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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.

##### MSC:
 2.2e+51 Representations of Lie and linear algebraic groups over local fields
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##### References:
  Bernstein, J.-N. (rédigé par Deligne, P.): Le ?centre? de Bernstein. Dans: Bernstein, J. et al. (eds.) Représentations des groupes réductifs sur un corps local. (Trav. Cours) Paris: Hermann 1984  Bernstein, J.-N., Zelevinskii, A.: Representations of the group GL (n, F) whereF is a non archimedean field. Usp. Mat. Nauk.31(3), 5-70 (1976); Lond. Math. Soc. Surv.31, 1-68 (1976)  Blondel, C.: Thèse du troisième cycle. Univ. Paris VII (1981)  Borel, A., de Siebenthal, J.: Les sous groupes fermés de rang maximum des groupes de Lie clos. Comment. Math. Helv.23, 200-221 (1949) · Zbl 0034.30701  Bourbaki, N.: Groupes et algèbres de Lie IV, V, VI. Paris: Hermann 1968  Bourbaki, N.: Groupes et algèbres de Lie VII, VIII. Paris: Masson 1990  Bruhat, F., Tits, J.: Groupes réductifs sur un corps local. I. Données radicielles valuées. Publ. Math., Inst. Hautes Étud. Sci.41, 5-251 (1972) · Zbl 0254.14017  Bruhat, F., Tits, J.: Groupes réductifs sur un corps local. II. Schémas en groupes. Existence d’une donnée radicielle valuée. Publ. Math., Inst. Hantes Étud. Sci.60, 5-184 (1984) · Zbl 0597.14041  Bushnell, C.J., Kutzko, P.C.: The admissible dual of GL n over a local field via compact open subgroups. Ann. Math. Studies no 129, Princeton NJ, 1993 · Zbl 0787.22016  Carter, R.: Finite groups of Lie type: conjugacy classes and complex characters. Chichester: Wiley Interscience 1985 · Zbl 0567.20023  Cartier, P.: Representations of p-adic groups. In: Borel, A., Casselman, W. (eds.) Automorphic forms, representations and L-functions. (Proc. Symp. Pure Math., vol. 33, part 1, pp. 111-155) Providence, RI: Am. Math. Soc. 1979  Casselman, W.: An assortment of results on representations of GL2(k). In: Deligne, P., Kuyk, W. (eds.) Modular functions of one variable II. (Lect. Notes Math., vol. 349) Berlin Heidelberg New York: Springer 1973 · Zbl 0301.22011  Dixmier, J.:C *-algebras. Amsterdam: North Holland, 1977 · Zbl 0372.46058  Flath, D.: Decomposition of representations into tensor products. In: Borel, A., Casselman, W. (eds.) Automorphic forms, representations and L-functions. (Proc. Symp. Pure Math., vol. 33, part 1, pp. 179-183) Providence, RI: Am. Math. Soc. 1979 · Zbl 0414.22019  Goldstein, D.: Thesis. University of Chicago (1990)  Howe, R.: (with the collaboration of Moy, A.), Harish-Chandra homomorphisms for p-adic groups. (CBMS Reg. Conf. Ser., vol. 59) Providence, RI: Am. Math. Soc. 1985 · Zbl 0593.22014  Howlett, R.: Normalizers of parabolic subgroups of reflection subgroups. J. Lond. Math. Soc.21, 62-80 (1980) · Zbl 0427.20040  Howlett, R., Lehrer, G.: Induced cuspidal representations and generalised Hecke rings. Invent. Math.58, 37-64 (1980) · Zbl 0435.20023  Iwahori, N., Matsumoto, H.: On some Bruhat decomposition and the structure of the Hecke ring ofp-adic Chevalley groups. Publ. Math., Inst. Hautes Étud. Sci.25, 5-48 (1965) · Zbl 0228.20015  Jacquet, H.: Sur les représentations des groupes réductifsp-adiques. C.R. Acad. Sci., Paris, Sér. A280, 1271-1272 (1975) · Zbl 0309.22012  Jacquet, H., Langlands, R.: Automorphic forms on GL(2) (Lect. Notes Math., vol. 260) Berlin Heidelberg New York: Springer 1972 · Zbl 0236.12010  Kutzko, P.: Mackey’s theorem for non unitary representations. Proc. Am. Math. Soc.64, 173-175 (1977) · Zbl 0375.22005  Kutzko, P.: On the restriction of supercuspidal representations to compact, open subgroups. Duke Math. J.52, 753-764 (1985) · Zbl 0604.22010  Lusztig, G.: Some examples of square integrable representations of semisimplep-adic groups. Trans. Am. Math. Soc.277, 623-653 (1983) · Zbl 0526.22015  Lusztig, G.: Characters of reductive groups over a finite field. (Ann. Math. Stud., vol. 107) Princeton: Princeton University Press 1984 · Zbl 0556.20033  Lusztig, G.: Representations of affine Hecke algebras. Dans: Orbites unipotentes et représentations II. Groupesp-adiques et réels. (Astérisque, vols. 171-172) Paris: Soc. Math. Fr. 1989  Lusztig, G.: Intersection cohomology methods in representation theory. In: Proceedings of the International Congress of Mathematicians. Kyoto, 1990 (to appear) · Zbl 0749.14010  Macdonald, I.G.: Affine root systems and Dedekind’s ?-function. Invent. Math.15, 91-143 (1972) · Zbl 0244.17005  Macdonald, I.G.: Spherical functions on a group ofp-adic type. Ramanujan Institute Lecture Notes, University of Madras (1971)  Morris, L.:P-cuspidal representations of level one. Proc. Lond. Math. Soc., III. Ser.58, 550-558 (1989) · Zbl 0678.22010  Moy, A.: Representations of U(2,1) over ap-adic field. J. Reine Angew. Math.372, 178-208 (1986) · Zbl 0589.22015  Moy, A.: Representations of GSp4 over ap-adic field. I. Compos. Math.66, 237-284 (1988) · Zbl 0662.22012  Springer, T.A.: Cusp forms for finite groups. In: Borel, A. et al. (eds.) Seminar on algebraic groups. (Lect. Notes Math., vol. 131) Berlin Heidelberg New York: Springer 1970 · Zbl 0263.20024  Tits, J.: Reductive groups over local fields. In: Borel, A., Casselman, W. (eds.) Automorphic forms, representations and L-functions. (Proc. Symp. Pure Math., vol. 33, part 1, pp. 29-69) Providence, RI: Am. Math. Soc. 1979  Tits, J.: Le problème des mots dans les groupes de Coxeter. Symp. Math.1, 175-185 (1968)
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