On Shalika germs

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_reg ⊂ G be the subset of regular elements. Let T ⊂ G be a maximal torus. We write T_reg = T ∩ G_reg. Let dg, dt be Haar measures on G and T . They define an invariant measure dg/dt on G/T. Let H be the space of complex valued locally constant functions on G with compact support. For any f ∈ H, t ∈ T_reg, we put I_t (f) = ∫_G/T f(ġtġ^-1)dg/dt. Let U be the set of conjugacy classes of unipotent elements in G. For any Ω ∈ U we fix an invariant measure ω on Ω. It is well known-see, e.g., Rao (Ann Math 96:505-510, 1972)—that for any f ∈ H the integral (Formula presented.) is absolutely convergent. Shalika (Ann Math 95:226–242, 1972) showed that there exist functions j_Ω(t), Ω ∈ U, on T ∩ G_reg, such that (Formula presented.) for any f ∈ H, t ∈ T near to e, where the notion of near depends on f . For any r ≥ 0 we define an open Ad(G)-invariant subset G_r of G, and a subspace H_r of H, as in Debacker (Ann Sci Ecole Norm Sup 35(4):391–422, 2002). Here I show that for any f ∈ H_r the equality (★) holds for all t ∈ T_reg ∩ G_r.