A bar operation on the double affine Hecke algebra

Abstract

We construct a bar-type operation for the affine Hecke algebra $\Ha$ attached to a symmetrizable Kac--Moody group. Our main tool is a geometric bimodule defined using the equivariant $K$-theory of a suitable version of the Steinberg variety for this Kac--Moody group. Using elements of this bimodule corresponding to standard and costandard Hodge $D$-modules on the Kashiwara flag variety, we define a bar involution on $\mathcal{H}_{W}^a$ by relating the left and right actions of $\mathcal{H}_{W}^a$ on this bimodule. A central issue is that the resulting formulas for the bar operation do not preserve the algebra itself and naturally produce infinite expansions. To address this, we introduce certain completions of $\mathcal{H}_{W}^a$ on which the bar operation is well-defined. For this construction, we prove a number of combinatorial finiteness results for products in $\mathcal{H}_{W}^a$. In affine type, we further analyze the level-zero part of the algebra and obtain an explicit combinatorial formula for a bar operation on a suitable localization of Cherednik's double affine Hecke algebra. In this case the bar operation itself is not an involution but we also show that this bar operation becomes an involution when the lattice parameter $q$ is set to $1$.