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On Shalika germs. (English) Zbl 1354.22014
Summary: Let $$G$$ be (the group of $$F$$-points of) a reductive group over a local field $$F$$ satisfying the assumptions of S. DeBacker [Ann. Sci. Éc. Norm. Supér. (4) 35, No. 3, 391–422 (2002; Zbl 0999.22013), Sections 2.2, 3.2, 4.3]. Let $$G_{\mathrm{reg}}\subset G$$ be the subset of regular elements. Let $$T\subset G$$ be a maximal torus. We write $$T_{\mathrm{reg}}=T\cap G_{\mathrm{reg}}$$. Let $$d_g$$, $$d_t$$ 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 $$f\in \mathcal {H}$$, $$t\in T_{\mathrm{reg}}$$, 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., [R. Ranga Rao, Ann. Math. (2) 96, 505–510 (1972; Zbl 0302.43002)] – that for any $$f\in \mathcal {H}$$ the integral $I_\Omega (f)=\int_\Omega f\omega$ is absolutely convergent. J. A. Shalika [Ann. Math. (2) 95, 226–242 (1972; Zbl 0281.22011)] showed that there exist functions $$j_\Omega (t)$$, $$\Omega \in \mathcal U$$, on $$T\cap G_{\mathrm{reg}}$$, such that $I_t(f)=\sum_{\Omega \in \mathcal U}j_\Omega (t)I_\Omega (f) \eqno{(\star)}$ for any $$f\in \mathcal {H}$$, $$t\in T$$ near to $$e$$, where the notion of near depends on $$f$$. For any $$r\geq 0$$ we define an open $$\mathrm{Ad}(G)$$-invariant subset $$G_r$$ of $$G$$, and a subspace $$\mathcal {H}_r$$ of $$\mathcal {H}$$, as in [Debacker, loc. cit.]. Here I show that for any $$f\in \mathcal {H}_r$$ the equality $$(\star)$$ holds for all $$t\in T_{\mathrm{reg}}\cap G_r$$.

##### MSC:
 2.2e+36 Analysis on $$p$$-adic Lie groups 2.2e+51 Representations of Lie and linear algebraic groups over local fields
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##### References:
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