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.[-]

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.[-]

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.[-]

In this talk we shall discuss our recent results on the Adams' conjecture on theta correspondence. In more words, given a representation of a classical group (in our case, symplectic or even orthogonal) belonging to a local Arthur packet, Adams predicts that, under certain assumptions, its theta lift (i.e. a corresponding irreducible representation of the other group in a dual reductive pair), provided it is non-zero, is also in A-packet which can be easily described in terms of the original one. Mœglin gave some partial results, specifically, in case when the original representation is square-integrable. We are able to extend her results to the case of so called Arthur packets with the discrete diagonal restriction. Moreover, it seems that Arthur packet encapsulates lot of additional information even in relation to theta correspondence, e.g. we can easily read of from it the first occurrence index for the given representation in it. Adams conjecture takes an unexpectedly elegant form for the representations in discrete diagonal restriction packets. Also, we are able to pinpoint exactly how low in theta towers we can go with this description of the theta lifts which belong to Arthur packets, we can also address some other related conjectures due to Mœglin. This is joint work with Petar Baki.[-]