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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References
P. N. Achar, A. Henderson, Orbit closures in the enhanced nilpotent cone, Adv. Math. 219 (2008), no. 1, 27–62.
H. Bao, J. Kujawa, Y. Li, W. Wang, Geometric Schur duality of classical type, To appear in Transform. Groups, arXiv1404.4000v3 (2014).
A. A. Beilinson, G. Lusztig, R. MacPherson, A geometric setting for the quantum deformation of GL n , Duke Math. J. 61 (1990), no. 2, 655–677.
M. Finkelberg, V. Ginzburg, Cherednik algebras for algebraic curves, in: Representation Theory of Algebraic Groups and Quantum Groups, Progr. Math., Vol. 284, Birkhäuser/Springer, New York, 2010, pp. 121–153.
Z. Fan, Y. Li, Geometric Schur duality of classical type, II, Trans. Amer. Math. Soc. Ser. B 2 (2015), 51–92.
I. Grojnowski, G. Lusztig, On bases of irreducible representations of quantum GL n , in: Kazhdan-Lusztig Theory and Related Topics (Chicago, IL, 1989), Contemp. Math., Vol. 139, Amer. Math. Soc., Providence, RI, pp. 167–174.
J. C. Jantzen, Lectures on Quantum Groups, Graduate Studies in Mathematics, Vol. 6, American Mathematical Society, Providence, RI, 1996.
M. Jimbo, A q-difference analogue of U \( \left(\mathfrak{g}\right) \) and the Yang-Baxter equation, Lett. Math. Phys. 10 (1985), no. 1, 63–69.
S. Kato, An exotic Deligne-Langlands correspondence for symplectic groups, Duke Math. J. 148 (2009), no. 2, 305–371.
P. Magyar, Bruhat order for two flags and a line, J. Algebraic Combin. 21 (2005), no. 1, 71–101.
P. Magyar, J. Weyman, A. Zelevinsky, Multiple flag varieties of finite type, Adv. Math. 141 (1999), no. 1, 97–118.
D. Rosso, The mirabolic Hecke algebra, J. Algebra 405 (2014), 179–212.
R. Travkin, Mirabolic Robinson-Schensted-Knuth correspondence, Selecta Math. (N.S.) 14 (2009), no. 3–4, 727–758.
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ROSSO, D. MIRABOLIC QUANTUM \( {\mathfrak{sl}}_2 \) . Transformation Groups 23, 217–255 (2018). https://doi.org/10.1007/s00031-017-9432-y
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DOI: https://doi.org/10.1007/s00031-017-9432-y