Mirabolic quantum ${\mathfrak{sl}}_2$

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 $G{L}_d$-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.

34 pages

References in corpus (2)

• Geometric Schur duality of classical type
• The mirabolic Hecke algebra