Algorithm for Iwahori-Matsumoto duality for tempered unipotent representations of geometric Hecke algebras of type B, C, D

Thesis

The Iwahori–Matsumoto involution is an involutive operation on the Grothendieck group of complex finite-dimensional representations of an affine Hecke algebra, or a graded Hecke algebra. The Aubert–Zelevinsky involution is another involutive operation on the Grothendieck group of complex finite-length representations of p-adic groups. The representation theory of p-adic groups can be described by certain affine and graded Hecke algebras, and the involutions mentioned above are related via this. The main goal of the thesis it to provide an explicit algorithm for the Iwahori–Matsumoto involution for irreducible tempered representations of certain affine Hecke algebras, for irreducible tempered representations of certain graded Hecke algebras which additionally have real infinitesimal character, and an algorithm for AZ for tempered unipotent representations for certian p-adic groups. The affine and geometric Hecke algebras that we consider are those coming from the p-adic groups SO(N) and Sp(2n). For AZ, we only consider the p-adic group SO(2n+1). We will consider the cases SO(2n) and Sp(2n) in a forthcoming paper. The algorithm will be obtained by completely different methods than the methods used by Atobe and Minguez, who obtained an algorithm for AZ for SO(2n + 1) and Sp(2n).

Authors: La, R

• DOI: 10.5287/ora-1rmzg6k9r
• Type of award: DPhil
• Level of award: Doctoral
• Awarding institution: University of Oxford
• Language: English
• Keywords: representation theory of p-adic groups; Iwahori-Matsumoto duality; Hecke algebras
• Subjects: Representation theory, algebra