Universal Weil module

Justin Trias

Vol. 323 (2023), No. 2, 359–399

DOI: 10.2140/pjm.2023.323.359

Abstract

The classical construction of the Weil representation, with complex coefficients, has long been expected to work for more general coefficient rings. This paper exhibits the minimal ring $\mathcal{𝒜}$ for which this is possible, the integral closure of $\mathbb{Z}\left[\frac{1}{p}\right]$ in a cyclotomic field, and carries out the construction of the Weil representation over $\mathcal{𝒜}$-algebras. As a leitmotif all along the work, most of the problems can actually be solved over the base ring $\mathcal{𝒜}$ and transferred to any $\mathcal{𝒜}$-algebra by scalar extension. The most striking fact is that all these Weil representations arise as the scalar extension of a single one with coefficients in $\mathcal{𝒜}$. In this sense, the Weil module obtained is universal. Building upon this construction, we speculate and make predictions about an integral theta correspondence.

Keywords

representation theory, reductive $p$-adic groups, integral representation, theta correspondence

Mathematical Subject Classification

Primary: 11F27, 11F70, 11S23, 20C20

Milestones

Received: 14 July 2022

Revised: 2 February 2023

Accepted: 26 February 2023

Published: 2 June 2023