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Affine Hecke algebras and their graded version. (English) Zbl 0715.22020
Affine Hecke algebras $$H_{v_ 0}$$ arise naturally in the representation theory of semisimple p-adic groups. Representations of $$H_{v_ 0}$$ correspond to representations with Iwahori-fixed vectors, which are important to understand for applications to number theory.
A classification of simple modules for $$H_{v_ 0}$$ is obtained by D. Kazhdan, G. Lusztig [Invent. Math. 87, 153-215 (1987; Zbl 0613.22004)] in the special case that the c(s) in the defining relations $$(T_ s+1)(T_ s-v_ 0^{2c(s)})=0$$ are independent of s. The approach, based on equivariant K-theory, is not suited to the general case.
The author [Publ. Math., Inst. Hautes Étud. Sci. 67, 145-202 (1988; Zbl 0699.22026)], defined a graded analogue $$\bar H_{r_ 0}$$ of $$H_{v_ 0}$$, for which the representation theory in the general case may be studied using equivariant homology and intersection cohomology. In the present paper, the author connects the representation theory of the $$H_{v_ 0}$$ to that of $$\bar H_{r_ 0}$$. The center $${\mathcal Z}$$ of $$H_{v_ 0}$$ is determined, generalizing unpublished work of Bernstein from the special case. A simple $$H_{v_ 0}$$-module M determines a unique maximal ideal T of $${\mathcal Z}$$ such that $$TM=0$$. The completion $$\hat {\mathcal Z}$$ of the center is taken with respect to the T-adic topology. Then the completion of the affine Hecke algebra is defined by $$\hat H_{v_ 0}=H_{v_ 0}\otimes_{{\mathcal Z}}\hat {\mathcal Z}.$$
Graded algebras $$\bar H_{r_ 0}$$ are defined via powers of a maximal ideal of a certain commutative subalgebra $${\mathcal O}$$ of $$H_{v_ 0}$$. The centers are determined and again completions are defined. The first reduction theorem is that the completion of an affine Hecke algebra with respect to a maximal ideal of the center is isomorphic to the ring of $$n\times n$$ matrices over the completion of a smaller affine Hecke algebra. The second reduction is that a certain natural homomorphism from an affine Hecke algebra into a suitable completion of its graded version becomes an isomorphism when the first algebra is completed. The classification of simple $$H_{v_ 0}$$-modules is then reduced essentially to the classification of simple $$\bar H_{r_ 0}$$-modules, in the case the parameter $$v_ 0\in {\mathbb{C}}^ x$$ has infinite order.
Reviewer: C.D.Keys

##### MSC:
 22E50 Representations of Lie and linear algebraic groups over local fields 22E35 Analysis on $$p$$-adic Lie groups 16W50 Graded rings and modules (associative rings and algebras) 20G05 Representation theory for linear algebraic groups 14G20 Local ground fields in algebraic geometry 11F85 $$p$$-adic theory, local fields 14F43 Other algebro-geometric (co)homologies (e.g., intersection, equivariant, Lawson, Deligne (co)homologies)
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##### References:
 [1] David Kazhdan and George Lusztig, Proof of the Deligne-Langlands conjecture for Hecke algebras, Invent. Math. 87 (1987), no. 1, 153 – 215. · Zbl 0613.22004 [2] Robert W. Kilmoyer, Principal series representations of finite Chevalley groups, J. Algebra 51 (1978), no. 1, 300 – 319. · Zbl 0389.20008 [3] George Lusztig, Singularities, character formulas, and a \?-analog of weight multiplicities, Analysis and topology on singular spaces, II, III (Luminy, 1981) Astérisque, vol. 101, Soc. Math. France, Paris, 1983, pp. 208 – 229. · Zbl 0561.22013 [4] George Lusztig, Some examples of square integrable representations of semisimple \?-adic groups, Trans. Amer. Math. Soc. 277 (1983), no. 2, 623 – 653. · Zbl 0526.22015 [5] George Lusztig, Cuspidal local systems and graded Hecke algebras. I, Inst. Hautes Études Sci. Publ. Math. 67 (1988), 145 – 202. · Zbl 0699.22026 [6] I. G. Macdonald, Spherical functions on a group of \?-adic type, Ramanujan Institute, Centre for Advanced Study in Mathematics,University of Madras, Madras, 1971. Publications of the Ramanujan Institute, No. 2. · Zbl 0302.43018
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