Representations of affine Hecke algebras. (English) Zbl 0699.22027

Orbites unipotentes et représentations. II: Groupes p-adiques et réels, Astérisque 171-172, 73-84 (1989).
[For the entire collection see Zbl 0694.00012.]
Let G denote a reductive group over a local field k. According to the Langlands conjectures the irreducible representations of G(k) should correspond to certain representations of the Weil-Deligne group of k. Let \(I\subset G(k)\) denote the Iwahori subgroup. For representations of G(k) admitting nonzero fixed vectors under I this correspondence was established in [D. Kazhdan, G. Lusztig, Invent. Math. 87, 153-215 (1987; Zbl 0613.22004)] using equivariant K-homology.
In the paper the correspondence is given in terms of the K-theory of coherent sheaves. This simplifies the formulation but the proofs still require equivariant K-homology. Further a bijection of the representations with I-fixed vectors and simple modules of the Weyl group is given. All results were known before but are presented in a short and comprehensive way.
Reviewer: A.Deitmar


22E50 Representations of Lie and linear algebraic groups over local fields
11F33 Congruences for modular and \(p\)-adic modular forms
14G20 Local ground fields in algebraic geometry
14C35 Applications of methods of algebraic \(K\)-theory in algebraic geometry