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Tamely ramified intertwining algebras. (English) Zbl 0854.22022
Let \(\mathbf G\) be a (not necessarily split) reductive group defined over a local non archimedean field \(F\); set \(G = {\mathbf G}(F)\). Suppose that \(P\) is a parahoric subgroup of \(G\) with pro-unipotent radical \(U\). Then \(M = U \setminus P\) is the group of points of a reductive group defined over the residue field \(\mathbb{F}_q\) of \(F\). Suppose that \((\sigma, V)\) is an irreducible cuspidal representation of \(M\); we can view \(\sigma\) as a representation of \(P\). We define \({\mathcal H} = {\mathcal H} (\sigma)\) to be the space of compactly supported functions from \(G\) to \(V\) which transform on the left and right under \(P\) by \(\sigma\). Then \({\mathcal H}(\sigma)\) is an algebra under convolution which can be identified with the algebra of intertwining operators of the (smooth) representation \(c\)-\(\text{Ind}^G_P (\sigma)\); it is an important problem to give an explicit description of \(\mathcal H\) by generators and relations. For example, if \(B = P\) is an Iwahori subgroup of \(G\) and \(\sigma\) is the trivial representation, the structure of \(\mathcal H\) is well understood [N. Iwahori and H. Matsumoto, Inst. Haut. Étud. Sci., Publ. Math. 25, 5-48 (1965; Zbl 0228.20015)]; it is also well understood when \(G = \text{GL}_n\) and \(P\), \(\sigma\) are arbitrary. (For more complete results in this case we refer the reader to [C. J. Bushnell and P. C. Kutzko, The admissible dual of \(\text{GL}_n\) over a local field via compact open subgroups. Ann. Math. Studies 129. (Princeton, NJ 1993; Zbl 0787.22016)].) Other cases that have been treated include \(B\) as above and \(\sigma\) non trivial [D. Goldstein, Thesis, Univ. Chicago 1990], and some groups of rank of at most 2 (see A. Moy [J. Reine Angew. Math. 372, 178-208 (1986; Zbl 0589.22015); Compos. Math. 66, 237-284 (1988; Zbl 0662.22012)]).
In this paper, we prove that in general \(\mathcal H\) has a structure that is similar to that for the Hecke algebra of an Iwahori subgroup with the trivial representation. This is analogous to the situation one encounters when one replaces \(G\) by a finite group of Lie type and \(P\) by a parabolic subgroup. Indeed, our results have been directly inspired by the paper of R. Howlett and G. Lehrer [Invent. Math. 58, 37-64 (1980; Zbl 0435.20023)], and the final proofs (Sect. 7) are the same once one has established an appropriate framework. We remark that this takes effort, and in particular one must overcome some combinatorial problems involving parahoric subgroups.

22E50 Representations of Lie and linear algebraic groups over local fields
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