L-packets and formal degrees for SL_2(K) with K a local function field of characteristic 2

TL;DR

This paper studies the representation theory of SL_2 over a local function field of characteristic 2, using Artin-Schreier theory to parametrize L-packets and compute formal degrees, connecting to geometric conjectures.

Contribution

It introduces a parametrization of L-packets for SL_2(K) in characteristic 2 using Artin-Schreier theory and relates it to a geometric conjecture, including explicit formal degree calculations.

Findings

L-packets in principal series parametrized by quadratic extensions

Supercuspidal L-packets parametrized by biquadratic extensions

Formal degrees computed for supercuspidal packets

Abstract

Let G = SL_2(K) with K a local function field of characteristic 2. We review Artin-Schreier theory for the field K, and show that this leads to a parametrization of L-packets in the smooth dual of G. We relate this to a recent geometric conjecture. The L-packets in the principal series are parametrized by quadratic extensions, and the supercuspidal L-packets by biquadratic extensions. We compute the formal degrees of the elements in the supercuspidal packets.