Beyond endoscopy is the strategy put forward by Langlands for applying the trace formula to the general principle of functoriality. Subsequent papers by Langlands (one in collaboration with Frenkel and Ngo), together with more recent papers by Altug, have refined the strategy. They all emphasize the importance of understanding the elliptic terms on the geometric side of the trace formula. We shall discuss the general strategy, and how it pertains to these terms.[-]

Let G be a connected reductive p-adic group that splits over a tamely ramified extension. Let H be the fixed points of an involution of G. An irreducible smooth H-distinguished representation of G is H-relatively supercuspidal if its relative matrix coefficients are compactly supported modulo H Z(G). (Here, Z(G) is the centre of G.) We will describe some relatively supercuspidal representations whose cuspidal supports belong to the supercuspidals constructed by J.K. Yu.[-]

The local Gan-Gross-Prasad conjectures concern certain branching or restriction problems between representations of real or p-adic Lie groups. In its simplest form it predicts certain multiplicity-one results for "extended" L-packets. In a recent series of papers, Waldspurger has settled the conjecture for special orthogonal groups over p-adic field. In this talk, I will present a proof of the conjecture for unitary groups which has the advantage of working equally well over archimedean and non-archimedean fields.[-]

Let $p$ be a prime number and $F$ be a non-archimedean field with finite residue class field of characteristic $p$. Understanding the category of Iwahori-Hecke modules for $SL_2(F)$ is of great interest in the study of $p$-modular smooth representations of $SL_2(F)$, as these modules naturally show up as spaces of invariant vectors under the action of the standard pro-$p$-Iwahori subgroup. In this talk, we will discuss a work in progress in which we aim to classify all non-trivial extensions between these modules and to compare them with their analogues for $p$-modular smooth representations of $SL_2(F)$ and with their Galois counterpart in the setting of the local Langlands correspondences in natural characteristic.[-]

I will discuss the transfer of Harish-Chandra characters under the local theta correspondence, in particular in the (almost) equal rank case. More precisely, if $G X H$ is a dual pair in the equal rank setting, it is known that discrete series (resp. tempered) representations lifts to discrete series (resp. tempered) representations. If two such representations correspond under theta lifting, one can ask how their Harish-Chandra characters are related. I will define a space of test functions on each group and a correspondence of their orbital integrals induced by the Weil representation, and show that the resulting transfer of invariant distribution carries the character of a tempered representation to that of its theta lift. I will also explain how the transfer of test functions can be understood geometrically, by relating it to the moment map arising in theta correspondence.[-]

The metaplectic covering Mp(2n) of Sp(2n) affords an accessible yet nontrivial instance of the Langlands-Weissman program for covering groups. In order to use Arthur's methods in this setting, one needs a stable trace formula for Mp(2n). Thus far, only the elliptic terms have been stabilized. In this talk, I will report an ongoing work on the full stabilization, which is nearing completion. It will hopefully grant access to the whole genuine discrete automorphic spectrum of Mp(2n). Time permitting, I will also try to explain the similarities and subtle differences with the case of linear groups solved by Arthur.[-]

In this talk, we consider the limit multiplicity question (and some variants): how many automorphic forms of fixed infinity-type and level N are there as N grows? The question is well-understood when the archimedean representation is a discrete series, and we focus on non-tempered cohomological representations on unitary groups. Using an inductive argument which relies on the stabilization of the trace formula and the endoscopic classification, we give asymptotic counts of multiplicities, and prove the Sarnak-Xue conjecture at split level for (almost!) all cohomological representations of unitary groups. Additionally, for some representations, we derive an average Sato-Tate result in which the measure is the one predicted by functoriality. This is joint work with Rahul Dalal.[-]