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On modules over the Hecke algebra of a p-adic group. (English) Zbl 0579.20037
Let I be an Iwahori subgroup of a group G of F-rational points of a connected, reductive algebraic group over a p-adic field F. It is known, by a theorem due to Bernstein, Borel, and Matsumoto, that the category of admissible representations of G that are generated by their spaces of I- fixed vectors is equivalent to the category of finite-dimensional modules over the Hecke algebra \({\mathcal H}\) of G with respect to I by associating to each such admissible representation its space of I-fixed vectors. To study \({\mathcal H}\)-modules for semi-simple groups of adjoint type, especially for the group \(GL_ n\) in detail, is the purpose of this paper. Zelevinsky gives a classification of all irreducible admissible representations of the group \(GL_ n\) modulo supercuspidal representations of Levi factors. In the present paper, an equivalent classification to it is given using the Kazhdan-Lusztig approach and a classification of standard modules of the group \(GL_ n\) is described. To the reviewer, it seems that this approach is very powerful for the groups of type \(A_ n\) and for other groups it is not sure that it goes well or not.
Reviewer: Y.Asoo

MSC:
20G05 Representation theory for linear algebraic groups
22E50 Representations of Lie and linear algebraic groups over local fields
20G25 Linear algebraic groups over local fields and their integers
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References:
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