Algebra Seminar: Muthiah -- The Iwahori-Hecke algebra for p-adic loop groups

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.