Orbits and characters associated to highest weight representations
HTML articles powered by AMS MathViewer
- by David H. Collingwood
- Proc. Amer. Math. Soc. 114 (1992), 1157-1165
- DOI: https://doi.org/10.1090/S0002-9939-1992-1074750-6
- PDF | Request permission
Abstract:
We relate two different orbit decompositions of the flag variety. This allows us to pass from the closed formulas of Boe, Enright, and Shelton for the formal character of an irreducible highest weight representation to closed formulas for the distributional character written as a sum of characters of generalized principal series representations. Otherwise put, we give a dictionary between certain Lusztig-Vogan polynomials arising in Harish-Chandra module theory and the Kazhdan-Lusztig polynomials associated to a relative category $\mathcal {O}$ of Hermitian symmetric type.References
- I. Bernstein, I. Gel’fand and S. Gel’fand, Schubert cells and the cohomology of the spaces $G/P$, Russian Math. Surveys, no. 28, (1973), 1-26.
- Brian D. Boe, Kazhdan-Lusztig polynomials for Hermitian symmetric spaces, Trans. Amer. Math. Soc. 309 (1988), no. 1, 279–294. MR 957071, DOI 10.1090/S0002-9947-1988-0957071-2 B. Boe and D. Collingwood, Enright-Shelton theory and Vogan’s problem for generalized principal series, preprint.
- Luis G. Casian and David H. Collingwood, The Kazhdan-Lusztig conjecture for generalized Verma modules, Math. Z. 195 (1987), no. 4, 581–600. MR 900346, DOI 10.1007/BF01166705
- 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
- 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
- Sigurdur Helgason, Differential geometry, Lie groups, and symmetric spaces, Graduate Studies in Mathematics, vol. 34, American Mathematical Society, Providence, RI, 2001. Corrected reprint of the 1978 original. MR 1834454, DOI 10.1090/gsm/034
- George Lusztig and David A. Vogan Jr., Singularities of closures of $K$-orbits on flag manifolds, Invent. Math. 71 (1983), no. 2, 365–379. MR 689649, DOI 10.1007/BF01389103
- Toshihiko Matsuki, The orbits of affine symmetric spaces under the action of minimal parabolic subgroups, J. Math. Soc. Japan 31 (1979), no. 2, 331–357. MR 527548, DOI 10.2969/jmsj/03120331
- Toshihiko Matsuki and Toshio Ōshima, Embeddings of discrete series into principal series, The orbit method in representation theory (Copenhagen, 1988) Progr. Math., vol. 82, Birkhäuser Boston, Boston, MA, 1990, pp. 147–175. MR 1095345
- Robert A. Proctor, Classical Bruhat orders and lexicographic shellability, J. Algebra 77 (1982), no. 1, 104–126. MR 665167, DOI 10.1016/0021-8693(82)90280-0
- R. W. Richardson and T. A. Springer, The Bruhat order on symmetric varieties, Geom. Dedicata 35 (1990), no. 1-3, 389–436. MR 1066573, DOI 10.1007/BF00147354 D. Vogan, Representations of reductive Lie groups, Birkhäuser, Boston, 1980.
- David A. Vogan, Irreducible characters of semisimple Lie groups. III. Proof of Kazhdan-Lusztig conjecture in the integral case, Invent. Math. 71 (1983), no. 2, 381–417. MR 689650, DOI 10.1007/BF01389104
- David A. Vogan Jr., Irreducible characters of semisimple Lie groups. IV. Character-multiplicity duality, Duke Math. J. 49 (1982), no. 4, 943–1073. MR 683010
Bibliographic Information
- © Copyright 1992 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 114 (1992), 1157-1165
- MSC: Primary 22E47; Secondary 17B10, 20G05, 22E46
- DOI: https://doi.org/10.1090/S0002-9939-1992-1074750-6
- MathSciNet review: 1074750