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Journal of the American Mathematical Society

Published by the American Mathematical Society, the Journal of the American Mathematical Society (JAMS) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6834 (online) ISSN 0894-0347 (print)

The 2020 MCQ for Journal of the American Mathematical Society is 4.79.

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On the Brylinski-Kostant filtration
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by Anthony Joseph, Gail Letzter and Shmuel Zelikson PDF
J. Amer. Math. Soc. 13 (2000), 945-970 Request permission

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
  • MR Author ID: 228201
  • 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
  • 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.
  • © Copyright 2000 American Mathematical Society
  • Journal: J. Amer. Math. Soc. 13 (2000), 945-970
  • MSC (2000): Primary 17B35
  • DOI: https://doi.org/10.1090/S0894-0347-00-00347-7
  • MathSciNet review: 1775740