On Shalika germs

Abstract

Let G be (the group of F-points of) a reductive group over a local field F satisfying the assumptions of Debacker (Ann Sci Ecole Norm Sup 35(4):391–422, 2002), sections 2.2, 3.2, 4.3. Let $G_{{\text {reg}}}\subset G$ be the subset of regular elements. Let $T\subset G$ be a maximal torus. We write $T_{{\text {reg}}}=T\cap G_{{\text {reg}}}$. Let dg, dt be Haar measures on G and T. They define an invariant measure on . Let $\mathcal {H}$ be the space of complex valued locally constant functions on G with compact support. For any $f\in \mathcal {H}$, $t\in T_{{\text {reg}}}$, we put $I_t(f)=\int _{G/T}f(\dot{g}t\dot{g}^{-1})dg/dt$. Let $\mathcal U$ be the set of conjugacy classes of unipotent elements in G. For any $\Omega \in \mathcal U$ we fix an invariant measure $\omega$ on $\Omega$. It is well known—see, e.g., Rao (Ann Math 96:505-510, 1972)—that for any $f\in \mathcal {H}$ the integral

$$\begin{aligned} I_\Omega (f)=\int _\Omega f\omega \end{aligned}$$

is absolutely convergent. Shalika (Ann Math 95:226–242, 1972) showed that there exist functions $j_\Omega (t)$, $\Omega \in \mathcal U$, on $T\cap G_{{\text {reg}}}$, such that

$$\begin{aligned} I_t(f)=\sum _{\Omega \in \mathcal U}j_\Omega (t)I_\Omega (f) \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \quad \qquad \qquad \qquad \qquad \qquad ({\star }) \end{aligned}$$

for any $f\in \mathcal {H}$, $t\in T$ near to e, where the notion of near depends on f. For any $r\ge 0$ we define an open ${\text {Ad}}(G)$-invariant subset $G_r$ of G, and a subspace $\mathcal {H}_r$ of $\mathcal {H}$, as in Debacker (Ann Sci Ecole Norm Sup 35(4):391–422, 2002). Here I show that for any $f\in \mathcal {H}_r$ the equality $(\star )$ holds for all $t\in T_{{\text {reg}}}\cap G_r$.