University of Sydney Algebra Seminar

Dinakar Muthiah (University of Alberta)

Friday 29 April, 12-1pm, Place: Carslaw 375

The Iwahori-Hecke algebra for p-adic loop groups: the double-coset basis and double-affine Bruhat order

Recently, Braverman, Kazhdan, and Patnaik have constructed Iwahori-Hecke algebras for p-adic loop groups. Perhaps unsurprisingly, the resulting algebra is a slight variation on Cherednik's DAHA. In addition to the relationship with the DAHA, the p-adic construction also comes with a basis (the double-coset basis) consisting of indicator functions of double-cosets. Braverman, Kazhdan, and Patnaik also proposed a (double affine) Bruhat preorder on the set of double cosets, which they conjectured to be a poset. I will describe a combinatorial presentation of the double-coset basis, and also an alternative way to develop the double affine Bruhat order that is closely related to the combinatorics of the double-coset basis and is manifestly a poset. One significant new feature is a length function that is compatible with the order. I will also discuss joint work in progress with Daniel Orr, where we give a positive answer to a question raised in a previous paper: namely, we prove that the length function can be specialized to take values in the integers. In particular, this proves finiteness of chains in the double-affine Bruhat order, and it gives an expected dimension formula for (yet to be defined) transversal slices in the double affine flag variety. If time remains, I will discuss a number of further open questions.