Mirabolic Hecke algebras, Schur-Weyl duality and Frobenius character formulas

We first introduce a new presentation for the mirabolic Hecke algebra ${\mathcal{H}}_{n,R}(q)$ over an arbitrary commutative ring $R$ and derive a new basis. Based on this presentation, specializing to the case of ${\mathcal{H}}_n(q)$ over the field $\mathbb{C}(q)$, we construct a basis for the cocenter of ${\mathcal{H}}_n(q)$, which facilitates the definition of its character table. We further establish a Schur–Weyl duality between ${\mathcal{H}}_n(q)$ and the quantum group $U_q({\mathfrak{g}\mathfrak{l}}_r)$. As an application, we obtain Frobenius character formulas for the irreducible characters of ${\mathcal{H}}_n(q)$ within the ring of symmetric functions. Finally, we derive a recursive Murnaghan–Nakayama rule for the computation of the character table.