Resolutions and Hilbert series of the unitary highest weight modules of the exceptional groups
Authors:
Thomas J. Enright and Markus Hunziker
Journal:
Represent. Theory 8 (2004), 1551
MSC (2000):
Primary 22E47, 17B10, 14M12, 13D02
Published electronically:
April 15, 2004
MathSciNet review:
2048586
Fulltext PDF Free Access
Abstract 
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Additional Information
Abstract: We give a sufficient criterion on a highest weight module of a semisimple Lie algebra to admit a resolution in terms of sums of modules induced from a parabolic subalgebra. In particular, we show that all unitary highest weight modules admit such a resolution. As an application of our results we compute (minimal) resolutions and explicit formulas for the Hilbert series of the unitary highest weight modules of the exceptional groups.
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 Davidson, M.G., Enright, T.J. and Stanke, R.J., Differential operators and highest weight representations, Memoirs AMS 94 (1991), 161187. MR 92c:22034
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 [EH]
 Enright, T.J. and Hunziker, M., Resolutions and Hilbert series of determinantal varieties and unitary highest weight modules, to appear in J. Algebra.
 [EJ]
 Enright, T.J. and Joseph, A., An intrinsic analysis of unitary highest weight modules, Math. Ann. 288 (1990), 571594. MR 91m:17005
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 Enright, T.J. and Shelton, B., Categories of highest weight modules: applications to classical Hermitian symmetric pairs, Memoirs AMS 67 (1987). MR 88f:22052
 [ES2]
 , Highest weight modules for Hermitian symmetric pairs of exceptional type, Proc. AMS 106 (1989), 807819. MR 89m:17010
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 Enright, T.J. and Willenbring, J.F., Hilbert series, Howe duality and branching for classical groups, to appear in Math. Ann.
 [L]
 Lepowsky, J., A generalization of the BernsteinGelfandGelfand resolution, J. Algebra 49 (1977), 496511. MR 57:16367
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 [W]
 Wallach, N.R., The analytic continuation of the discrete series I, II, Trans. Amer. Math. Soc. 251 (1979), 117, 1937. MR 81a:22009
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Additional Information
Thomas J. Enright
Affiliation:
Department of Mathematics, University of California at San Diego, La Jolla, California 920930112
Email:
tenright@math.ucsd.edu
Markus Hunziker
Affiliation:
Department of Mathematics, University of Georgia, Athens, Georgia 306027403
Email:
hunziker@math.uga.edu
DOI:
http://dx.doi.org/10.1090/S1088416504002158
PII:
S 10884165(04)002158
Keywords:
Highest weight modules,
minimal resolutions,
Hilbert series
Received by editor(s):
October 22, 2003
Published electronically:
April 15, 2004
Article copyright:
© Copyright 2004
American Mathematical Society
