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), 15-51

MSC (2000):
Primary 22E47, 17B10, 14M12, 13D02

DOI:
https://doi.org/10.1090/S1088-4165-04-00215-8

Published electronically:
April 15, 2004

MathSciNet review:
2048586

Full-text PDF

Abstract | References | Similar Articles | 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.

**[BGG]**Bernstein, I.N., Gelfand I.M. and Gelfand, S.I.,*Differential equations on the base affine space and a study of**-modules*, Lie Groups and their Representations, Summer School of the Bolyai János Math. Soc. edited by I. M. Gelfand, Halsted Press, Division of John Wiley & Sons, New York, 1975, 21-64.**[C]**Collingwood, D.,*The**-homology of Harish-Chandra modules: generalizing a theorem of Kostant*, Math. Ann.**272**(1985). MR**87a:22027****[DES]**Davidson, M.G., Enright, T.J. and Stanke, R.J.,*Differential operators and highest weight representations*, Memoirs AMS**94**(1991), 161-187. MR**92c:22034****[E]**Enright, T.J.,*Analogues of Kostant's**-cohomology formulas for unitary highest weight modules*, J. Reine Angew. Math.**392**(1988), 27-36. MR**89m:22022****[EHW]**Enright, T.J., Howe, R. and Wallach, N.R.,*A classification of unitary highest weight modules*, Representation theory of reductive groups (Park City, Utah, 1982), Progr. Math.**40**(1983), 97-143. MR**86c:22028****[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), 571-594. MR**91m:17005****[ES1]**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), 807-819. MR**89m:17010****[EW]**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 Bernstein-Gelfand-Gelfand resolution*, J. Algebra**49**(1977), 496-511. MR**57:16367****[NOTYK]**Nishiyama, K., Ochiai, H., Taniguchi, K., Yamashita, H., Kato, S.,*Nilpotent orbits, associated cycles and Whittaker models for highest weigh representations*, Asterisque**273**(2001). MR**2002b:22025****[W]**Wallach, N.R.,*The analytic continuation of the discrete series I, II*, Trans. Amer. Math. Soc.**251**(1979), 1-17, 19-37. MR**81a:22009**

Retrieve articles in *Representation Theory of the American Mathematical Society*
with MSC (2000):
22E47,
17B10,
14M12,
13D02

Retrieve articles in all journals with MSC (2000): 22E47, 17B10, 14M12, 13D02

Additional Information

**Thomas J. Enright**

Affiliation:
Department of Mathematics, University of California at San Diego, La Jolla, California 92093-0112

Email:
tenright@math.ucsd.edu

**Markus Hunziker**

Affiliation:
Department of Mathematics, University of Georgia, Athens, Georgia 30602-7403

Email:
hunziker@math.uga.edu

DOI:
https://doi.org/10.1090/S1088-4165-04-00215-8

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