Double affine Bruhat order and Iwahori-Hecke algebras for $p$-adic loop groups

Recently, Braverman, Kazhdan, and Patnaik have constructed Iwahori-Hecke algebras for p-adic loop groups. Unsurprisingly, the resulting algebra is a variation on Cherednik's DAHA. The p-adic construction also comes with a basis (the double-coset basis) consisting of indicator functions of Iwahori 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 this combinatorics; from this perspective the order is manifestly a poset. One new feature is a length function that is compatible with the order. I will also discuss joint work with Daniel Orr, where we give a positive answer to a question raised in a previous paper: we prove that the length function can be specialized to take values in the integers. 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 how these results are stepping stones for developing Kazhdan-Lusztig theory in this setting and a number of further open questions.Non UBCUnreviewedAuthor affiliation: University of AlbertaPostdoctora