Journal of the American Mathematical Society

Author(s): Arkady Berenstein; Reyer Sjamaar
Journal: J. Amer. Math. Soc. 13 (2000), 433-466.
MSC (2000): Primary 53D20; Secondary 14L24
Posted: January 31, 2000
Retrieve article in: PDF
This article is available free of charge

Abstract: Consider a compact Lie group and a closed subgroup. Generalizing a result of Klyachko, we give a necessary and sufficient criterion for a coadjoint orbit of the subgroup to be contained in the projection of a given coadjoint orbit of the ambient group. The criterion is couched in terms of the ``relative'' Schubert calculus of the flag varieties of the two groups.

1.: S. Agnihotri and C. Woodward, Eigenvalues of products of unitary matrices and quantum Schubert calculus, Math. Res. Lett. 5 (1998), no. 6, 817-836. MR 2000a:14066
2.: P. Belkale, Local systems on $\mathbb P^1-S$ for a finite set, preprint, University of Chicago, 1999.
3.: I. Bernstein, I. Gelfand, and S. Gelfand, Schubert cells and the cohomology of the spaces , Uspehi Mat. Nauk 28 (1973), no. 3(171), 3-26 (Russian), English translation in Russian Math. Surveys 28 (1973), no. 3, 1-26. MR 55:2941
4.: N. Bourbaki, Groupes et algèbres de Lie, Éléments de mathématique, Masson, Paris, 1982. MR 84i:22001
5.: M. Brion, On the general faces of the moment polytope, Internat. Math. Res. Notices (1999), no. 4, 185-201. CMP 99:09
6.: M. Demazure, Désingularisation des variétés de Schubert généralisées, Ann. Sci. École Norm. Sup. (4) 7 (1974), 53-88. MR 50:7174
7.: E. B. Dynkin, Semisimple subalgebras of semisimple Lie algebras, Mat. Sbornik N. S. 30(72) (1952), 349-462 (Russian), English translation in American Mathematical Society Translations, Series 2, vol. 6, Amer. Math. Soc., Providence, R.I., 1957, pp. 111-244. MR 13:904c
8.: W. Fulton, Young tableaux, London Mathematical Society Student Texts, vol. 35, Cambridge University Press, Cambridge, 1997. MR 99f:05119
9.: -, Eigenvalues of sums of Hermitian matrices (after A. Klyachko), Séminaire Bourbaki, 50ème année, exposé no. 845 (Paris, 1997-98), Astérisque, vol. 252, Société Mathématique de France, Paris, 1998, pp. 255-269. CMP 99:13
10.: V. Guillemin and S. Sternberg, Convexity properties of the moment mapping, Invent. Math. 67 (1982), no. 3, 491-513. MR 83m:58037
11.: R. Hartshorne, Algebraic geometry, Graduate Texts in Mathematics, vol. 52, Springer-Verlag, New York, 1977. MR 57:3116
12.: G. J. Heckman, Projections of orbits and asymptotic behavior of multiplicities for compact connected Lie groups, Invent. Math. 67 (1982), no. 2, 333-356. MR 84d:22019
13.: U. Helmke and J. Rosenthal, Eigenvalue inequalities and Schubert calculus, Math. Nachr. 171 (1995), 207-225. MR 96b:15039
14.: F. Kirwan, Convexity properties of the moment mapping, III, Invent. Math. 77 (1984), no. 3, 547-552. MR 86b:58042b
15.: A. Klyachko, Stable bundles, representation theory and Hermitian operators, Selecta Math. (N.S.) 4 (1998), no. 3, 419-445. MR 2000b:14054
16.: A. Knutson and T. Tao, The honeycomb model of $\mathrm{GL}_n(\mathbf C)$ tensor products. I. Proof of the saturation conjecture, J. Amer. Math. Soc. 12 (1999), no. 4, 1055-1090. CMP 99:15
17.: M. Krämer, Über Untergruppen kompakter Liegruppen als Isotropiegruppen bei linearen Aktionen, Math. Z. 147 (1976), no. 3, 207-224. MR 53:8330
18.: P. Littelmann, A generalization of the Littlewood-Richardson rule, J. Algebra 130 (1990), no. 2, 328-368. MR 91f:22023
19.: P.-L. Montagard, Sur les faces du cône associé au pléthysme, Comm. Algebra 26 (1998), no. 7, 2321-2336. MR 99e:20057
20.: D. Mumford, J. Fogarty, and F. Kirwan, Geometric invariant theory, third ed., Ergebnisse der Mathematik und ihrer Grenzgebiete, 2. Folge, vol. 34, Springer-Verlag, Berlin, 1994. MR 95m:14012
21.: P. Pragacz, Algebro-geometric applications of Schur - and -polynomials, Topics in Invariant Theory (Paris, 1989-1990) (M.-P. Malliavin, ed.), Lecture Notes in Mathematics, vol. 1478, Springer-Verlag, Berlin, 1991, pp. 130-191. MR 93h:05170
22.: R. Sjamaar, Convexity properties of the moment mapping re-examined, Adv. Math. 138 (1998), no. 1, 46-91. MR 2000a:53148

Retrieve articles in Journal of the American Mathematical Society with MSC (2000): 53D20, 14L24

Arkady Berenstein
Affiliation: Department of Mathematics, Cornell University, Ithaca, New York 14853-4201
Address at time of publication: Department of Mathematics, Harvard University, Cambridge, Massachusetts 02138-2901
Email: arkadiy@math.harvard.edu

			Journals Home Search Author Info Subscribe Tech Support Help
ISSN: 1088-6834(e) ISSN: 0894-0347(p)

Previous issue \| Table of contents \| Next issue Articles in press \| Recently posted articles \| Most recent issue \| All issues Previous Article


	Comments: webmaster@ams.org © Copyright 2009, American Mathematical Society Privacy Statement		Search the AMS