zbMATH — the first resource for mathematics

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

22E35 Analysis on \(p\)-adic Lie groups
22E50 Representations of Lie and linear algebraic groups over local fields
Full Text: DOI arXiv
[1] Bezrukavnikov, R., Kazhdan, D., Varshavsky, Y.: On the depth \(r\) Bernstein projector. arXiv:1504.01353 · Zbl 1370.22013
[2] Debacker, S, Homogeneity results for invariant distributions of a reductive p-adic group, Ann. Sci. Ecole Norm. Sup., 35, 391-422, (2002) · Zbl 0999.22013
[3] Debacker, S, Some applications of Bruhat-Tits theory to harmonic analysis on a reductive p-adic group, Michigan Math. J., 50, 241-261, (2002) · Zbl 1018.22014
[4] Harish-Chandra: Admissible invariant distributions on reductive \(p\)-adic groups. Lie theories and their applications (Proc. Ann. Sem. Canad. Math. Congr., Queen’s Univ., Kingston, Ont., 1977), pp. 281-347. Queen’s Papers in Pure Appl. Math. 48 · Zbl 0860.22006
[5] Moy, A; Prasad, G, Jacquet functors and unrefined minimal K-types, Comment. Math. Helv., 71, 98-121, (1996) · Zbl 0860.22006
[6] Rao, R, Orbital integrals in reductive groups, Ann. Math., 96, 505-510, (1972) · Zbl 0302.43002
[7] Shalika, J, A theorem on semi-simple \({\cal P}\)-adic groups, Ann. Math., 95, 226-242, (1972) · Zbl 0281.22011
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.