Coadjoint orbits, moment polytopes, and the Hilbert-Mumford criterion

Berenstein, Arkady; Sjamaar, Reyer

doi:10.1090/S0894-0347-00-00327-1

Coadjoint orbits, moment polytopes, and the Hilbert-Mumford criterion
HTML articles powered by AMS MathViewer

by Arkady Berenstein and Reyer Sjamaar

J. Amer. Math. Soc. 13 (2000), 433-466

DOI: https://doi.org/10.1090/S0894-0347-00-00327-1

Published electronically: January 31, 2000

PDF | Request permission

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.

References

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 1671192, DOI 10.4310/MRL.1998.v5.n6.a10
P. Belkale, Local systems on $\mathbb P^1-S$ for $S$ a finite set, preprint, University of Chicago, 1999.
I. N. Bernšteĭn, I. M. Gel′fand, and S. I. Gel′fand, Schubert cells, and the cohomology of the spaces $G/P$, Uspehi Mat. Nauk 28 (1973), no. 3(171), 3–26 (Russian). MR 0429933
Nicolas Bourbaki, Éléments de mathématique: groupes et algèbres de Lie, Masson, Paris, 1982 (French). Chapitre 9. Groupes de Lie réels compacts. [Chapter 9. Compact real Lie groups]. MR 682756
M. Brion, On the general faces of the moment polytope, Internat. Math. Res. Notices (1999), no. 4, 185–201.
Michel Demazure, Désingularisation des variétés de Schubert généralisées, Ann. Sci. École Norm. Sup. (4) 7 (1974), 53–88 (French). MR 354697
C. J. Everett Jr., Annihilator ideals and representation iteration for abstract rings, Duke Math. J. 5 (1939), 623–627. MR 13
William Fulton, Young tableaux, London Mathematical Society Student Texts, vol. 35, Cambridge University Press, Cambridge, 1997. With applications to representation theory and geometry. MR 1464693
—, 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.
V. Guillemin and S. Sternberg, Convexity properties of the moment mapping, Invent. Math. 67 (1982), no. 3, 491–513. MR 664117, DOI 10.1007/BF01398933
Robin Hartshorne, Algebraic geometry, Graduate Texts in Mathematics, No. 52, Springer-Verlag, New York-Heidelberg, 1977. MR 0463157
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 665160, DOI 10.1007/BF01393821
Uwe Helmke and Joachim Rosenthal, Eigenvalue inequalities and Schubert calculus, Math. Nachr. 171 (1995), 207–225. MR 1316359, DOI 10.1002/mana.19951710113
V. Guillemin and S. Sternberg, Convexity properties of the moment mapping. II, Invent. Math. 77 (1984), no. 3, 533–546. MR 759258, DOI 10.1007/BF01388837
Alexander A. Klyachko, Stable bundles, representation theory and Hermitian operators, Selecta Math. (N.S.) 4 (1998), no. 3, 419–445. MR 1654578, DOI 10.1007/s000290050037
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.
Manfred Krämer, Über Untergruppen kompakter Liegruppen als Isotropiegruppen bei linearen Aktionen, Math. Z. 147 (1976), no. 3, 207–224. MR 404530, DOI 10.1007/BF01214079
Peter Littelmann, A generalization of the Littlewood-Richardson rule, J. Algebra 130 (1990), no. 2, 328–368. MR 1051307, DOI 10.1016/0021-8693(90)90086-4
P.-L. Montagard, Sur les faces du cône associé au pléthysme, Comm. Algebra 26 (1998), no. 7, 2321–2336 (French). MR 1626618, DOI 10.1080/00927879808826278
D. Mumford, J. Fogarty, and F. Kirwan, Geometric invariant theory, 3rd ed., Ergebnisse der Mathematik und ihrer Grenzgebiete (2) [Results in Mathematics and Related Areas (2)], vol. 34, Springer-Verlag, Berlin, 1994. MR 1304906
Piotr Pragacz, Algebro-geometric applications of Schur $S$- and $Q$-polynomials, Topics in invariant theory (Paris, 1989/1990) Lecture Notes in Math., vol. 1478, Springer, Berlin, 1991, pp. 130–191. MR 1180989, DOI 10.1007/BFb0083503
Reyer Sjamaar, Convexity properties of the moment mapping re-examined, Adv. Math. 138 (1998), no. 1, 46–91. MR 1645052, DOI 10.1006/aima.1998.1739