Local coefficients and Gelfand-Graev representations for non-split covers on SL(2)

TL;DR

This paper extends local coefficient computation methods to non-split covers of SL(2) and explores the algebraic structure of Gelfand-Graev representations over non-archimedean fields, broadening previous results.

Contribution

It generalizes the local coefficient computation approach to non-split cases and analyzes Gelfand-Graev representations without characteristic restrictions.

Findings

Extended local coefficient computation to non-split covers of SL(2).

Analyzed algebraic structure of Gelfand-Graev representations in broader settings.

Broadened applicability of previous results to all characteristics.

Abstract

A purely local approach has been developed by Krishnamurthy and Kutzko to compute Langlands-Shahidi local coefficient for ${\rm SL}(2)$ via types and covers \`{a} la Bushnell-Kutzko. In this paper, we extend their method to the non-split case and complete their project. We also study the algebraic structure of Gelfand-Graev representations, which generalizes the results of Chan-Savin and Mishra-Pattanayak to ${\rm SL}(2)$ over non-archimidean local fields without any restriction on the characteristic.