zbMATH — the first resource for mathematics

Restrictions of generalized Verma modules to symmetric pairs. (English) Zbl 1257.22014
The branching problem for generalized Verma modules with respect to reductive symmetric pairs \((g,g')\) is studied. This problem in representation theory means to determine how irreducible modules decompose when restricted to subalgebras. A necessary and sufficient condition on the triple \((g,g',p)\) such that the restriction \(X/{}_g{}_{\prime}\) always contains \(g'\)-modules for any \(g\)-module \(X\) in a parabolic Bernstein-Gelfand-Gelfand category \(O^p\) is given. The results are obtained for the Gelfand-Kirillov dimension of any simple module occurring in a simple generalized Verma module. The cases of parabolic subalgebras with \(p\) or Heisenberg nilpotent radicals are presented as illustrative examples. A complete classification of the triples \((g,p,g^\tau)\), with \(\tau\) an involutive automorphism of the Lie algebra \(g\), is given. It is shown that the restriction \(X/{}_g{}_{\prime}\) is generically multiplicity free for any \(p\) and \(X\in\) \(O^p\) if and only if the pair \((g,g')\) is isomorphic to \((A_n,A{}_n{}_-{}_1)\), \((B_n,D_n)\) or \((D_n{}_+{}_1,B{}_n)\). Explicit branching laws for such pairs are also presented.

22E47 Representations of Lie and real algebraic groups: algebraic methods (Verma modules, etc.)
22F30 Homogeneous spaces
53C35 Differential geometry of symmetric spaces
Full Text: DOI arXiv
[1] C. Benson, G. Ratcliff, A classification of multiplicity free actions, J. Algebra 181 (1996), 152–186. · Zbl 0869.14021
[2] И. Н. Бернштейн, И. М. Гельфанд, С. И. Гельфанд, Об одной каmегорuu $ \(\backslash\)mathfrak{g} $ -модулей, Функц. анализ и его прилож, 10 (1976), no. 2, 1–8. Engl. transl.: I. N. Bernshtein, I. M. Gel’fand, S. I. Gel’fand, Category of $ \(\backslash\)mathfrak{g} $ -modules, Funct. Anal. Appl. 10 (1976), no. 2, 87–92. · Zbl 1222.11084
[3] N. Conze-Berline, M. Duo, Sur les représentations induites des groupes semisimples complexes, Compositio Math. 34 (1977), 307–336. · Zbl 0389.22016
[4] T. J. Enright, F. Willenbring, Hilbert series, Howe duality and branching for classical groups, Ann. Math. (2) 159 (2004), 337–375. · Zbl 1087.22011
[5] J. C. Jantzen, Einhüllende Algebren Halbeinfacher Lie-Algebren, Ergebnisse der Mathematik und ihrer Grenzgebiete (3), 3, Springer–Verlag, Berlin, 1983. · Zbl 0541.17001
[6] T. Kobayashi, Discrete decomposability of the restriction of $ {{\(\backslash\)text{A}}_\(\backslash\)mathfrak{q}}\(\backslash\)left( \(\backslash\)lambda \(\backslash\)right) $ with respect to reductive subgroups II–micro-local analysis and asymptotic K-support, Ann. Math. (2) 147 (1998), 709–729. · Zbl 0910.22016
[7] T. Kobayashi, Discrete decomposability of the restriction of $ {{\(\backslash\)text{A}}_\(\backslash\)mathfrak{q}}\(\backslash\)left( \(\backslash\)lambda \(\backslash\)right) $ with respect to reductive subgroups III–restriction of Harish-Chandra modules and associated varieties, Invent. Math. 131 (1998), 229–256. · Zbl 0907.22016
[8] T. Kobayashi, Multiplicity-free theorems of the restrictions of unitary highest weight modules with respect to reductive symmetric pairs, in: Representation Theory and Automorphic Forms, Progress in Mathematics, Vol. 255, Birkhäuser Boston, Boston, MA, 2008, pp. 45–109. · Zbl 1304.22013
[9] T. Kobayashi, Visible actions on symmetric spaces, Transformation Groups 12 (2007), pp. 671–694. · Zbl 1147.53041
[10] T. Kobayashi, B. sted, P. Somberg, V. Souček, Branching laws for Verma modules and applications in parabolic geometry, Part I, in preparation.
[11] T. Matsuki, The orbits of affine symmetric spaces under the action of minimal parabolic subgroups, J. Math. Soc. Japan 31 (1979), 331–357. · Zbl 0396.53025
[12] T. Matsuki, Orbits on affine symmetric spaces under the action of parabolic subgroups, Hiroshima Math. J. 12 (1982), 307–320. · Zbl 0495.53049
[13] R. Richardson, G. Röhrle, R. Steinberg, Parabolic subgroup with abelian unipotent radical, Invent. Math. 110 (1992), 649–671. · Zbl 0786.20029
[14] W. Schmid, Die Randwerte holomorphe Funktionen auf hermetisch symmetrischen Raumen, Invent. Math. 9 (1969–70), 61–80. · Zbl 0219.32013
[15] D. Vogan, Associated varieties and unipotent representations, in: Harmonic Analysis on Reductive Groups (Brunswick, ME, 1989), 315–388, Progress in Mathematics, Vol. 101, Birkhaüser Boston, Boston, MA, 1991, pp. 315–388. · Zbl 0832.22019
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.