Mirabolic quantum $\mathfrak{sl}_2$

TL;DR

This paper extends the quantum algebra framework to the mirabolic setting involving triples of flags and vectors, classifies irreducible representations for n=2, and establishes a mirabolic quantum Schur-Weyl duality.

Contribution

It introduces a new mirabolic quantum algebra, classifies its finite-dimensional irreducible representations for n=2, and develops a mirabolic quantum Schur-Weyl duality with the mirabolic Hecke algebra.

Findings

Classified finite-dimensional irreducible representations for n=2

Proved the semisimplicity of the representation category

Established a mirabolic quantum Schur-Weyl duality

Abstract

The quantum enveloping algebra of $\mathfrak{sl}_n$ (and the quantum Schur algebras) was constructed by Beilinson-Lusztig-MacPherson as the convolution algebra of $GL_d$-invariant functions over the space of pairs of partial $n$-step flags over a finite field. In this paper we expand the construction to the mirabolic setting of triples of two partial flags and a vector, and examine the resulting convolution algebra. In the case of $n=2$, we classify the finite dimensional irreducible representations of the mirabolic quantum algebra and we prove that the category of such representations is semisimple. Finally, we describe a mirabolic version of the quantum Schur-Weyl duality, which involves the mirabolic Hecke algebra.