Iwahori-Hecke algebras and their representation theory: lectures given at the C.I.M.E. summer school held in Martina Franca, Italy, June 28-July 6, 1999

Ivan Cherednik, Yavor Markov, Roger Howe, George Lusztig, Dan Barbasch, M. Welleda Baldoni3540002243, 9783540002246

Two basic problems of representation theory are to classify irreducible representations and decompose representations occuring naturally in some other context. Algebras of Iwahori-Hecke type are one of the tools and were, probably, first considered in the context of representation theory of finite groups of Lie type. This volume consists of notes of the courses on Iwahori-Hecke algebras and their representation theory, given during the CIME summer school which took place in 1999 in Martina Franca, Italy.

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Table of contents :
vga27aakld1557k2.pdf……Page 0
Table of Contents……Page 9
Hankel transform via double Hecke algebra……Page 10
1 L-operator……Page 11
2 Hankel transform……Page 12
3 Dunkl operator……Page 14
4 Nonsymmetric eigenfunctions……Page 16
5 Master formula……Page 18
6 Double H double prime……Page 20
7 Algebraization……Page 23
8 Inverse transform and Plancherel formula……Page 25
9 Finite-dimensional case……Page 27
10 Truncated Bessel functions……Page 30
References……Page 33
1 Introduction……Page 35
2 Structure of $p$-adic $mathrm { GL } (V )$ and $mathrm { Sp } (V )$……Page 37
2.1 Preliminaries……Page 38
2.2 Bruhat decomposition of $mathrm { GL } (V )$……Page 39
2.3 Iwahori-Bruhat decomposition of $mathrm { GL } (V )$……Page 41
2.4 Bruhat decomposition of $mathrm { Sp } (V )$……Page 46
2.5 Iwahori-Bruhat decomposition of $mathrm { Sp } (V ) $……Page 49
3.1 $mathcal { H } (G//K)$……Page 54
3.2 $mathcal { H } (G//J)$……Page 55
4.1 Fundamental techniques……Page 59
4.3 Category equivalence……Page 61
5 Spherical Function Algebras……Page 62
5.1 Structure……Page 63
5.2 Lie lattices and characters……Page 64
5.3 Harish-Chandra homomorphism for $mathrm { SL } _2$……Page 66
5.4 General minimal principal series……Page 70
6 Consequences……Page 73
References……Page 75
1 The affine Hecke algebra……Page 78
2 $mathcal { H }$ and equivariant K-theory……Page 80
3 Convolution……Page 84
4 Subregular case……Page 86
5 Subregular case: type $A$……Page 90
6 Subregular case: types $C, D,E, F,G$……Page 104
References……Page 110