Recently, Braverman, Kazhdan, and Patnaik have constructed Iwahori-Hecke algebras for p-adic loop groups. They also constructed a basis (the double-coset basis) of such algebras via indicator functions of double-cosets, but the combinatorics and structure coefficients of the basis remained mysterious. I will describe a combinatorial presentation of this basis. 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 discuss an alternative way to develop this 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.