Genericity and contragredience in the local Langlands correspondence

We prove the recent conjectures of Adams-Vogan and D. Prasad on the behavior of the local Langlands correspondence with respect to taking the contragredient of a representation. The proof holds for tempered representations of quasi-split real K-groups and quasi-split p-adic classical groups (in the sense of Arthur). We also prove a formula for the behavior of the local Langlands correspondence for these groups with respect to changes of the Whittaker data.Comment: Minor changes to the introduction and references to place the paper in the proper context. Corollary 4.10 added. An inaccuracy in the treatment of even orthogonal groups fixe

On the Kottwitz conjecture for local Shimura varieties

Kottwitz’s conjecture describes the contribution of a supercuspidal represention to the cohomology of a local Shimura variety in terms of the local Langlands correspondence. Using a Lefschetz-Verdier fixedpoint formula, we prove a weakened generalized version of Kottwitz’s conjecture. The weakening comes from ignoring the action of the Weil group and only considering the actions of the groups G and Jb up to non-elliptic representations. The generalization is that we allow arbitrary connected reductive groups G and non-minuscule coweights µ

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).Comment: 18 page