Germ expansion for SL(2) in arbitrary characteristics
Representation Theory
2025-07-23 v2
Abstract
Let be a local field of characteristic and be a connected reductive group over . 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 . Observe that if this set is always compact; it is finite if while it is uncountable if . As a consequence, Shalika's germ expansion for elliptic elements does not make sense if . On the other hand the endoscopic expansion of elliptic orbital integrals always exists and yields a germ expansion equivalent if (up to a Fourier transform) to Shalika's germ expansion but is new if . A conjecture for arbitrary groups is stated.
Cite
@article{arxiv.2507.05003,
title = {Germ expansion for SL(2) in arbitrary characteristics},
author = {Jean-Pierre Labesse},
journal= {arXiv preprint arXiv:2507.05003},
year = {2025}
}