Affine Hecke algebras and their graded version. (English) Zbl 0715.22020

Lusztig, George

J. Am. Math. Soc. 2, No. 3, 599-635 (1989).

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 The center \({\mathcal Z}\) of \(H_{v_ 0}\) is determined, generalizing unpublished work of Bernstein from the special case.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)

Keywords:

representation theory of semisimple p-adic groups; simple modules; equivariant homology; intersection cohomology; affine Hecke algebra; Graded algebras; completions

Citations:

Zbl 0613.22004; Zbl 0699.22026

References:

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