The annihilation theorem for the completely reducible Lie superalgebras |
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Authors: | Maria Gorelik Emmanuel Lanzmann |
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Institution: | (1) Laboratoire de Mathématiques Fondamentales, Université Pierre et Marie Curie, 4, place Jussieu, F-75252 Paris Cedex 05, France (e-mail: Gorelik@math.jussieu.fr), FR;(2) Department of Theoretical Mathematics, The Weizmann Institute of Sciences, Rehovot 76100, Israel (e-mail: Lanzmann@wisdom.weizmann.ac.il), IL |
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Abstract: | A well known theorem of Duflo claims that the annihilator of a Verma module in the enveloping algebra of a complex semisimple
Lie algebra is generated by its intersection with the centre. For a Lie superalgebra this result fails to be true. For instance,
in the case of the orthosymplectic Lie superalgebra osp(1,2), Pinczon gave in Pi] an example of a Verma module whose annihilator
is not generated by its intersection with the centre of universal enveloping algebra. More generally, Musson produced in Mu1]
a family of such “singular” Verma modules for osp(1,2l) cases. In this article we give a necessary and sufficient condition
on the highest weight of a osp(1,2l)-Verma module for its annihilator to be generated by its intersection with the centre.
This answers a question of Musson. The classical proof of the Duflo theorem is based on a deep result of Kostant which uses
some delicate algebraic geometry reasonings. Unfortunately these arguments can not be reproduced in the quantum and super
cases. This obstruction forced Joseph and Letzter, in their work on the quantum case (see JL]), to find an alternativeapproach
to the Duflo theorem. Following their ideas, we compute the factorization of the Parthasarathy–Ranga-Rao–Varadarajan (PRV)
determinants. Comparing it with the factorization of Shapovalov determinants we find, unlike to the classical and quantum
cases, that the PRV determinant contains some extrafactors. The set of zeroes of these extrafactors is precisely the set of
highest weights of Verma modules whose annihilators are not generated by their intersection with the centre. We also find
an analogue of Hesselink formula (see He]) giving the multiplicity of every simple finite dimensional module in the graded
component of the harmonic space in the symmetric algebra.
Oblatum 1-IX-1998 & 4-XII-1998 / Published online: 10 May 1999 |
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