Invariant norms on the p-adic Schrödinger representation

Abstract: Motivated by questions about a p-adic Fourier transform, we study invariant norms on the p-adic Schrödinger representations of Heisenberg groups. These Heisenberg groups are p-adic, and the Schrodinger representations are explicit irreducible smooth representations that play an important role in their representation theory. Classically, the field of coefficients is taken to be the complex numbers and, among other things, one studies the unitary completions of the representations (which are well understood). By taking the field of coefficients to be an extension of the p-adic numbers, we can consider completions that better capture the p-adic topology, but at the cost of losing the Haar measure and the $L^2$-norm. Nevertheless, we establish a rigidity property for a family of norms (parametrized by a Grassmannian) that are invariant under the action of the Heisenberg group. The irreducibility of some Banach representations follows as a result. The proof uses "q-arithmetics".

number theory

Audience: researchers in the topic

Comments: pre-talk

UCSD number theory seminar

Series comments: Most talks are preceded by a pre-talk for graduate students and postdocs. The pre-talks start 40 minutes prior to the posted time (usually at 1:20pm Pacific) and last about 30 minutes.