove a metaplectic Casselman-Shalika formula along the lines of the non-metplectic formula of Sakellaridis. As an application we consider a new Godement-Jacquet integral for the symmetric square L-function. This is a joint work with Jan Möllers.

September 23: For a reductive p-adic group H, the description of the Bernstein center of the category of smooth H-representations is equivalent to the description of H x H-endomorphisms of the space of smooth, compactly supported functions on H, and the Bernstein decomposition corresponds to an H x H-equivariant decomposition of this space. The goal of this talk is to generalize this decomposition when H is replaced by a spherical variety X satisfying some strong assumptions (which cover all symmetric cases). In parallel, we will discuss the spectral transform of the space of Harish-Chandra Schwartz functions on X, generalizing Harish-Chandra's description for X=H. This is joint work with Patrick Delorme and Pascale Harinck.

September 30: We present a proof of Voronoi formula for coefficients of a large class of L-functions, in the style of the classical converse theorem of Weil. Our formula applies to Maass cusp forms, Rankin-Selberg convolutions, and certain isobaric sums. Our proof is based on the functional equations of L-functions twisted by Dirichlet characters and does not directly depend on automorphy and hence has wider application than previous proofs. The key ingredient is the construction of a double Dirichlet series associated with these coefficients. This is joint work with Eren Mehmet Kiral.

October 14: We shall highlight recent advancements in the study of sup-norms for Hecke-Maass cusp forms on higher rank groups. In particular, this talk will provide a summary of joint work with G. Ricotta and E. Royer regarding an explicit non-trivial bound in the eigenvalue aspect for the sup-norm of an SL3(Z) Hecke-Maass cusp form restricted to a compact set.

October 21: We describe the construction of Deligne's category $Rep(O_{\delta})$, classify the indecomposable objects and explain how to decompose their tensor products. We then classify thick ideals in $Rep(O_{\delta})$. As an application we classify the indecomposable summands of tensor powers of the standard representation of the orthosymplectic supergroup up to isomorphism.

November 4: After introducing some necessary definitions, we will formulate and discuss the tantalizing conjecture due to Sarnak which, roughly, states that the classical Möbius function is orthogonal to any deterministic bounded sequence.

November 18: One of the pillars of the Langlands program is L-functions. We will recall Artin's L-functions, which were a precursor of Langlands', then briefly describe the modern automorphic point of view. One of the tools frequently used in the study of group representations and L-functions is called a model. Roughly speaking, a model is a unique realization of a representation in a convenient space of functions on the group. We will present a novel model: the metaplectic Shalika model. This is the analog of the Shalika model of GL(2n) of Jacquet and Shalika. One interesting representation having this model is the so-called exceptional representation of Kazhdan and Patterson, which is the analog for linear groups of the Weil representation. This representation is truly exceptional. We will describe it and its role in the study of the symmetric square L-function, and related problems.

December 2: I will survey the structure of families of global integrals of Rankin-Selberg type, which were predicted to represent partial L-functions for pairs of irreducible, automorphic, cuspidal representations (\pi, \tau) on (G, GL_n), where G is a classical group. I will focus on split orthogonal groups. In the global integrals, we integrate a Fourier coefficient "of Bessel type" applied to a cusp form on G against an Eisenstein series on a related classical group H, induced from a maximal parabolic subgroup, or vice versa. These families of integrals contain all known ones which represent the partial L-functions above. They were first introduced by Ginzburg, Piatetski-Shapiro and Rallis, and were calculated in the so-called "spherical case" (of Bessel models). I will present the calculation of the unramified local integrals at all cases. It is done by "analytic continuation" from the generic cases above. The global integrals above are useful in locating poles of L-functions of representations \pi with a given type of Bessel models.

January 27: This will be a hands-on talk about covering groups, with plenty of examples and computations. I'll survey some of the basics, discuss references to fundamental works, and try to give a taste of what's out there.

February 3: Part II of the talk from January 27.

February 10: Part III of the talk from January 27.

February 24: Part IV of the talk from January 27, we shall now focus on the covering groups of GL(n).

March 2: In a joint work with Jan Möllers, we present a novel integral representation. This is part II of the talk from September 16.

March 9: I will present a joint work with Cai, Friedberg and Ginzburg. In a series of constructions, we apply the "doubling method" from the theory of automorphic forms to covering groups. We obtain partial tensor product L-functions attached to generalized Shimura lifts, which may be defined in a natural way since at almost all places the representations are unramified principal series.

March 30: As part of his overall program, R. P. Langlands conjectured a certain correspondence between the set of (equivalence classes of) irreducible smooth representations of a $p$-adic reductive group and its set of $L$-parameters, i.e., certain admissible homomorphisms from the Weil group of the $p$-adic field to a certain complex group, called the $L$-group. This parameterization is to satisfy various number theoretic and representation theoretic properties involving $L$-functions and $\epsilon$-factors among other things, and lies, along with its global conjectural analogue, at the heart of the modern theory of automorphic forms and representations. When the group is $GL_n$ the correspondence is actually a bijection, now a theorem due to Michael Harris & Richard Taylor as well as Guy Henniart (around 2001) and Peter Scholze again more recently (around 2013). For symplectic and special orthogonal groups, it is a consequence of Arthur's recent work on endoscopic classification of local and global representations of these groups. I will review some background material and report on a recent work, joint with Kwangho Choiy, establishing the Local Langlands Correspondence for small rank general spin groups and their inner forms.

April 6: Kazhdan and Patterson constructed generalized theta representations on covers of general linear groups as multi-residues of the Borel Eisenstein series. These representations and their unique models play important roles in the Rankin-Selberg constructions of the symmetric square L-functions for GL(r). In this talk, we will discuss the two other types of models that the theta representations may support. We also determine the covers when these models are unique. Time permitting, we will discuss applications in Rankin-Selberg constructions.

April 20: Let G be a semisimple split reductive group over a finite extension k of Q_p. In recent work Reeder and Yu have given a new construction of supercuspidal representations of G(k). Their construction is uniform for all p but requires as input stable vectors (in the sense of geometric invariant theory) in a certain representation coming from a Moy-Prasad filtration. In joint work, Jessica Fintzen and I have given a criterion for this representation to contain stable vectors. Our methods do not depend on p and thus give new representations when p is small. In my talk, I will give an overview of this work, as well as explicit examples for the case when G = G_2. For these examples, I will explicitly describe the locus of all stable vectors, as well as the parameters which correspond under the local Langlands correspondence to the representations given by the construction of Reeder-Yu.