Proceedings of the Japan Academy, Series A, Mathematical Sciences

Branching rules of Dolbeault cohomology groups over indefinite Grassmannian manifolds

Hideko Sekiguchi

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Abstract

We consider a family of singular unitary representations which are realized in Dolbeault cohomology groups over indefinite Grassmannian manifolds, and find a closed formula of irreducible decompositions with respect to reductive symmetric pairs $(A_{2n-1}, D_{n})$. The resulting branching rule is a multiplicity-free sum of infinite dimensional, irreducible representations.

Article information

Source
Proc. Japan Acad. Ser. A Math. Sci., Volume 87, Number 3 (2011), 31-34.

Dates
First available in Project Euclid: 3 March 2011

Permanent link to this document
https://projecteuclid.org/euclid.pja/1299161392

Digital Object Identifier
doi:10.3792/pjaa.87.31

Mathematical Reviews number (MathSciNet)
MR2802604

Zentralblatt MATH identifier
1227.22017

Subjects
Primary: 22E46: Semisimple Lie groups and their representations
Secondary: 05E15: Combinatorial aspects of groups and algebras [See also 14Nxx, 22E45, 33C80] 20G05: Representation theory

Keywords
Branching rule symmetric pair Penrose transform singular unitary representation

Citation

Sekiguchi, Hideko. Branching rules of Dolbeault cohomology groups over indefinite Grassmannian manifolds. Proc. Japan Acad. Ser. A Math. Sci. 87 (2011), no. 3, 31--34. doi:10.3792/pjaa.87.31. https://projecteuclid.org/euclid.pja/1299161392


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References

  • M. Berger, Les espaces symétriques noncompacts, Ann. Sci. École Norm. Sup. (3) 74 (1957), 85–177.
  • M. Duflo and J. A. Vargas, Branching laws for square integrable representations, Proc. Japan Acad. Ser. A Math. Sci. 86 (2010), no. 3, 49–54.
  • B. Gross and N. Wallach, Restriction of small discrete series representations to symmetric subgroups, in The mathematical legacy of Harish-Chandra (Baltimore, MD, 1998), 255–272, Proc. Sympos. Pure Math., 68 Amer. Math. Soc., Providence, RI.
  • M. Ishikawa and M. Wakayama, Minor summation formula of Pfaffians and Schur function identities, Proc. Japan Acad. Ser. A Math. Sci. 71 (1995), no. 3, 54–57.
  • H. P. Jakobsen and M. Vergne, Restrictions and expansions of holomorphic representations, J. Funct. Anal. 34 (1979), no. 1, 29–53.
  • T. Kobayashi, The restriction of $A_{\mathfrak{q}}(\lambda)$ to reductive subgroups, Proc. Japan Acad. Ser. A Math. Sci. 69 (1993), no. 7, 262–267.
  • T. Kobayashi, Discrete decomposability of the restriction of $A_{\mathfrak{q}}(\lambda)$ with respect to reductive subgroups and its applications, Invent. Math. 117 (1994), no. 2, 181–205.
  • T. Kobayashi, Harmonic analysis on reduced homogeneous manifolds and unitary representation theory, Sūgaku 46 (1994), no. 2, 124–143.
  • T. Kobayashi, Discrete decomposability of the restriction of $A_{\mathfrak{q}}(\lambda)$ with respect to reductive subgroups II, –-micro-local analysis and asymptotic $K$-support, Ann. of Math. (2) 147 (1998), no. 3, 709–729.
  • T. Kobayashi, Discrete decomposability of the restriction of $A_{\mathfrak{q}}(\lambda)$ with respect to reductive subgroups III, –-restriction of Harish-Chandra modules and associated varieties, Invent. Math. 131 (1998), no. 2, 229–256.
  • T. Kobayashi, Harmonic analysis on homogeneous manifolds of reductive type and unitary representation theory, Amer. Math. Soc. Transl. (2) 183 (1998) 1–31 (English translation of [8]).
  • T. Kobayashi, Discrete series representations for the orbit spaces arising from two involutions of real reductive Lie groups, J. Funct. Anal. 152 (1998), no. 1, 100–135.
  • T. Kobayashi, Branching problems of unitary representations, in Proceedings of the International Congress of Mathematicians, Vol. II (Beijing, 2002), 615–627, Higher Ed. Press, Beijing.
  • T. Kobayashi, Multiplicity-free representations and visible actions on complex manifolds, Publ. Res. Inst. Math. Sci. 41 (2005), no. 3, 497–549, special issue commemorating the fortieth anniversary of the founding of RIMS.
  • T. Kobayashi, Restrictions of unitary representations of real reductive groups, in Lie theory, Progr. Math., 229 Birkhäuser, Boston, MA 2005, 139–207.
  • 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, Progr. Math., 255 Birkhäuser, Boston, MA 2007, 45–109.
  • T. Kobayashi, Visible actions on symmetric spaces, Transform. Groups 12 (2007), no. 4, 671–694.
  • K. Koike and I. Terada, Young-diagrammatic methods for the representation theory of the classical groups of type $B_{n}, C_{n}, D_{n}$, J. Algebra 107 (1987), no. 2, 466–511.
  • S.-T. Lee and H.-Y. Loke, Degenerate principal series representations of $U(p,q)$ and $\mathrm{Spin}_{0}(p,q)$, Compositio Math. 132 (2002), no. 3, 311–348.
  • S. Okada, Applications of minor summation formulas to rectangular-shaped representations of classical groups, J. Algebra 205 (1998), no. 2, 337–367.
  • B. Ørsted and B. Speh, Branching laws for some unitary representations of $\mathrm{SL}(4,\mathbf{R})$, SIGMA Symmetry Integrability Geom. Methods Appl. 4 (2008), Paper 017, 19 pp.
  • B. Ørsted and J. A. Vargas, Restriction of square integrable representations: discrete spectrum, Duke Math. J. 123 (2004), no. 3, 609–633.
  • W. Schmid, Die Randwerte holomorpher Funktionen auf hermitesch symmetrischen Räumen, Invent. Math. 9 (1969/1970), 61–80.
  • H. Sekiguchi, The Penrose transform for certain non-compact homogeneous manifolds of $U(n,n)$, J. Math. Sci. Univ. Tokyo 3 (1996), no. 3, 655–697.
  • D. A. Vogan, Jr., Unitarizability of certain series of representations, Ann. of Math. (2) 120 (1984), no. 1, 141–187.
  • D. A. Vogan, Jr., Irreducibility of discrete series representations for semisimple symmetric spaces, in Representations of Lie groups, Kyoto, Hiroshima, 1986, Adv. Stud. Pure Math., 14, 1988, 191–221.
  • R. O. Wells, Jr., Complex manifolds and mathematical physics, Bull. Amer. Math. Soc. (N.S.) 1 (1979), no. 2, 296–336.
  • H.-W. Wong, Dolbeault cohomological realization of Zuckerman modules associated with finite rank representations, J. Funct. Anal. 129 (1995), no. 2, 428–454.