On Shalika germs

TL;DR

This paper proves that Shalika's germ expansion formula for orbital integrals holds uniformly on certain subsets of a reductive group over a local field, extending its validity beyond a neighborhood of the identity.

Contribution

The paper extends the validity of Shalika's germ expansion formula to larger subsets of the group, specifically for functions in \\mathcal{H}_r, not just near the identity.

Findings

Shalika's germ expansion formula holds on larger subsets for functions in \\mathcal{H}_r

The extension applies to all t in T_{reg} \\cap G_r

The result broadens understanding of orbital integrals in harmonic analysis

Abstract

Let $G$ be a reductive group over a local field $F$ satisfying the assumptions of \cite{Deb1}, $G_{reg}\subset G$ the subset of regular elements. Let $T\subset G$ be a maximal torus. We write $T_{reg}=T\cap G_{reg}$. Let $dg ,dt$ be Haar measures on $G$ and $T$. They define an invariant measure $dg/dt$ on $G/T$. Let $\mathcal{H}$ be the space of complex valued locally constant functions on $G$ with compact support. For any $f\in \mathcal{H} ,t\in T_{reg}$ we define $I_t(f)=\int_{G/T}f(\bar gt\bar g^{-1})dg/dt$. Let $P$ be the set of conjugacy classes of unipotent elements in $G$. For any $\Omega \in P$ we fix an invariant measure $\omega$ on $\Omega$. As well known \cite {R} for any $f\in \mathcal{H}$ the integral $$I_\Omega (f)=\int_\Omega f\omega$$ is absolutely convergent. Shalika \cite{Sh} has shown that there exist functions $\tilde{j}_\Omega (t),\Omega \in P$ on $T\cap G_{reg}$ such that $$I_t(f) = \sum_{\Omega \in P} \tilde{j}_\Omega(t) I_\Omega(f)\qquad\qquad (\star)$$ for any $f\in \mathcal{H} ,t\in T$ {\it near} to $e$ where the notion of {\it near} depends on $f$. For any positive real number $r$ one defines an open $Ad$-invariant subset $G_r$ of $G$ and a subspace $\mathcal{H}_r$ as in \cite{Deb1}. In this paper I show that for any $f\in \mathcal{H}_r$ the equality $(\star)$ is true for all $t\in T_{reg}\cap G_r$.