On Shalika germs

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 $Ω\in P$ we fix an invariant measure $ω$ on $Ω$. As well known \cite {R} for any $f\in \mathcal{H}$ the integral $I_Ω(f)=\int_Ωfω$ is absolutely convergent. Shalika \cite{Sh} has shown that there exist functions $\tilde{j}_Ω(t),Ω\in P$ on $T\cap G_{reg}$ such that $I_t(f) = \sum_{Ω\in P} \tilde{j}_Ω(t) I_Ω(f)\qquad\qquad (\star)$ 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 $t\in T_{reg}\cap G_r$.

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