On Iwahori-Hecke algebras for p-adic loop groups: double coset basis and Bruhat order

We study the $p$-adic loop group Iwahori-Hecke algebra $\scr{H}(G^+,I)$ constructed by Braverman, Kazhdan, and Patnaik and give positive answers to two of their conjectures. First, we algebraically develop the “double coset basis” of $\scr{H}(G^+,I)$ 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 $\scr{W}_{\scr{T}}$ 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 $\scr{W}_{\scr{T}}$ which carries 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 $\Bbb{Z}\oplus\Bbb{Z}\varepsilon$ where $\varepsilon$ is “infinitesimally” small.