Plancherel measures for coverings of p-adic SL(2,F)

TL;DR

This paper computes Plancherel measures for n-fold covers of p-adic SL(2,F), introduces a higher-dimensional metaplectic analog of Shahidi local coefficients, and proves an irreducibility theorem using new functional equations.

Contribution

It provides explicit formulas for Plancherel measures and local coefficients in the metaplectic setting, advancing understanding of covering groups over p-adic fields.

Findings

Computed Plancherel measures for coverings of p-adic SL(2,F)

Derived a metaplectic analog of Shahidi local coefficients

Proved an irreducibility criterion for principal series representations

Abstract

In these notes we compute the Plancherel measures associated with genuine principal series representations of n-fold covers of p-adic SL(2,F). Along the way we also compute a higher dimensional metaplectic analog of Shahidi local coefficients. Our method involves new functional equations utilizing the Tate Gamma-factor and a metaplectic counterpart. As an application we prove an irreducibility theorem.