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

Muthiah, D. (2018) On Iwahori-Hecke algebras for p-adic loop groups: double coset basis and Bruhat order. American Journal of Mathematics, 140(1), pp. 221-244. (doi: 10.1353/ajm.2018.0004)

Abstract

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 Z⊕Zε where ε is “infinitesimally” small.

• Item Type: Articles
• Status: Published
• Refereed: Yes
• Glasgow Author(s) Enlighten ID: Muthiah, Dr Dinakar
• Authors: Muthiah, D.
• College/School: College of Science and Engineering > School of Mathematics and Statistics > Mathematics
• Journal Name: American Journal of Mathematics
• Publisher: John Hopkins University Press
• ISSN: 0002-9327
• ISSN (Online): 1080-6377