On structure constants of Iwahori–Hecke algebras for Kac–Moody groups
Algebraic Combinatorics, Volume 4 (2021) no. 3, pp. 465-490.

We consider the Iwahori–Hecke algebra I 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 ) wW + indexed by a sub-semigroup W + of the affine Weyl group W. The multiplication is given by structure constants a w,v u = 0 : T w *T v = uP 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 , 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,vW + if G is of affine or strictly hyperbolic type. In the split affine case (where q i =q i =q, i) we get a universal Iwahori–Hecke algebra with the same basis (T w ) wW + over a polynomial ring [Q]; it specializes to I when one sets Q=q.

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DOI: https://doi.org/10.5802/alco.163
Classification: 20G44,  20C08,  20G25,  20E42,  51E24
Keywords: Building, Hecke algebra, Kac–Moody group, masure, local field.
@article{ALCO_2021__4_3_465_0,
     author = {Bardy-Panse, Nicole and Rousseau, Guy},
     title = {On structure constants of {Iwahori{\textendash}Hecke} algebras for {Kac{\textendash}Moody} groups},
     journal = {Algebraic Combinatorics},
     pages = {465--490},
     publisher = {MathOA foundation},
     volume = {4},
     number = {3},
     year = {2021},
     doi = {10.5802/alco.163},
     language = {en},
     url = {https://alco.centre-mersenne.org/articles/10.5802/alco.163/}
}
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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