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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:
22E35 Analysis on \(p\)-adic Lie groups
22E50 Representations of Lie and linear algebraic groups over local fields
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