The mirabolic Hecke algebra

The Iwahori–Hecke algebra of the symmetric group is the convolution algebra of GLnGLn-invariant functions on the variety of pairs of complete flags over a finite field. Considering convolution on the space of triples of two flags and a vector we obtain the mirabolic Hecke algebra RnRn, which had originally been described by Solomon. In this paper we give a new presentation for RnRn, which shows that it is a quotient of a cyclotomic Hecke algebra as defined by Ariki and Koike. From this we recover the results of Siegel about the representations of RnRn. We use Jucys–Murphy elements to describe the center of RnRn and to give a gl∞gl∞-structure on the Grothendieck group of the category of its representations, giving ‘mirabolic’ analogues of classical results about the Iwahori–Hecke algebra. We also outline a strategy towards a proof of the conjecture that the mirabolic Hecke algebra is a cellular algebra.