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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

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
Full Text: DOI EuDML
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