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Journal of the American Mathematical Society
Journal of the American Mathematical Society
ISSN 1088-6834(online) ISSN 0894-0347(print)

 

On the Brylinski-Kostant filtration


Authors: Anthony Joseph, Gail Letzter and Shmuel Zelikson
Journal: J. Amer. Math. Soc. 13 (2000), 945-970
MSC (2000): Primary 17B35
Published electronically: July 20, 2000
MathSciNet review: 1775740
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Abstract:

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.


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Additional Information

Anthony Joseph
Affiliation: 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
Email: joseph@wisdom.weizmann.ac.il

Gail Letzter
Affiliation: Department of Mathematics, Virginia Tech, Blacksburg, Virginia 24061-0123
Email: letzter@calvin.math.vt.edu

Shmuel Zelikson
Affiliation: Laboratoire S.D.A.D., Département de Mathématiques, Campus II, Université de Caen, Boite Postale 5186, 14032 Caen Cedex, France
Email: zelik@unicaen.fr

DOI: http://dx.doi.org/10.1090/S0894-0347-00-00347-7
PII: S 0894-0347(00)00347-7
Received by editor(s): September 27, 1999
Received by editor(s) in revised form: May 1, 2000
Published electronically: July 20, 2000
Additional Notes: This work was supported in part by EC TMR network “Algebraic Lie Representations” grant no. ERB FMRX-CT97-0100 and grant no. 7773 from the United States-Israel Binational Science Foundation (BSF), Jerusalem, Israel. The second author was also supported in part by NSF grant no. DMS-9753211 and NSA grant no. MDA 904-99-1-0033.
Article copyright: © Copyright 2000 American Mathematical Society