Search arXiv⌕ Search
Home/Archive/arXiv

arXiv · 2609.23866

A bar operation on the double affine Hecke algebra

Dougal Davis·Ivan Losev·Calder Morton-Ferguson
View original source ↗Read source document ↗Export metadata ↓

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 $1$.

Explore related subjects

math.RT

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Dougal Davis, Ivan Losev, Calder Morton-Ferguson. 2026-09-20. A bar operation on the double affine Hecke algebra. https://arxiv.org/abs/2609.23866

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

A local relative trace formula for F*\SL(2,F)

In this note, we derive explicitly the local relative trace formula for the symmetric space F*\SL(2,F) at the level of Lie algebras, where F is a p-adic field of residue characteristic greater than two and F* is the set of invertible elements in F. This is perhaps one of the simplest non-trivial analogs of the trace formula, and also a motivating example for the author's work (in preparation) on the relative trace formula.

math.RT↗
arXiv2026

Semi-infinite parabolic IC-sheaf

Let G be a connected reductive group, P its parabolic subgroup. We consider the parabolic semi-infinite category of sheaves on the affine Grassmanian of G and construct the parabolic version of the semi-infinite IC-sheaf of each orbit. We establish some of its properties and relate it to sheaves on the Drinfeld compactification of the moduli stack Bun_P of P-torsors on a curve. We also relate the parabolic semi-infinite IC-sheaf with the dual baby Verma object on the spectral side.

math.RT↗
arXiv2026

The Grothendieck group of an extriangulated category

In this paper, we investigate the split Grothendieck group $K^{\rm sp}_{0}(\mathcal{M})$ of a $d$-rigid subcategory $\mathcal{M}$ in an extriangulated category $\mathscr{C}$. As applications, we prove the following results: (1) If $\mathcal{M}$ is a silting subcategory, then the Grothendieck group $K_{0}(\mathscr{C})$ is isomorphic to $K_{0}^{\rm sp}(\mathcal{M})$; (2) If $\mathcal{M}$ is a $d$-cluster tilting subcategory, then $K_{0}(\mathscr{C})$ is isomorphic to the index Grothendieck group $K_{0}^{\rm in}(\mathcal{M})$; (3) Let $\mathcal{C}_{A_{n}}^{d}$ be the $d$-cluster category of type $A_n$. If $d$ is even, then $K_0(\mathcal{C}_{A_{n}}^{d})\cong \mathbb{Z}/(n+1)\mathbb{Z}$. If $d$ is odd, then $K_0(\mathcal{C}_{A_{n}}^{d})\cong \mathbb{Z}$ if $n$ is odd; $K_0(\mathcal{C}_{A_{n}}^{d})\cong 0$ if $n$ is even.

math.RT↗