Germ expansion for SL(2) in arbitrary characteristics

TL;DR

This paper investigates the behavior of Shalika's germ expansion for orbital integrals in the group SL(2) over local fields of characteristic p, highlighting differences between characteristic 2 and other primes, and proposes a conjecture for general groups.

Contribution

It provides a detailed analysis of germ expansions in SL(2) across different characteristics and introduces a new endoscopic expansion approach for characteristic 2.

Findings

Shalika's germ expansion is finite for p≠2 but uncountable for p=2.

Endoscopic expansion exists for elliptic elements in all characteristics.

A new conjecture is proposed for germ expansions in arbitrary groups.

Abstract

Let $F$ be a local field of characteristic $p$ and $G$ be a connected reductive group over $F$. Recall that Shalika's germ expansion of orbital integrals of regular semi-simple elements near the identity, when it exists, is a sum indexed by the set of unipotent conjugacy classes in $G(F)$. Observe that if $G=SL(2)$ this set is always compact; it is finite if $p\ne2$ while it is uncountable if $p= 2$. As a consequence, Shalika's germ expansion for elliptic elements does not make sense if $p=2$. On the other hand the endoscopic expansion of elliptic orbital integrals always exists and yields a germ expansion equivalent if $p\ne2$ (up to a Fourier transform) to Shalika's germ expansion but is new if $p=2$. A conjecture for arbitrary groups is stated.