Mirabolic affine Grassmannian and character sheaves
  We compute the Frobenius trace functions of mirabolic character sheaves defined over a finite field. The answer is given in terms of the character values of general linear groups over the finite field, and the structure constants of multiplication in the mirabolic Hall-Littlewood basis of symmetric functions, introduced by Shoji. 
Michael Finkelberg, Victor Ginzburg, Roman Travkin