2024/03/21 by La, Ruben
Mathematics · #20C08 #20C30 #20G05 #22E50 #Advanced Algebra and Geometry #Advanced Topics in Algebra #Algebraic structures and combinatorial models #FOS: Mathematics #Representation Theory (math.RT)
paper · pdf · doi:10.48550/arxiv.2403.14528
openalex publication_date 2024/03/21 · openalex created_date 2024/03/24 · openalex updated_date 2026/07/28
The graded Iwahori--Matsumoto involution \mathbbIM is an algebra involution on a graded Hecke algebra closely related to the more well-known Iwahori--Matsumoto involution on an affine Hecke algebra. It induces an involution on the Grothendieck group of complex finite-dimensional representations of ℍ. When ℍ is a geometric graded Hecke algebra (in the sense of Lusztig) associated to a connected complex reductive group G, the irreducible representations of ℍ are parametrised by a set M consisting of certain G-conjugacy classes of quadruples (e,s,r0,ψ) where r0 ∈ ℂ, e ∈ Lie(G) is nilpotent, s ∈ Lie(G) is semisimple, and ψ is some irreducible representation of the group of components of the simultaneous centraliser of (e,s) in G. Let Y be an irreducible tempered representation of ℍ with real infinitesimal character. Then \mathbbIM( Y) = Y(e',s,r0,ψ') for some (e',s,r0,ψ') ∈ M. The main result of this paper is to give an explicit algorithm that computes the G-orbit of e' for G = Sp(2n,ℂ) and G = SO(N,ℂ). As a key ingredient of the main result, we also prove a generalisation of the main theorems of Waldspurger 2019 (for Sp(2n,ℂ)) and La 2024 (for SO(N,ℂ)) regarding certain maximality properties of generalised Springer representations.