Saved in:
| Main Author: | |
|---|---|
| Format: | Preprint |
| Published: |
2025
|
| Subjects: | |
| Online Access: | https://arxiv.org/abs/2507.05003 |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866912495940141056 |
|---|---|
| author | Labesse, Jean-Pierre |
| author_facet | Labesse, Jean-Pierre |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2507_05003 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Germ expansion for SL(2) in arbitrary characteristics Labesse, Jean-Pierre Representation Theory 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. |
| title | Germ expansion for SL(2) in arbitrary characteristics |
| topic | Representation Theory |
| url | https://arxiv.org/abs/2507.05003 |