Local metaplectic correspondence and applications

TL;DR

This paper revisits and generalizes the local metaplectic correspondence, providing new criteria and classifications for representations of Kazhdan-Patterson covering groups over p-adic fields, with applications to irreducibility and Whittaker models.

Contribution

It introduces a generalized local metaplectic correspondence and applies it to classify representations and determine irreducibility criteria for covering groups.

Findings

Full criterion for irreducibility of Bernstein-Zelevinsky products

Classification of square integrable and tempered representations

Calculation of Whittaker dimensions

Abstract

We revisit the local metaplectic correspondence previously constructed and studied by Flicker and Kazhdan. After restoring and generalizing some of their results, we get several interesting applications to the representation theory of a Kazhdan-Patterson covering group over a $p$-adic field, including a full criterion of the irreducibility of the Bernstein-Zelevinsky product of two cuspidal representations, the classification of essentially square integrable and tempered representations, and more interestingly, the calculation of the Whittaker dimension of an irreducible representation.