Tasho Kaletha

Quantifying residual finiteness of arithmetic groups

The normal Farb growth of a group quantifies how well-approximated the group is by its finite quotients. We show that any S-arithmetic subgroup of a higher rank Chevalley group G has normal Farb growth n dim(G) .

Global rigid inner forms and multiplicities of discrete automorphic representations

We study the cohomology of certain Galois gerbes over number fields. This cohomology provides a bridge between refined local endoscopy, as introduced in Kaletha (Ann Math (2) 184(2):559–632, 2016), and classical global endoscopy. As particular applications, we express the canonical adelic transfer factor that governs the stabilization of the Arthur–Selberg trace formula as a product of normalized local transfer factors, we give an explicit constriction of the pairing between an adelic L-packet and the corresponding S-group (based on the conjectural pairings in the local setting) that is the essential ingredient in the description of the discrete automorphic spectrum of a reductive group, and we give a proof of some expectations of Arthur.

Simple wild -packets

In a recent paper, Gross and Reeder study arithmetic properties of discrete Langlands parameters for semi-simple p-adic groups and conjecture that a special class of these -- the simple wild parameters -- should correspond to L-packets consisting of simple supercuspidal representations. We provide a construction of this correspondence and show that the simple wild L-packets satisfy many expected properties. In particular, they admit a description in terms of the Langlands dual group and contain a unique generic element for a fixed Whittaker datum. Moreover, we prove their stability on an open subset of the regular semi-simple elements, and show that they satisfy a natural compatibility with respect to unramified base-change.

Rigid inner forms vs isocrystals

We compare two statements of the refined local Langlands correspondence for connected reductive groups defined over a p-adic field -- one involving Kottwitzs set B(G) of isocrystals with additional structure, and one involving the cohomology set H^1(u -> W,Z -> G) introduced in arXiv:1304.3292. We show that if either statement is valid for all connected reductive groups, then so is the other. We also discuss how the second statement depends on the choice of element of H^1(u -> W,Z -> G).

Endoscopic character identities for depth-zero supercuspidal L-packets

We prove the conjectural endoscopic transfer of L-packets for the local Langlands correspondence for pure inner forms of unramified p-adic groups and depth-zero parameters established by DeBacker and Reeder. More precisely, we show that under mild conditions on the residual characteristic, endoscopic induction identifies an unstable character of such an L-packet with the stable character of the corresponding endoscopic L-packet.

The Local Langlands Conjectures for Non-quasi-split Groups

We present different statements of the local Langlands conjectures for non-quasi-split groups that currently exist in the literature and provide an overview of their historic development. Afterwards, we formulate the conjectural multiplicity formula for discrete automorphic representations of non-quasi-split groups.

Decomposition of splitting invariants in split real groups

To a maximal torus in a quasi-split semi-simple simply-connected group over a local field of characteristic 0, Langlands and Shelstad construct a cohomological invariant called the splitting invariant, which is an important component of their endoscopic transfer factors. We study this invariant in the case of a split real group and prove a decomposition theorem which expresses this invariant for a general torus as a product of the corresponding invariants for simple tori. We also show how this reduction formula allows for the comparison of splitting invariants between different tori in the given real group.