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The representation theory of p-adic GL(n) and Deligne-Langlands parameters: Chapter 4

The representation theory of p-adic GL(n) and Deligne-Langlands parameters: Chapter 4
The representation theory of p-adic GL(n) and Deligne-Langlands parameters: Chapter 4
In this article we cover an episode in the representation theory of GL(n) defined over a p-adic field with finite residue class field. We concentrate on the irreducible tempered representations admitting non-zero Iwahori-fixed vectors. We describe the space of these representations in terms of Deligne-Langlands parameters. In [6], Kazdhan and Lusztig prove the Deligne-Langlands conjecture for split reductive p-adic groups with connected centre. For GL(n), this conjecture amounts to a parametrization of such representations by certain pairs (s, N) satisfying the equation sNs−1 = qN where q is the cardinality of the residue field. We discuss these parameters in §3. In §4 and §5 we discuss the theory of Zelevinsky segments and prove results concerning the form of irreducible representations of GL(n) admitting non-zero Iwahori-fixed vectors. In the final section we define the Brylinski quotient Bryl(n) for the space T n equipped with the natural action of the symmetric group S n and prove that the space of Deligne-Langlands parameters of these representations is homeomorphic to Bryl(n).
2366-8717
54-72
Hindustan Book Agency
Hodgins, J. E.
9e137af2-db2e-4ad9-bc6d-264a727d25f4
Plymen, R. J.
76de3dd0-ddcb-4a34-98e1-257dddb731f5
Bhatia, R.
Hodgins, J. E.
9e137af2-db2e-4ad9-bc6d-264a727d25f4
Plymen, R. J.
76de3dd0-ddcb-4a34-98e1-257dddb731f5
Bhatia, R.

Hodgins, J. E. and Plymen, R. J. (1996) The representation theory of p-adic GL(n) and Deligne-Langlands parameters: Chapter 4. In, Bhatia, R. (ed.) Texts and Readings in Mathematics: Analysis, Geometry and Probability. (Analysis, Geometry and Probability, , (doi:10.1007/978-93-80250-87-8_4), 10) Hindustan Book Agency, pp. 54-72. (doi:10.1007/978-93-80250-87-8_4).

Record type: Book Section

Abstract

In this article we cover an episode in the representation theory of GL(n) defined over a p-adic field with finite residue class field. We concentrate on the irreducible tempered representations admitting non-zero Iwahori-fixed vectors. We describe the space of these representations in terms of Deligne-Langlands parameters. In [6], Kazdhan and Lusztig prove the Deligne-Langlands conjecture for split reductive p-adic groups with connected centre. For GL(n), this conjecture amounts to a parametrization of such representations by certain pairs (s, N) satisfying the equation sNs−1 = qN where q is the cardinality of the residue field. We discuss these parameters in §3. In §4 and §5 we discuss the theory of Zelevinsky segments and prove results concerning the form of irreducible representations of GL(n) admitting non-zero Iwahori-fixed vectors. In the final section we define the Brylinski quotient Bryl(n) for the space T n equipped with the natural action of the symmetric group S n and prove that the space of Deligne-Langlands parameters of these representations is homeomorphic to Bryl(n).

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Published date: 1996

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Local EPrints ID: 421993
URI: https://eprints.soton.ac.uk/id/eprint/421993
ISSN: 2366-8717
PURE UUID: da04ae97-90e0-452e-9819-ea7094114ab8

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Date deposited: 12 Jul 2018 16:30
Last modified: 13 Mar 2019 18:20

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