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 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 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 WT 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 WT 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 ZⴲZε where ε is "infinitesimally" small.