## Arithmetic Geometry and Number Theory RTG Seminar: Towards a $p$-adic Deligne–Lusztig theory

Seminar | November 13 | 3:10-5 p.m. | 891 Evans Hall

Charlotte Chan, University of Michigan

Department of Mathematics

The cuspidal representations of $GL_2(\mathbf F_q)$ can be realized in the cohomology of the Drinfeld curve. This is an example of Deligne–Lusztig theory, which gives a geometric construction of the representations of finite reductive groups. In 1979, Lusztig proposed a conjectural analogue of this story for p-adic groups. We verify this conjecture in the setting of division algebras and show that this gives a geometric construction of automorphic induction.

There is a closely related unipotent analogue of Deligne–Lusztig theory, whose study was initiated by Boyarchenko in 2012. We prove Boyarchenko's conjectures: torus-eigenspaces of the cohomology are concentrated in a single degree and the relevant varieties are maximal in the sense that the number of rational points attains its Weil–Deligne bound.

Seminar Format: The seminar consists of two 50-minute talks, a pre-talk (3:10-4:00) and an advanced talk (4:10-5:00), with a 10-minute break (4:00-4:10) between them. The advanced talk is a regular formal presentation about recent research results to general audiences in arithmetic geometry and number theory; the pre-talk (3:10-4:00) is to introduce some prerequisites or background for the advanced talk to audiences consisting of graduate students.

yxy@berkeley.edu