Advertisement

Inventiones mathematicae

, Volume 200, Issue 3, pp 849–923 | Cite as

Dimensions of irreducible modules over W-algebras and Goldie ranks

  • Ivan LosevEmail author
Article

Abstract

The main goal of this paper is to compute two related numerical invariants of a primitive ideal in the universal enveloping algebra of a semisimple Lie algebra. The first one, very classical, is the Goldie rank of an ideal. The second one is the dimension of an irreducible module corresponding to this ideal over an appropriate finite W-algebra. We concentrate on the integral central character case. We prove, modulo a conjecture, that in this case the two are equal. Our conjecture asserts that there is a one-dimensional module over the W-algebra with certain additional properties. The conjecture is proved for the classical types. Also, modulo the same conjecture, we compute certain scale factors introduced by Joseph, this allows to compute the Goldie ranks of the algebras of locally finite endomorphisms of simples in the BGG category \(\mathcal {O}\). This completes a program of computing Goldie ranks proposed by Joseph in the 80’s (for integral central characters and modulo our conjecture). We also provide an essentially Kazhdan–Lusztig type formula for computing the characters of the irreducibles in the Brundan–Goodwin–Kleshchev category \(\mathcal {O}\) for a W-algebra again under the assumption that the central character is integral. In particular, this allows to compute the dimensions of the finite dimensional irreducible modules. The formula is based on a certain functor from an appropriate parabolic category \(\mathcal {O}\) to the W-algebra category \(\mathcal {O}\). This functor can be regarded as a generalization of functors previously constructed by Soergel and by Brundan-Kleshchev. We prove a number of properties of this functor including the quotient property and the double centralizer property. We develop several side topics related to our generalized Soergel functor. For example, we discuss its analog for the category of Harish–Chandra modules. We also discuss generalizations to the case of categories \(\mathcal {O}\) over Dixmier algebras. The most interesting example of this situation comes from the theory of quantum groups: we prove that an algebra that is a mild quotient of Luszitg’s form of a quantum group at a root of unity is a Dixmier algebra. For this we check that the quantum Frobenius epimorphism splits.

Mathematics Subject Classification

17B35 16G99 

List of Symbols

\(\mathcal {A}^{opp}\)

The opposite algebra of \(\mathcal {A}\)

\(\widehat{\otimes }\)

The completed tensor product of complete topological vector spaces/ modules

\((a_1,\ldots ,a_k)\)

The two-sided ideal in an associative algebra generated by elements \(a_1,\ldots ,a_k\)

\(A^{\wedge _\chi }\)

The completion of a commutative (or “almost commutative”) algebra \(A\) with respect to the maximal ideal of a point \(\chi \in \mathrm{Spec }(A)\)

\(\mathrm{Ann }_\mathcal {A}(\mathcal {M})\)

The annihilator of an \(\mathcal {A}\)-module \(\mathcal {M}\) in an algebra \(\mathcal {A}\)

\(D(X)\)

The algebra of differential operators on a smooth variety \(X\)

\(G^\circ \)

The connected component of unit in an algebraic group \(G\)

\((G,G)\)

The derived subgroup of a group \(G\)

\(G_x\)

The stabilizer of \(x\) in \(G\)

\(\mathrm{Grk }(\mathcal {A})\)

The Goldie rank of a prime Noetherian algebra \(\mathcal {A}\)

\(\mathrm{gr }\mathcal {A}\)

The associated graded vector space of a filtered vector space \(\mathcal {A}\)

\(R_\hbar (\mathcal {A})\)

\(:=\bigoplus _{i\in \mathbb {Z}}\hbar ^i \mathrm{F }_i\mathcal {A}\): the Rees \({\mathbb {K}}[\hbar ]\)-module of a filtered vector space \(\mathcal {A}\)

\(S(V)\)

The symmetric algebra of a vector space \(V\)

\(\mathfrak {X}(H)\)

The group of characters of an algebraic group \(H\)

Notes

Acknowledgments

I would like to thank J. Adams, R. Bezrukavnikov, J. Brundan, I. Gordon, A. Joseph, G. Lusztig, V. Ostrik, A. Premet, C. Stroppel, D. Vogan and W. Wang for stimulating discussions related to various parts of this paper. Also I would like to thank the referee for many useful comments that allowed me to improve the exposition.

References

  1. 1.
    Antonyan, L.V.: On the classification of homogeneous elements of \({\mathbb{Z}}_2\)-graded semisimple Lie algebras. Vestn. Mosk. Univ., Ser. 1, 29–34 (1982, in Russian) [English translation. Mosc. Univ. Math. Bull. 37, 36–43 (1982)]Google Scholar
  2. 2.
    Barbasch, D., Vogan, D.: Unipotent representations. Ann. Math. 121, 41–110 (1985)CrossRefzbMATHMathSciNetGoogle Scholar
  3. 3.
    Bernstein, J., Gelfand, S.: Tensor products of finite and infinite dimensional representations of semisimple Lie algebras. Compositio Math. 41(2), 245–285 (1980)zbMATHMathSciNetGoogle Scholar
  4. 4.
    Bezrukavnikov, R., Finkelberg, M., Ostrik, V.: On tensor categories attached to cells in affine Weyl groups III. Israel J. Math. 170, 207–234 (2009)CrossRefzbMATHMathSciNetGoogle Scholar
  5. 5.
    Bezrukavnikov, R., Losev, I.: Etingof conjecture for quantized quiver varieties. arXiv:1309.1716
  6. 6.
    Bezrukavnikov, R., Mirkovic, I.: Representations of semisimple Lie algebras in prime characteristic and noncommutative Springer resolution. Ann. Math. 178(3), 835–919 (2013)CrossRefzbMATHMathSciNetGoogle Scholar
  7. 7.
    Borho, W., Gabriel, P., Rentschler, R.: Primideale in Einhüllenden auflösbarer Lie-Algebren. In: Lecture Notes in Mathematics, vol 357. Springer, Berlin (1973)Google Scholar
  8. 8.
    Borho, W., Kraft, H.: Über die Gelfand–Kirillov-dimension. Math. Ann. 220, 1–24 (1976)CrossRefzbMATHMathSciNetGoogle Scholar
  9. 9.
    Brundan, J.: Moeglin’s theorem and Goldie rank polynomials in Cartan type A. Compos. Math. 147(6), 1741–1771 (2011)CrossRefzbMATHMathSciNetGoogle Scholar
  10. 10.
    Brown, J., Brundan, J., Goodwin, S.: Principal W-algebras for \(\text{ GL }(m|n)\). Alg. Numb. Theory 7, 1849–1882 (2013)CrossRefzbMATHMathSciNetGoogle Scholar
  11. 11.
    Brundan, J., Goodwin, S., Kleshchev, A.: Highest weight theory for finite \(W\)-algebras. Int. math. Res. Not. 2008(15), 1–53 (2008). Art ID rnn05Google Scholar
  12. 12.
    Brundan, J., Kleshchev, A.: Representations of shifted Yangians and finite W-algebras, vol. 196, no. 918. Memoirs of the American Mathematical Society (2008)Google Scholar
  13. 13.
    Brundan, J., Kleshchev, A.: Schur–Weyl duality for higher levels. Selecta Math. 14, 1–57 (2008)CrossRefzbMATHMathSciNetGoogle Scholar
  14. 14.
    Chriss, N., Ginzburg, V.: Representation theory and complex geometry. Birkhäuser, Basel (1997)Google Scholar
  15. 15.
    Collingwood, D., McGovern, W.: Nilpotent Orbits in Semisimple Lie Algebras. Chapman and Hall, London (1993)zbMATHGoogle Scholar
  16. 16.
    Dixmier, J.: Enveloping Algebras. AMS, Amsterdam (1977)Google Scholar
  17. 17.
    Ginzburg, V.: Harish–Chandra bimodules for quantized Slodowy slices. Repres. Theory 13, 236–271 (2009)CrossRefzbMATHMathSciNetGoogle Scholar
  18. 18.
    Ginzburg, V., Guay, N., Opdam, E., Rouquier, R.: On the category \({\cal O}\) for rational Cherednik algebras. Invent. Math. 154, 617–651 (2003)CrossRefzbMATHMathSciNetGoogle Scholar
  19. 19.
    Gordon, I., Losev, I.: On category \({\cal O}\) for cyclotomic rational Cherednik algebras. J. Eur. Math. Soc. 16, 1017–1079 (2014). arXiv:1109.2315
  20. 20.
    Jantzen, J.C.: Einhüllende Algebren halbeinfacher Lie-Algebren. Ergebnisse der Math., vol. 3. Springer, New York (1983)Google Scholar
  21. 21.
    Joseph, A.: Kostant’s problem, Goldie rank and the Gelfand–Kirillov conjecture. Invent. Math. 56(3), 191–213 (1980)CrossRefzbMATHMathSciNetGoogle Scholar
  22. 22.
    Joseph, A.: Goldie rank in the enveloping algebra of a semisimple Lie algebra. I. J. Algebra 65, 269–283 (1980)CrossRefzbMATHMathSciNetGoogle Scholar
  23. 23.
    Joseph, A.: Goldie rank in the enveloping algebra of a semisimple Lie algebra. II. J. Algebra 65, 284–306 (1980)CrossRefGoogle Scholar
  24. 24.
    Joseph, A.: Primitive ideals in enveloping algebras. In: Proceedings of the International Congress of Mathematicians (1983)Google Scholar
  25. 25.
    Joseph, A.: On the cyclicity of vectors associated with Duflo involutions. In: Lecture Notes in Mathematics, vol. 1243, pp. 144–188. Springer, Berlin (1987)Google Scholar
  26. 26.
    Joseph, A.: On the associated variety of a primitive ideal. J. Algebra 93, 509–523 (1985)CrossRefzbMATHMathSciNetGoogle Scholar
  27. 27.
    Joseph, A.: A sum rule for the scale factors in the Goldie rank polynomials. J. Algebra 118, 276–311 (1988) [Addendum. J. Algebra 118, 312–321 (1988)]Google Scholar
  28. 28.
    Losev, I.: Quantized symplectic actions and \(W\)-algebras. J. Am. Math. Soc 23, 34–59 (2010)CrossRefMathSciNetGoogle Scholar
  29. 29.
    Losev, I.: Finite dimensional representations of W-algebras. Duke Math. J. 159(1), 99–143 (2011)CrossRefzbMATHMathSciNetGoogle Scholar
  30. 30.
    Losev, I.: On the structure of the category \({\cal O}\) for W-algebras. Séminaires et Congrès 24, 351–368 (2012)MathSciNetGoogle Scholar
  31. 31.
    Losev, I.: Parabolic induction and one-dimensional representations of \(W\)-algebras. Adv. Math. 226(6), 4841–4883 (2011)CrossRefzbMATHMathSciNetGoogle Scholar
  32. 32.
    Losev, I.: Primitive ideals in W-algebras of type A. J. Algebra 359, 80–88 (2012)CrossRefzbMATHMathSciNetGoogle Scholar
  33. 33.
    Losev, I., Ostrik, V.: Classification of finite dimensional irreducible modules over W-algebras. Compos. Math. 150(6), 1024–1076 (2014)CrossRefzbMATHMathSciNetGoogle Scholar
  34. 34.
    Lusztig, G.: Characters of reductive groups over a finite field. In: Annals of Mathematics Studies, vol. 107. Princeton University Press, Princeton (1984)Google Scholar
  35. 35.
    Lusztig, G.: Leading coefficients of character values of Hecke algebras. In: Proc. Symp. Pure Math., vol. 47(2), pp. 235–262. Am. Math.Soc., Amsterdam (1987)Google Scholar
  36. 36.
    Lusztig, G.: Quantum groups at roots of 1. Geom. Dedicata 35, 89–114 (1990)CrossRefzbMATHMathSciNetGoogle Scholar
  37. 37.
    Lusztig, G., Spaltenstein, N.: Induced unipotent classes. J. Lond. Math. Soc. (2), 19, 41–52 (1979)Google Scholar
  38. 38.
    McGovern, W.: Completely Prime Maximal Ideals and Quantization, p. 519. Mem. Amer. Math. Soc., USA (1994)Google Scholar
  39. 39.
    Moeglin, C.: Modèles de Whittaker et idéaux primitifs complètement premiers dans les algèbres enveloppantes II. Math. Scand. 63, 5–35 (1988)zbMATHMathSciNetGoogle Scholar
  40. 40.
    Premet, A.: Irreducible representations of Lie algebras of reductive groups and the Kac–Weisfeiler conjecture. Invent. Math. 121, 79–117 (1995)CrossRefzbMATHMathSciNetGoogle Scholar
  41. 41.
    Premet, A.: Special transverse slices and their enveloping algebras. Adv. Math. 170, 1–55 (2002)CrossRefzbMATHMathSciNetGoogle Scholar
  42. 42.
    Premet, A.: Enveloping algebras of Slodowy slices and the Joseph ideal. J. Eur. Math. Soc. 9(3), 487–543 (2007)CrossRefzbMATHMathSciNetGoogle Scholar
  43. 43.
    Premet, A.: Enveloping algebras of Slodowy slices and Goldie rank. Transform. Groups 16(3), 857–888 (2011)CrossRefzbMATHMathSciNetGoogle Scholar
  44. 44.
    Premet, A.: Multiplicity-free primitive ideals associated with rigid nilpotent orbits. Transform. Groups 19(2), 569–641 (2014)CrossRefzbMATHMathSciNetGoogle Scholar
  45. 45.
    Soergel, W.: Kategorie \(\cal O\), perverse Garben und Moduln den Koinvarianten zur Weyl-grouppe. J. Am. Math. Soc. 3, 421–445 (1990)zbMATHMathSciNetGoogle Scholar
  46. 46.
    Stroppel, C.: Der Kombinatorikfunktor \(\mathbb{V}\): Graduierte Kategorie \(\cal O\), Hauptserien und primitive Ideale. Dissertation Universität Freiburg i. Br. (2001)Google Scholar
  47. 47.
    Stroppel, C.: Category \(\cal O\): quivers and endomorphisms of projectives. Repres. Theory 7, 322–345 (2003)CrossRefzbMATHMathSciNetGoogle Scholar
  48. 48.
    Warfield, R.B.: Prime ideals in ring extensions. J. Lond. Math. Soc. 28, 453–460 (1983)CrossRefzbMATHMathSciNetGoogle Scholar
  49. 49.
    Zhao, L.: Finite W-superalgebras for queer Lie algebras. J. Pure Appl. Algebra 218(7), 1184–1194 (2014)CrossRefzbMATHMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2014

Authors and Affiliations

  1. 1.Department of MathematicsNortheastern UniversityBostonUSA

Personalised recommendations