The mirabolic Hecke algebra
The Iwahori-Hecke algebra of the symmetric group is the convolution algebra of $\gl_n$-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 $R_n$, which had originally been described by Solomon. In this paper we give a new presentation for $R_n$ 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 $R_n$. We use Jucys-Murphy elements to describe the center of $R_n$ and to give a $\mathfrak{gl}_\infty$-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.