Presented by: 
Dinakar Muthiah (University of Alberta)
Tue 3 May, 3:00 pm - 4:00 pm
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.