On Shahidi local coefficients matrix

TL;DR

This paper introduces the Shahidi local coefficients matrix for genuine principal series representations of n-fold covers of p-adic SL(2,F), relating its entries to gamma-factors and providing new formulas for the Plancherel measure.

Contribution

It defines and studies the Shahidi local coefficients matrix for covering groups, connecting its entries to gamma-factors and deriving formulas for the Plancherel measure.

Findings

The Shahidi local coefficients matrix is an invariant of the inducing data.

Entries of the matrix relate to Tate-type gamma-factors.

New formulas for the Plancherel measure are derived.

Abstract

In these notes we define and study the Shahidi local coefficients matrix associated with a genuine principal series representation I({\sigma}) of an n-fold cover of p-adic SL(2,F) and an additive character {\psi}. The conjugacy class of this matrix is an invariant of the inducing representation {\sigma} and {\psi} and its entries are linear combinations of Tate or Tate type {\gamma}-factors. We relate these entries to functional equations associated with linear maps defined on the dual of the space of Schwartz functions. As an application we give new formulas for the Plancherel measure and use these to relate principal series representations of different coverings of SL(2,F). While we do not assume that the residual characteristic of F is relatively prime to n we do assume that n is not divisible by 4.