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Transformation Groups

, Volume 23, Issue 1, pp 217–255 | Cite as

MIRABOLIC QUANTUM \( {\mathfrak{sl}}_2 \)

  • DANIELE ROSSO
Article

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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Copyright information

© Springer Science+Business Media New York 2017

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of California, RiversideRiversideUSA

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