Resolutions and Hilbert series of the unitary highest weight modules of the exceptional groups
HTML articles powered by AMS MathViewer
- by Thomas J. Enright and Markus Hunziker
- Represent. Theory 8 (2004), 15-51
- DOI: https://doi.org/10.1090/S1088-4165-04-00215-8
- Published electronically: April 15, 2004
- PDF | Request permission
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.References
- [BGG]BGG Bernstein, I. N., Gelfand I. M. and Gelfand, S. I., Differential equations on the base affine space and a study of ${\mathfrak {g}}$-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.
- David H. Collingwood, The ${\mathfrak {n}}$-homology of Harish-Chandra modules: generalizing a theorem of Kostant, Math. Ann. 272 (1985), no. 2, 161–187. MR 796245, DOI 10.1007/BF01450563
- Mark G. Davidson, Thomas J. Enright, and Ronald J. Stanke, Differential operators and highest weight representations, Mem. Amer. Math. Soc. 94 (1991), no. 455, iv+102. MR 1081660, DOI 10.1090/memo/0455
- Thomas J. Enright, Analogues of Kostant’s ${\mathfrak {u}}$-cohomology formulas for unitary highest weight modules, J. Reine Angew. Math. 392 (1988), 27–36. MR 965055, DOI 10.1515/crll.1988.392.27
- Thomas Enright, Roger Howe, and Nolan Wallach, A classification of unitary highest weight modules, Representation theory of reductive groups (Park City, Utah, 1982) Progr. Math., vol. 40, Birkhäuser Boston, Boston, MA, 1983, pp. 97–143. MR 733809, DOI 10.1007/978-1-4684-6730-7_{7} [EH]EH Enright, T. J. and Hunziker, M., Resolutions and Hilbert series of determinantal varieties and unitary highest weight modules, to appear in J. Algebra.
- Thomas J. Enright and Anthony Joseph, An intrinsic analysis of unitarizable highest weight modules, Math. Ann. 288 (1990), no. 4, 571–594. MR 1081264, DOI 10.1007/BF01444551
- Thomas J. Enright and Brad Shelton, Categories of highest weight modules: applications to classical Hermitian symmetric pairs, Mem. Amer. Math. Soc. 67 (1987), no. 367, iv+94. MR 888703, DOI 10.1090/memo/0367
- Thomas J. Enright and Brad Shelton, Highest weight modules for Hermitian symmetric pairs of exceptional type, Proc. Amer. Math. Soc. 106 (1989), no. 3, 807–819. MR 961404, DOI 10.1090/S0002-9939-1989-0961404-7 [EW]EW Enright, T. J. and Willenbring, J. F., Hilbert series, Howe duality and branching for classical groups, to appear in Math. Ann.
- J. Lepowsky, A generalization of the Bernstein-Gelfand-Gelfand resolution, J. Algebra 49 (1977), no. 2, 496–511. MR 476813, DOI 10.1016/0021-8693(77)90254-X
- Kyo Nishiyama, Hiroyuki Ochiai, Kenji Taniguchi, Hiroshi Yamashita, and Shohei Kato, Nilpotent orbits, associated cycles and Whittaker models for highest weight representations, Société Mathématique de France, Paris, 2001. Astérisque No. 273 (2001). MR 1845713
- Nolan R. Wallach, The analytic continuation of the discrete series. I, II, Trans. Amer. Math. Soc. 251 (1979), 1–17, 19–37. MR 531967, DOI 10.1090/S0002-9947-1979-0531967-2
Bibliographic 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
- MR Author ID: 601797
- Email: hunziker@math.uga.edu
- Received by editor(s): October 22, 2003
- Published electronically: April 15, 2004
- © Copyright 2004 American Mathematical Society
- 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
- MathSciNet review: 2048586