Abstract
The convolution algebra of GL d -invariant functions over the space of pairs of partial n-step flags over a finite field was studied by Beilinson-Lusztig-MacPherson. They used it to give a construction of the quantum Schur algebras and, using a stabilization procedure, of the idempotented version of the quantum enveloping algebra of $$ {\mathfrak{gl}}_n $$ . 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.
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