Abstract
Let G be a symplectic or special orthogonal group, let H be a connected reductive subgroup of G, and let X be a flag variety of G. We classify all triples (G, H, X) such that the natural action of H on X is spherical. For each of these triples, we determine the restrictions to H of all irreducible representations of G realized in spaces of sections of homogeneous line bundles on X.
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D. Akhiezer, D. Panyushev, Multiplicities in the branching rules and the complexity of homogeneous spaces, Mosc. Math. J. 2 (2002), no. 1, 17–33.
Р. С. Авдеев, А. В. Петухов, Сферические действия на многообразиях флагов, Матем. сб. 205 (2014), вып. 9, 1223–1263. Engl. transl.: R. S. Avdeev, A. V. Petukhov, Spherical actions on flag varieties, Sb. Math. 205 (2014), no. 9, 1223–1263.
R. Avdeev, A. Petukhov, Branching rules related to spherical actions on flag varieties, Algebr. Represent. Theory 23 (2020), no. 3, 541–581.
C. Benson, G. Ratcliff, A classiffication of multiplicity free actions, J. Algebra 181 (1996), no. 1, 152–186.
N. Bourbaki, Groupes et Algèbres de Lie, Chap. IV: Groupes de Coxeter et systèmes de Tits, Chap. V: Groupes engendrés par des réflexions, Chap. VI: Systèmes de racines, Actualités Scientifiques et Industrielles, No. 1337, Hermann, Paris, 1968.
N. Bourbaki, Groupes et Algèbres de Lie, Chap. VII: Sous-algèbres de Cartan, éléments réguliers, Chap. VIII: Algèbres de Lie semi-simples déployées, Actualités Scientifiques et Industrielles, No. 1364, Hermann, Paris, 1975.
M. Brion, Représentations exceptionnelles des groupes semi-simples, Ann. Sci. École Norm. Sup. (4) 18 (1985), no. 2, 345–387.
D. H. Collingwood, W. M. McGovern, Nilpotent Orbits in Semisimple Lie Algebras, Van Nostrand Reinhold, New York, 1993.
M. Gerstenhaber, Dominance over the classical groups, Ann. of Math. 74 (1961), no. 3, 532–569.
X. He, K. Nishiyama, H. Ochiai, Y. Oshima, On orbits in double flag varieties for symmetric pairs, Transform. Groups 18 (2013), no. 4, 1091–1136.
W. Hesselink, Singularities in the nilpotent scheme of a classical group, Trans. Amer. Math. Soc. 222 (1976), 1–32.
R. Howe, T. Umeda, The Capelli identity, the double commutant theorem, and multiplicity-free actions, Math. Ann. 290 (1991), no. 3, 565–619.
J. C. Jantzen, Representations of Algebraic Groups, Pure and Applied Mathematics, Vol. 131, Academic Press, Inc., Boston, MA, 1987.
D. S. Johnston, R. W. Richardson, Conjugacy classes in parabolic subgroups of semisimple algebraic groups, II, Bull. London Math. Soc. 9 (1977), no. 3, 245–250.
V. G. Kac, Some remarks on nilpotent orbits, J. Algebra 64 (1980), no. 1, 190–213.
G. Kempken, Induced conjugacy classes in classical Lie-algebras, Abh. Math. Sem. Univ. Hamburg 53 (1983), 53–83.
B. Kimelfeld, Homogeneous domains on ag manifolds, J. Math. Anal. Appl. 121 (1987), no. 2, 506–588.
F. Knop, Some remarks on multiplicity free spaces, in: Representation Theories and Algebraic Geometry, NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci., Vol. 514, Dordrecht, Springer Netherlands, 1998, pp. 301–317.
A. S. Leahy, A classification of multiplicity free representations, J. Lie Theory 8 (1998), no. 2, 367–391.
LiE, A Computer Algebra Package for Lie Group Computations, available at http://wwwmathlabo.univ-poitiers.fr/~maavl/LiE.
LiE Manual, available at http://wwwmathlabo.univ-poitiers.fr/~maavl/pdf/LiE-manual.pdf.
P. Littelmann, On spherical double cones, J. Algebra 166 (1994), no. 1, 142–157.
I. V. Losev, Algebraic Hamiltonian actions, Math. Z. 263 (2009), no. 3, 685–723.
P. Magyar, J. Weyman, A. Zelevinsky, Multiple flag varieties of finite type, Adv. Math. 141 (1999), 97–118.
P. Magyar, J. Weyman, A. Zelevinsky, Symplectic multiple flag varieties of finite type, J. Algebra 230 (2000), no. 1, 245–265.
А. И. Мальцев, О полупростых подгруппах групп Ли, Изв. АН СССР. Сер. матем. 8 (1944), вып. 4, 143–174. Engl. transl.: A. I. Malcev, On semi-simple subgroups of Lie groups, Amer. Math. Soc. transl., 1950, no. 33, 43 pp.
B. Niemann, Spherical affine cones in exceptional cases and related branching rules, preprint (2011), see arXiv:1111.3823 (2011).
D. I. Panyushev, On the conormal bundle of a G-stable subvariety, Manuscripta Math. 99 (1999), no. 2, 185–202.
D. I. Panyushev, O. S. Yakimova, Poisson-commutative subalgebras and complete integrability on non-regular coadjoint orbits and flag varieties, Math. Z. 295 (2020), no. 1-2, 101–127.
A. V. Petukhov, Bounded reductive subalgebras of \( {\mathfrak{sl}}_n \), Transform. Groups 16 (2011), no. 4, 1173–1182.
Е. В. Пономарева, Классификация двойных многообразий флагов сложности 0 и 1, Изв. РАН. Сер. матем. 77 (2013), вып. 5, 155–178. Engl. transl.: E. V. Ponomareva, Classification of double flag varieties of complexity 0 and 1, Izv. Math. 77 (2013), no. 5, 998–1020.
Е. В. Пономарёва, Инварианты колец Кокса двойных многообразий флагов малой сложности классических групп, Труды Моск. матем. общ-ва 76 (2015), вып. 1, 85–150. Engl. transl.: E. V. Ponomareva, Invariants of the Cox rings of low-complexity double flag varieties for classical groups, Trans. Moscow Math. Soc. 2015, 71–133.
Е. В. Пономарёва, Инварианты колец Кокса двойных многообразий флагов малой сложности для особых групп, матем. сб. 208 (2017), вып. 5, 129–166. Engl. transl.: E. V. Ponomareva, Invariants of the Cox rings of double flag varieties of low complexity for exceptional groups, Sb. Math. 208 (2017), no. 5, 707–742.
R. W. Richardson, Conjugacy classes in parabolic subgroups of semisimple algebraic groups, Bull. London Math. Soc. 6 (1974), no. 1, 21–24.
T. Springer, R. Steinberg, Conjugacy classes, in: Seminar on Algebraic Groups and Related Finite Groups, Lecture Notes in Math., Vol. 131, Springer, New York, 1970, pp. 167–266.
R. Steinberg, Endomorphisms of Linear Algebraic Groups, Memoirs of the American Mathematical Society, No. 80, American Mathematical Society, Providence, R.I., 1968.
J. R. Stembridge, Multiplicity-free products and restrictions of Weyl characters, Represent. Theory 7 (2003), 404–439.
D. A. Timashev, Homogeneous Spaces and Equivariant Embeddings, Encycl. Math. Sci., Vol. 138, Springer-Verlag, Berlin, 2011.
Э. Б. Винберг, Сложность действий редуктивных групп, Функц. анализ. и его прилож. 20 (1986), вып. 1, 1–13. Engl. transl.: E. B. Vinberg, Complexity of actions of reductive groups, Funct. Anal. Appl. 20 (1986), no. 1, 1–11.
Э. Б. Винберг, Б. Н. Кимельфельд, Однородные обласми на флаговых многообразиях и сферические подгруппы полупростых групп Ли, Функц. анализ и его прилож. 12 (1978), вып. 3, 12–19. Engl. transl.: E. B. Vinberg, B. N. Kimel’fel’d, Homogeneous domains on flag manifolds and spherical sub-groups of semisimple Lie groups, Funct. Anal. Appl. 12 (1978), no. 3, 168–174.
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AVDEEV, R., PETUKHOV, A. SPHERICAL ACTIONS ON ISOTROPIC FLAG VARIETIES AND RELATED BRANCHING RULES. Transformation Groups 26, 719–774 (2021). https://doi.org/10.1007/s00031-020-09593-1
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DOI: https://doi.org/10.1007/s00031-020-09593-1