Transformation Groups

, Volume 22, Issue 2, pp 403–451 | Cite as




In the present article, we combine some techniques in harmonic analysis together with the geometric approach given by modules over sheaves of rings of twisted differential operators (\( \mathcal{D} \) -modules), and reformulate the composition series and branching problems for objects in the Bernstein–Gelfand–Gelfand parabolic category \( {\mathcal{O}}^{\mathfrak{p}} \) geometrically realized on certain orbits in the generalized flag manifolds. The general framework is then applied to the scalar generalized Verma modules supported on the closed Schubert cell of the generalized flag manifold G / P for G = SL(n + 2, ℂ) and P the Heisenberg parabolic subgroup; and algebraic analysis gives a complete classification of \( {\mathfrak{g}}_r^{\prime } \) -singular vectors for all \( {\mathfrak{g}}_r^{\prime }=\mathfrak{s}\mathfrak{l}\left(n-r+2,\mathbb{C}\right)\kern1em \subset \mathfrak{g}=\mathfrak{s}\mathfrak{l}\left(n+2,\mathbb{C}\right),n-r>21 \). A consequence of our results is that we classify SL(n − r + 2, ) -covariant differential operators acting on homogeneous line bundles over the complexification of the odd-dimensional CR-sphere S 2n+1 and valued in homogeneous vector bundles over the complexification of the CR-subspheres S 2(n-r)+1.


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  1. [1]
    A. Čap, J. Slovák, Parabolic Geometries I: Background and General Theory, Mathematical Surveys and Monographs, Vol. 154, American Mathematical Society, Providence, 2009.Google Scholar
  2. [2]
    L. Barchini, A. C. Kable, R. Zierau, Conformally invariant systems of differential equations and prehomogeneous vector spaces of Heisenberg parabolic type, Publ. Res. Inst. Math. Sci. 44 (2008), no. 3, 749–835.MathSciNetCrossRefzbMATHGoogle Scholar
  3. [3]
    A. A. Beilinson, J. N. Bernstein, Localisation de \( \mathfrak{g}- modules \), C. R. Acad. Sci. Paris Sér. I Math. 292 (1981), 15–18.Google Scholar
  4. [4]
    И. H. Бepнштeйн, И. M. Гeльфaнд, C. И. Гeльфaнд, Cтpуктуpa пpeдcтaвлeний, пopoждённыx вeктopaми cтapшeгo вeca, Функц aнaлиз и eгo пpил. 5 (1971), vyp. 1, 1–9. Engl. transl.: J. N. Bernstein, I. M. Gelfand, S. I. Gelfand, Structure of representations generated by vectors of highest weight, Functional Anal. Appl. 5 (1971), no. 1, 1–8.Google Scholar
  5. [5]
    T. P. Branson, L. Fontana, C. Morpurgo, Moser–Trudinger and Beckner–Onofris inequalities on the CR sphere, Ann. of Math. (2) 177 (2013), no. 1, 1–52.Google Scholar
  6. [6]
    D. H. Collingwood, B. Shelton, A duality theorem for extensions of induced highest weight modules, Pacific J. Math. 146 (1990), no. 2, 227–237.MathSciNetCrossRefzbMATHGoogle Scholar
  7. [7]
    C. L. Fefferman, Monge–Ampere equations, the Bergman kernel, and geometry of pseudoconvex domains, Ann. of Math. (2) 103 (1976), no. 3, 395–416.Google Scholar
  8. [8]
    E. Fischer, Über die Differentiationsprozesse der Algebra, J. Reine Angew. Math. 148 (1918), 1–78.MathSciNetzbMATHGoogle Scholar
  9. [9]
    A. Gyoja, Highest weight modules and b-functions of semi-invariants, Publ. Res. Inst. Math. Sci. 30 (1994), no. 3, 353–400.MathSciNetCrossRefzbMATHGoogle Scholar
  10. [10]
    R. F. Harvey, H. Blaine Lawson, Jr., On boundaries of complex analytic varieties, I, Ann. of Math. (2) 102 (1975), no. 2, 223–290.Google Scholar
  11. [11]
    R. Hotta, K. Takeuchi, T. Tanisaki, \( \mathcal{D} \) -Modules, Perverse Sheaves, and Representation Theory, Progress in Mathematics, Vol. 236, Birkhäuser, Boston, 2008.Google Scholar
  12. [12]
    J. E. Humphreys, Representations of Semisimple Lie Algebras in the BGG Category \( \mathcal{O} \), Graduate Studies in Mathematics, Vol. 94, American Mathematical Society, Providence, 2008.Google Scholar
  13. [13]
    A. Juhl, Families of Conformally Covariant Differential Operators, Q-Curvature and Holography, Progress in Mathematics, Vol. 275, Birkhäuser, Basel, 2009.Google Scholar
  14. [14]
    M. Kashiwara, Representation theory and \( \mathcal{D} \) -modules on flag varieties, Astérisque 173–174 (1989), 55–109.Google Scholar
  15. [15]
    M. Kashiwara, \( \mathcal{D} \) -modules and Microlocal Calculus, Translations of Mathematical Monographs, Vol. 217, American Mathematical Society, Providence, 2003.Google Scholar
  16. [16]
    T. Kobayashi, Discrete decomposability of the restriction of \( {A}_{\mathfrak{q}}\left(\leftthreetimes \right) \) with respect to reductive subgroups and its applications, Invent. Math. 117 (1994), no. 1, 181–205.Google Scholar
  17. [17]
    T. Kobayashi, Discrete decomposability of the restriction of \( {A}_{\mathfrak{q}}\left(\leftthreetimes \right) \) with respect to reductive subgroups, II: Micro-local analysis and asymptotic K-support, Ann. of Math. (2) 147 (1998), no. 2, 709–729.Google Scholar
  18. [18]
    T. Kobayashi, Discrete decomposability of the restriction of \( {A}_{\mathfrak{q}}\left(\leftthreetimes \right) \) with respect to reductive subgroups, III: Restriction of Harish-Chandra modules and associated varieties, Invent. Math. 131 (1998), no. 2, 229–256.Google Scholar
  19. [19]
    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, 2007, pp. 45–109.Google Scholar
  20. [20]
    T. Kobayashi, Restrictions of generalized Verma modules to symmetric pairs, Trans-formation Groups 17 (2012), no. 2, 523–546.MathSciNetCrossRefzbMATHGoogle Scholar
  21. [21]
    T. Kobayashi, B. Ørsted, P. Somberg, V. Souček, Branching laws for Verma modules and applications in parabolic geometry, I, Adv. Math. 285 (2015), 1796–1852.Google Scholar
  22. [22]
    T. Kobayashi, M. Pevzner, Differential symmetry breaking operators: I. General theory and F-method, Sel. Math. New Ser. 22 (2016), no. 2, 801–845, doi: 10.1007/s00029-15-0207-9; Differential symmetry breaking operators: II. Rankin–Cohen operators for symmetric pairs, Sel. Math. New Ser. 22 (2016), no. 2, 847–911, doi: 10.1007/s00029-015-0208-8.
  23. [23]
    J. Lepowsky, A generalization of the Bernstein–Gelfand–Gelfand resolution, J. Algebra 49 (1977), no. 2, 496–511.MathSciNetCrossRefzbMATHGoogle Scholar
  24. [24]
    N. Tanaka, A Differential Geometric Study on Strongly Pseudoconvex Manifolds, Lectures in Mathematics, Kinokuniya Book-Store, Tokyo, 1975.Google Scholar

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© Springer Science+Business Media New York 2016

Authors and Affiliations

  1. 1.Mathematical Institute Faculty of Mathematics and PhysicsCharles University in PraguePraha 8Czech Republic

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