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

Abstract

We consider the Iwahori-Hecke algebra 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 $(T_{\mathbf{w}}{)}_{\mathbf{w}\in W^{+}}$ indexed by a sub-semigroup $W^{+}$ of the affine Weyl group $W$. The multiplication is given by structure constants $a_{\mathbf{w},\mathbf{v}}^{\mathbf{u}}\in N=Z_{\ge 0}$ : $T_{\mathbf{w}}\ast T_{\mathbf{v}}={\sum }_{\mathbf{u}\in P_{\mathbf{w},\mathbf{v}}}{a}_{\mathbf{w},\mathbf{v}}^{\mathbf{u}}{T}_{\mathbf{u}}$. A conjecture, by Bravermann, Kazhdan, Patnaik, Gaussent and the authors, tells that $a_{\mathbf{w},\mathbf{v}}^{\mathbf{u}}$ is a polynomial, with coefficients in $N$, in the parameters $q_i-1,q_i^{\prime }-1$ of $G$ over $K$. We prove this conjecture when $\mathbf{w}$ and $\mathbf{v}$ are spherical or, more generally, when they are said generic: this includes all cases of $\mathbf{w},\mathbf{v}\in W^{+}$ if $G$ is of affine or strictly hyperbolic type. In the split affine case (where $q_i=q_i^{\prime }=q$, $\forall i$) we get a universal Iwahori-Hecke algebra with the same basis $(T_{\mathbf{w}}{)}_{\mathbf{w}\in W^{+}}$ over a polynomial ring $Z[Q]$; it specializes to our Iwahori-Hecke algebra when one sets $Q=q$.

References in corpus (2)

• Macdonald's formula for Kac-Moody groups over local fields
• A new axiomatics for masures