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Studies in Lie Theory pp 377–403Cite as

Differential operators and cohomology groups on the basic affine space

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

Summary

We study the ring of differential operators \( \mathcal{D} \) (X) on the basic affine space X = G/U of a complex semisimple group G with maximal unipotent subgroup U. One of the main results shows that the cohomology group H*(X \( \mathcal{O} \) X) decomposes as a finite direct sum of nonisomorphic simple \( \mathcal{D} \) (X)-modules, each of which is isomorphic to a twist of \( \mathcal{O} \) (X) by an automorphism of \( \mathcal{D} \) (X).

We also use \( \mathcal{D} \) (X) to study the properties of \( \mathcal{D} \) (Z) for highest weight varieties Z. For example, we prove that Z is \( \mathcal{D} \) -simple in the sense that \( \mathcal{O} \) (Z) is a simple \( \mathcal{D} \) (Z)-module and produce an irreducible G-module of differential operators on Z of degree −1 and specified order.

This paper is dedicated to Tony Joseph on the occasion of his 60thbirthday.

The second author was supported in part by the NSF through the grants DMS-9801148 and DMS-0245320. Part of this work was done while he was visiting the Mittag-Leffler Institute and he would like to thank the Institute for its financial support and hospitality.

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Author information

Authors and Affiliations

  1. Département de Mathématiques, Université de Brest, 29238, Brest cedex 3, France

    Thierry Levasseur

  2. Department of Mathematics, University of Michigan, Ann Arbor, MI, 48109-1043, USA

    J. T. Stafford

Authors
  1. Thierry Levasseur
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  2. J. T. Stafford
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Editor information

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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Levasseur, T., Stafford, J.T. (2006). Differential operators and cohomology groups on the basic affine space. 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_14

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