On Iwahori-Hecke Algebras for p-adic Loop Groups: Double Coset Basis and Bruhat Order

TL;DR

This paper advances the understanding of p-adic loop group Iwahori-Hecke algebras by establishing a basis, proving polynomial structure coefficients, and confirming a conjectured Bruhat order generalization.

Contribution

It algebraically develops the double coset basis, proves a generalized Iwahori-Matsumoto formula, and confirms the conjecture that a certain preorder is a partial order.

Findings

Structure coefficients are polynomials in the residue field size.

The defined order on the semi-group coincides with the Bruhat order generalization.

The length function takes values in an extended integer set.

Abstract

We study the $p$-adic loop group Iwahori-Hecke algebra $\mathcal{H}(G^+,I)$ constructed by Braverman, Kazhdan, and Patnaik and give positive answers to two of their conjectures. First, we algebraically develop the "double coset basis" of $\mathcal{H}(G^+,I)$ given by indicator functions of double cosets. We prove a generalization of the Iwahori-Matsumoto formula, and as a consequence, we prove that the structure coefficients of the double coset basis are polynomials in the order of the residue field. The basis is naturally indexed by a semi-group $\mathcal{W}_\mathcal{T}$ on which Braverman, Kazhdan, and Patnaik define a preorder. Their preorder is a natural generalization of the Bruhat order on affine Weyl groups, and they conjecture that the preorder is a partial order. We define another order on $\mathcal{W}_\mathcal{T}$ which is graded by a length function and is manifestly a partial order. We prove the two definitions coincide, which implies a positive answer to their conjecture. Interestingly, the length function seems to naturally take values in $\mathbb{Z} \oplus \mathbb{Z} \varepsilon$ where $\varepsilon$ is "infinitesimally" small.