On structure constants of Iwahori–Hecke algebras for Kac–Moody groups

Bardy-Panse, Nicole; Rousseau, Guy

doi:10.5802/alco.163

Bardy-Panse, Nicole ¹; Rousseau, Guy ¹

¹ Université de Lorraine CNRS, IECL Nancy F-54000, France

Algebraic Combinatorics, Volume 4 (2021) no. 3, pp. 465-490.

Abstract

We consider the Iwahori–Hecke algebra $^{I} ℋ$ associated to an almost split Kac–Moody group $G$ (affine or not) over a nonarchimedean local field $𝒦$ . It has a canonical double-coset basis ${(T_{w})}_{w \in W^{+}}$ indexed by a sub-semigroup $W^{+}$ of the affine Weyl group $W$ . The multiplication is given by structure constants $a_{w, v}^{u} \in ℕ = ℤ_{\geq 0}$ : $T_{w} * T_{v} = \sum_{u \in P_{w, v}} a_{w, v}^{u} T_{u}$ . A conjecture, by Braverman, Kazhdan, Patnaik, Gaussent and the authors, tells that $a_{w, v}^{u}$ is a polynomial, with coefficients in $ℕ$ , in the parameters $q_{i} - 1, q_{i}^{'} - 1$ of $G$ over $𝒦$ . We prove this conjecture when $w$ and $v$ are spherical or, more generally, when they are said to be generic: this includes all cases of $w, v \in W^{+}$ if $G$ is of affine or strictly hyperbolic type. In the split affine case (where $q_{i} = q_{i}^{'} = q$ , $\forall i$ ) we get a universal Iwahori–Hecke algebra with the same basis ${(T_{w})}_{w \in W^{+}}$ over a polynomial ring $ℤ [Q]$ ; it specializes to $^{I} ℋ$ when one sets $Q = q$ .

Received: 2020-03-09
Revised: 2020-11-25
Accepted: 2020-12-04
Published online: 2021-06-22

DOI: 10.5802/alco.163

Classification: 20G44, 20C08, 20G25, 20E42, 51E24
Keywords: Building, Hecke algebra, Kac–Moody group, masure, local field.

Author's affiliations:

Bardy-Panse, Nicole ¹; Rousseau, Guy ¹

¹ Université de Lorraine CNRS, IECL Nancy F-54000, France

License:

CC-BY 4.0

Copyrights: The authors retain unrestricted copyrights and publishing rights

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     title = {On structure constants of {Iwahori{\textendash}Hecke} algebras for {Kac{\textendash}Moody} groups},
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Bardy-Panse, Nicole; Rousseau, Guy. On structure constants of Iwahori–Hecke algebras for Kac–Moody groups. Algebraic Combinatorics, Volume 4 (2021) no. 3, pp. 465-490. doi : 10.5802/alco.163. https://alco.centre-mersenne.org/articles/10.5802/alco.163/

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