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 Open image in new window on Open image in new window. 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 \(t\in T_{{\text {reg}}}\cap G_r\).