## 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

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

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 \(t\in T_{{\text {reg}}}\cap G_r\).

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*Dedicated to J. Bernstein on the occasion of his 70th birthday.*

This research is supported by ERC grant No. 247049-GLC.