English

Germ expansion for SL(2) in arbitrary characteristics

Representation Theory 2025-07-23 v2

Abstract

Let FF be a local field of characteristic pp and GG be a connected reductive group over FF. 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)G(F). Observe that if G=SL(2)G=SL(2) this set is always compact; it is finite if p2p\ne2 while it is uncountable if p=2p= 2. As a consequence, Shalika's germ expansion for elliptic elements does not make sense if p=2p=2. On the other hand the endoscopic expansion of elliptic orbital integrals always exists and yields a germ expansion equivalent if p2p\ne2 (up to a Fourier transform) to Shalika's germ expansion but is new if p=2p=2. A conjecture for arbitrary groups is stated.

Keywords

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}
}
R2 v1 2026-07-01T03:49:30.468Z