## Number Theory Seminar at Johns Hopkins University |

## Fall 2017 |

Except where (frequently) noted to the contrary, the seminar meets from 4:30-5:30 every other Tuesday in Krieger 306.

If time permits, we will present the underlying general set-up that allows us to compare the Moy-Prasad filtration representations for different primes p. This provides a tool to transfer results about the Moy-Prasad filtration from one prime to arbitrary primes and also yields new descriptions of the Moy-Prasad filtration representations.

SPECIAL TIME AND ROOM:

SPECIAL TIME AND ROOM:

SPECIAL TIME AND ROOM:

The study of the interaction between the Bruhat decomposition and the conjugation action is an important and very active area. In this talk, we focus on the affine Deligne-Lusztig variety, which describes the interaction between the Bruhat decomposition and the Frobenius-twisted conjugation action of loop groups. The affine Deligne-Lusztig variety was introduced by Rapoport around 20 years ago and it has found many applications in arithmetic geometry and number theory.

In this talk, we will discuss some recent progress on the study of affine Deligne-Lusztig varieties, and some applications to Shimura varieties.

More precisely, for a spherical variety X=H\G of rank one, I will prove that there is an explicit "transfer operator" which transforms the orbital integrals of the relative trace formula for X x X/G to the orbital integrals of the Kuznetsov formula for GL(2) or SL(2), equipped with suitable non-standard test functions. The operator is determined by the L-value associated to the square of the H-period integral, and the proof uses a deep theory of Friedrich Knop on the cotangent bundles of spherical varieties. This is part of an ongoing joint project with Daniel Johnstone and Rahul Krishna, who are proving instances of the fundamental lemma. Globally, this transfer will induce an identity of relative trace formulas and global relative characters, translating to an Ichino-Ikeda type formula that relates the square of the H-period to the said L-value.

This can be viewed as part of the program of relative functoriality, a generalization of the Langlands functoriality conjecture, predicting relations between the automorphic spectra of two spherical varieties when there is a map between their dual groups. The case under consideration here is the simplest non-abelian case of this, when the dual groups are equal and of rank one. If time permits, I will discuss how the transfer operator here and in a few examples of higher rank where it is known is a "deformation" of an abelian transfer operator obtained by replacing the spherical variety by its asymptotic cone (or boundary degeneration).

There will also be following special series of three lectures: