Cellular bases of mirabolic Hecke algebras

Abstract

We prove that the mirabolic Hecke algebra $\mathscr{H}_{n,R}(q)$ over a commutative ring $R$ is cellular by constructing explicit cellular bases. When $R=\mathbb{K}$ is a field, we also establish a Morita equivalence between $\mathscr{H}_{n,\mathbb{K}}(q)$ and a direct sum of Hecke algebras associated with symmetric groups. As applications, we classify the irreducible $\mathscr{H}_{n,\mathbb{K}}(q)$-modules and obtain a semisimplicity criterion. Finally, when $\mathbb{K}=\mathbb{C}(q)$, we show that the cell modules arising from one of our cellular bases coincide with Siegel's irreducible $\mathscr{H}_{n,\mathbb{K}}(q)$-modules.