On the Brylinski-Kostant filtration

Joseph, Anthony¹ ; Letzter, Gail² ; Zelikson, Shmuel³

¹ Department of Theoretical Mathematics, The Weizmann Institute of Science, Rehovot 76100, Israel and Institut de Mathématiques Fondamentales, Université Pierre et Marie Curie, 175 rue du Chevaleret, Plateau 7D, 75013 Paris Cedex, France
² Department of Mathematics, Virginia Tech, Blacksburg, Virginia 24061-0123
³ Laboratoire S.D.A.D., Département de Mathématiques, Campus II, Université de Caen, Boite Postale 5186, 14032 Caen Cedex, France

Journal of the American Mathematical Society, Tome 13 (2000) no. 4, pp. 945-970

Résumé

Let $\mathfrak g$ be a semisimple Lie algebra and $V$ a finite dimensional simple $\mathfrak g$ module. The Brylinski-Kostant (simply, BK) filtration on weight spaces of $V$ is defined by applying powers of a principle nilpotent element. It leads to a $q$-character of $V$. Through a result of B. Kostant the BK filtration of the zero weight space is determined by the so-called generalized exponents of $\mathfrak g$. Later R. K. Brylinski calculated the BK filtration on dominant weights of $V$ assuming a vanishing result for cohomology later established by B. Broer. The result could be expressed in terms of $q$ polynomials introduced by G. Lusztig. In the present work, Verma module maps are used to determine the BK filtration for all weights. To do this several filtrations are introduced and compared, a key point being the graded injectivity of the ring of differential operators on the open Bruhat cell viewed as a $\mathfrak g$ module under diagonal action. This replaces cohomological vanishing and thereby Brylinski’s result is given a new proof. The calculation for non-dominant weights uses the fact that the corresponding graded ring is a domain as well as a positivity result of G. Lusztig which ensures that there are no accidental cancellations. This method allows one to compare the BK filtrations in adjacent chambers.

Détail

DOI : 10.1090/S0894-0347-00-00347-7

Bibliographie

[1] Andersen, Henning Haahr, Jantzen, Jens Carsten Cohomology of induced representations for algebraic groups Math. Ann. 1984 487 525

[2] Bernå¡Teä­N, I. N., Gel′Fand, I. M., Gel′Fand, S. I. Structure of representations that are generated by vectors of highest weight Funkcional. Anal. i Priložen. 1971 1 9

[3] Borho, Walter, Brylinski, Jean-Luc Differential operators on homogeneous spaces. I. Irreducibility of the associated variety for annihilators of induced modules Invent. Math. 1982 437 476

[4] Broer, Bram Line bundles on the cotangent bundle of the flag variety Invent. Math. 1993 1 20

[5] Brylinski, Ranee Kathryn Limits of weight spaces, Lusztig’s 𝑞-analogs, and fiberings of adjoint orbits J. Amer. Math. Soc. 1989 517 533

[6] Brylinski, Ranee Kathryn Twisted ideals of the nullcone 1990 289 316

[7] Conze, Nicole Algèbres d’opérateurs différentiels et quotients des algèbres enveloppantes Bull. Soc. Math. France 1974 379 415

[8] Dixmier, Jacques Algèbres enveloppantes 1974

[9] Gabber, O., Joseph, A. On the Bernšteĭn-Gel′fand-Gel′fand resolution and the Duflo sum formula Compositio Math. 1981 107 131

[10] Griffiths, Phillip A. Hermitian differential geometry, Chern classes, and positive vector bundles 1969 185 251

[11] Hesselink, Wim H. Characters of the nullcone Math. Ann. 1980 179 182

[12] Jantzen, Jens Carsten Moduln mit einem höchsten Gewicht 1979

[13] Joseph, Anthony Enveloping algebras: problems old and new 1994 385 413

[14] Joseph, Anthony Quantum groups and their primitive ideals 1995

[15] Joseph, Anthony Orbital varieties, Goldie rank polynomials and unitary highest weight modules 1997 53 98

[16] Joseph, Anthony Sur l’annulateur d’un module de Verma 1998 237 300

[17] Joseph, Anthony A surjectivity theorem for rigid highest weight modules Invent. Math. 1988 567 596

[18] Joseph, A. A generalization of the Gelfand-Kirillov conjecture Amer. J. Math. 1977 1151 1165

[19] Kostant, Bertram The principal three-dimensional subgroup and the Betti numbers of a complex simple Lie group Amer. J. Math. 1959 973 1032

[20] Kostant, Bertram Lie group representations on polynomial rings Amer. J. Math. 1963 327 404

[21] Lusztig, George Singularities, character formulas, and a 𝑞-analog of weight multiplicities 1983 208 229

[22] Lusztig, George Introduction to quantum groups 1993