Transformation Groups

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

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



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.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [AH08]
    P. N. Achar, A. Henderson, Orbit closures in the enhanced nilpotent cone, Adv. Math. 219 (2008), no. 1, 27–62.MathSciNetCrossRefMATHGoogle Scholar
  2. [BKLW]
    H. Bao, J. Kujawa, Y. Li, W. Wang, Geometric Schur duality of classical type, To appear in Transform. Groups, arXiv1404.4000v3 (2014).Google Scholar
  3. [BLM90]
    A. A. Beilinson, G. Lusztig, R. MacPherson, A geometric setting for the quantum deformation of GLn, Duke Math. J. 61 (1990), no. 2, 655–677.MathSciNetCrossRefMATHGoogle Scholar
  4. [FG10]
    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.Google Scholar
  5. [FL]
    Z. Fan, Y. Li, Geometric Schur duality of classical type, II, Trans. Amer. Math. Soc. Ser. B 2 (2015), 51–92.MathSciNetCrossRefMATHGoogle Scholar
  6. [GL92]
    I. Grojnowski, G. Lusztig, On bases of irreducible representations of quantum GLn, in: Kazhdan-Lusztig Theory and Related Topics (Chicago, IL, 1989), Contemp. Math., Vol. 139, Amer. Math. Soc., Providence, RI, pp. 167–174.Google Scholar
  7. [Jan96]
    J. C. Jantzen, Lectures on Quantum Groups, Graduate Studies in Mathematics, Vol. 6, American Mathematical Society, Providence, RI, 1996.Google Scholar
  8. [Jim85]
    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.Google Scholar
  9. [Kat09]
    S. Kato, An exotic Deligne-Langlands correspondence for symplectic groups, Duke Math. J. 148 (2009), no. 2, 305–371.MathSciNetCrossRefMATHGoogle Scholar
  10. [Mag05]
    P. Magyar, Bruhat order for two flags and a line, J. Algebraic Combin. 21 (2005), no. 1, 71–101.MathSciNetCrossRefMATHGoogle Scholar
  11. [MWZ99]
    P. Magyar, J. Weyman, A. Zelevinsky, Multiple flag varieties of finite type, Adv. Math. 141 (1999), no. 1, 97–118.MathSciNetCrossRefMATHGoogle Scholar
  12. [Ros14]
    D. Rosso, The mirabolic Hecke algebra, J. Algebra 405 (2014), 179–212.MathSciNetCrossRefMATHGoogle Scholar
  13. [Tra09]
    R. Travkin, Mirabolic Robinson-Schensted-Knuth correspondence, Selecta Math. (N.S.) 14 (2009), no. 3–4, 727–758.MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2017

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of California, RiversideRiversideUSA

Personalised recommendations