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Modules with a Demazure flag

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Studies in Lie Theory

Part of the book series: Progress in Mathematics ((PM,volume 243))

Summary

A Demazure module can be described as the space of global sections of a suitable line bundle on a Schubert variety. A problem posed by the author in 1985 was to show that the tensor product of a one-dimensional Demazure module with an arbitrary one admits a Demazure flag, that is, a filtration whose quotients are Demazure modules. This was shown by P. Polo (who called such filtrations “excellent”) in a large number of cases including positive characteristic and by O. Mathieu for all semisimple algebraic groups first in zero characteristic and later in arbitrary characteristic.

This paper settles this question in the context of a Kac-Moody algebra with symmetric simply-laced Cartan datum and in arbitrary characteristic. The method combines the corresponding “combinatorial excellent filtration” established independently by P. Littelmann et al. and the author with the globalization techniques of G. Lusztig and M. Kashiwara. In principle the method applies to an arbitrary symmetrizable Kac-Moody algebra; but for technical reasons it is necessary to use a positivity result of Lusztig which applies to only the simply-laced case.

To Denise

Work supported by European Community RTN network “Liegrits”, Grant No. MRTN-CT-2003-505078.

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Authors and Affiliations

  1. Department of Mathematics, The Weizmann Institute of Science, Rehovot, 76100, Israel

    Anthony Joseph (The Donald Frey Professorial Chair)

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  1. Anthony Joseph
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Editors and Affiliations

  1. School of Mathematical Sciences, Tel Aviv University, Ramat Aviv, Tel Aviv, 69978, Israel

    Joseph Bernstein

  2. Department of Mathematics, University of Haifa, Mount Carmel, Haifa, 31905, Israel

    Vladimir Hinich  & Anna Melnikov  & 

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© 2006 Birkhäuser Boston

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Joseph, A. (2006). Modules with a Demazure flag. In: Bernstein, J., Hinich, V., Melnikov, A. (eds) Studies in Lie Theory. Progress in Mathematics, vol 243. Birkhäuser Boston. https://doi.org/10.1007/0-8176-4478-4_8

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