Remote Access Representation Theory
Green Open Access

Representation Theory

ISSN 1088-4165

   
 
 

 

Twisting functors on ${\mathcal O}$


Authors: Henning Haahr Andersen and Catharina Stroppel
Journal: Represent. Theory 7 (2003), 681-699
MSC (2000): Primary 17B10, 17B35, 20F29
DOI: https://doi.org/10.1090/S1088-4165-03-00189-4
Published electronically: December 3, 2003
MathSciNet review: 2032059
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: This paper studies twisting functors on the main block of the Bernstein-Gelfand-Gelfand category ${\mathcal O}$ and describes what happens to (dual) Verma modules. We consider properties of the right adjoint functors and show that they induce an auto-equivalence of derived categories. This allows us to give a very precise description of twisted simple objects. We explain how these results give a reformulation of the Kazhdan-Lusztig conjectures in terms of twisting functors.


References [Enhancements On Off] (What's this?)

  • [AL02] H. H. Andersen and N. Lauritzen, Twisted Verma modules, Studies in Memory of Issai Schur, Progress in Math., vol. 210, Birkhäuser, Basel, 2002, pp. 1-26.
  • [And86] H. H. Andersen, An inversion formula for the Kazhdan-Lusztig polynomials for affine Weyl groups, Adv. in Math. 60 (1986), no. 2, 125-153. MR 87j:22025
  • [And03] -, Twisted Verma modules and their quantized analogues, Combinatorial and geometric representation theory, Contemporary Mathematics, vol. 325, AMS, 2003, pp. 1-10.
  • [Ark] S. Arkhipov, Algebraic construction of contragradient quasi-Verma modules in positive characteristic, math. AG/0105042.
  • [BB93] A. Beilinson and J. Bernstein, A proof of Jantzen conjectures, I. M. Gelfand Seminar, Adv. Soviet Math., vol. 16, Amer. Math. Soc., Providence, RI, 1993, pp. 1-50. MR 95a:22022
  • [BG80] J. N. Bernstein and S. I. Gelfand, Tensor products of finite- and infinite-dimensional representations of semisimple Lie algebras, Compositio Math. 41 (1980), no. 2, 245-285. MR 82c:17003
  • [BGG76] I. N. Bernstein, I. M. Gelfand, and S. I. Gelfand, A certain category of ${\mathfrak{g}}$-modules, Funkcional. Anal. i Prilozen. 10 (1976), no. 2, 1-8. MR 53:10880
  • [BK81] J.-L. Brylinski and M. Kashiwara, Kazhdan-Lusztig conjecture and holonomic systems, Invent. Math. 64 (1981), no. 3, 387-410. MR 83e:22020
  • [CPS93] E. Cline, B. Parshall, and L. Scott, Abstract Kazhdan-Lusztig theories, Tohoku Math. J. (2) 45 (1993), no. 4, 511-534. MR 94k:20079
  • [Dix96] J. Dixmier, Enveloping algebras, Graduate Studies in Mathematics, vol. 11, American Mathematical Society, 1996, Revised reprint of the 1977 translation. MR 97c:17010
  • [FF90] B. L. Feigin and E. V. Frenkel, Affine Kac-Moody algebras and semi-infinite flag manifolds, Comm. Math. Phys. 128 (1990), no. 1, 161-189. MR 92f:17026
  • [Hum90] J. E. Humphreys, Reflection groups and Coxeter groups, Cambridge Studies in Advanced Mathematics, vol. 29, Cambridge University Press, 1990.
  • [Irv93] R. Irving, Shuffled Verma modules and principal series modules over complex semisimple Lie algebras, J. London Math. Soc. (2) 48 (1993), no. 2, 263-277. MR 94i:17013
  • [Jan79] J. C. Jantzen, Moduln mit einem höchsten Gewicht, Lecture Notes in Mathematics, vol. 750, Springer, 1979. MR 81m:17011
  • [Jan83] -, Einhüllende Algebren halbeinfacher Lie-Algebren, Proceedings of the International Congress of Mathematicians, Vol. 1, 2, PWN, 1983. MR 86c:17011
  • [Jan87] -, Representations of algebraic groups, Pure and Applied Mathematics, vol. 131, Academic Press Inc., Boston, MA, 1987. MR 89g:20076
  • [Jos83] A. Joseph, Completion functors in the ${{\mathcal O}}$ category, Noncommutative harmonic analysis and Lie groups (Marseille, 1982), Lecture Notes in Math., vol. 1020, Springer, Berlin, 1983, pp. 80-106. MR 85i:17012
  • [Jos94] -, Enveloping algebras: problems old and new, Lie theory and geometry, Progr. Math., vol. 123, Birkhäuser Boston, Boston, MA, 1994, pp. 385-413. MR 96e:17022
  • [KL79] D. Kazhdan and G. Lusztig, Representations of Coxeter groups and Hecke algebras, Invent. Math. 53 (1979), no. 2, 165-184. MR 81j:20066
  • [KM] O. Khomenko and V. Mazorchuk, On Arkhipov's and Enright's functors, Technical Report 2003:7, University of Uppsala (Sweden).
  • [KS02] M. Khovanov and P. Seidel, Quivers, Floer cohomology, and braid group actions, J. Amer. Math. Soc. 15 (2002), no. 1, 203-271 (electronic). MR 2003d:53155
  • [McC85] J. McCleary, User's guide to spectral sequences, Mathematics Lecture Series, vol. 12, Publish or Perish Inc., Wilmington, DE, 1985. MR 87f:55014
  • [Soe98] W. Soergel, Character formulas for tilting modules over Kac-Moody algebras, Represent. Theory 2 (1998), 432-448 (electronic). MR 2000c:17048
  • [Vog79] D. A. Vogan, Jr., Irreducible characters of semisimple Lie groups. II. The Kazhdan-Lusztig conjectures, Duke Math. J. 46 (1979), no. 4, 805-859. MR 81f:22024
  • [Vor99] A. A. Voronov, Semi-infinite induction and Wakimoto modules, Amer. J. Math. 121 (1999), no. 5, 1079-1094. MR 2000g:17025

Similar Articles

Retrieve articles in Representation Theory of the American Mathematical Society with MSC (2000): 17B10, 17B35, 20F29

Retrieve articles in all journals with MSC (2000): 17B10, 17B35, 20F29


Additional Information

Henning Haahr Andersen
Affiliation: Department of Mathematics, University of Aarhus, Dk-8000 Aarhus C, Denmark
Email: mathha@imf.au.dk

Catharina Stroppel
Affiliation: Department of Mathematics and Computer Science, Leicester University, GB Leicester LE1 7RH
Address at time of publication: University of Aarhus, Ny Munkegade 530, Dk-8000 Aarhus C, Denmark
Email: cs93@le.ac.uk, stroppel@imf.au.dk

DOI: https://doi.org/10.1090/S1088-4165-03-00189-4
Received by editor(s): February 27, 2003
Received by editor(s) in revised form: July 10, 2003
Published electronically: December 3, 2003
Article copyright: © Copyright 2003 by the authors

American Mathematical Society