The Langlands-Shahidi method for pairs via types and covers

TL;DR

This paper computes the local coefficient for pairs of supercuspidal representations of GL(n) using types and covers, providing a new proof of Shahidi's formula and potential applications to classical groups.

Contribution

It introduces a novel approach employing types and covers to compute local coefficients, offering an alternative proof and extending the method's applicability.

Findings

New proof of Shahidi's Plancherel constant formula

Method adaptable to classical groups and other arithmetic contexts

Enhanced understanding of local coefficients for supercuspidal pairs

Abstract

We compute the local coefficient attached to a pair $(\pi_1,\pi_2)$ of supercuspidal (complex) representations of the general linear group using the theory of types and covers \`{a} la Bushnell-Kutzko. In the process, we obtain another proof of a well-known formula of Shahidi for the corresponding Plancherel constant. The approach taken here can be adapted to other situations of arithmetic interest within the context of the Langlands-Shahidi method, particularly, to that of a Siegel Levi subgroup inside a classical group.