Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 4 (2008), 017, 19 pages      arXiv:0802.0974
Contribution to the Proceedings of the 2007 Midwest Geometry Conference in honor of Thomas P. Branson

Branching Laws for Some Unitary Representations of SL(4,R)

Bent Ørsted a and Birgit Speh b
a) Department of Mathematics, University of Aarhus, Aarhus, Denmark
b) Department of Mathematics, 310 Malott Hall, Cornell University, Ithaca, NY 14853-4201, USA

Received September 10, 2007, in final form January 27, 2008; Published online February 07, 2008

In this paper we consider the restriction of a unitary irreducible representation of type Aq(λ) of GL(4,R) to reductive subgroups H which are the fixpoint sets of an involution. We obtain a formula for the restriction to the symplectic group and to GL(2,C), and as an application we construct in the last section some representations in the cuspidal spectrum of the symplectic and the complex general linear group. In addition to working directly with the cohmologically induced module to obtain the branching law, we also introduce the useful concept of pseudo dual pairs of subgroups in a reductive Lie group.

Key words: semisimple Lie groups; unitary representation; branching laws.

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  1. Barbasch D., Sahi S., Speh B., Degenerate series representations for GL(2n,R) and Fourier analysis, Symposia Mathematica, Vol. XXXI (1988, Rome), Sympos. Math., Vol. XXXI, Academic Press, London, 1990, 45-69.
  2. Clozel L., Venkataramana T.N., Restriction of the holomorphic cohomology of a Shimura variety to a smaller Shimura variety, Duke Math. J. 95 (1998), 51-106.
  3. Gross B., Prasad D., On the decomposition of a representation of SOn when restricted to SOn-1, Canadian J. Math. 44 (1992), 974-1002.
  4. Gross B., Wallach N., Restriction of small discrete series representations to symmetric subgroups, in The Mathematical Legacy of Harish-Chandra (1998, Baltimore, MD), Proc. Sympos. Pure Math., Vol. 68, Amer. Math. Soc., Providence, RI, 2000, 255-272.
  5. Harder G., On the cohomology of SL(2,O), in Lie Groups and Their Representations (Proceedings Summer School on Groups Representations of the Bolyai Janos Math. Soc. Budapest, 1971), Halsted, New York, 1975, 139-150.
  6. Hecht H., Schmid W., A proof of Blatner's conjecture, Invent. Math. 31 (1976), 129-154.
  7. Howe R., Reciprocity laws in the theory of dual pairs, in Representation Theory of Reductive Groups, Editor P. Trombi, Progr. Math., Vol. 40, Birkhäuser Boston, Boston, MA, 1983, 159-175.
  8. Jacobsen H.P., Vergne M., Restriction and expansions of holomorphic representations, J. Funct. Anal. 34 (1979), 29-53.
  9. Kim H., The residual spectrum of Sp4, Compositio Math. 99 (1995), 129-151.
  10. Knapp A.W., Vogan D.A. Jr., Cohomological induction and unitary representations, Princeton Mathematical Series, Vol. 45, Princeton University Press, Princeton, NJ, 1995.
  11. Kobayashi T., Harmonic analysis on homogeneous manifolds of reductive type and unitary representation theory, in Selected Papers on Harmonic Analysis, Groups and Invariants, Editor K. Nomizu, Amer. Math. Soc. Transl. Ser. 2, Vol. 183, Amer. Math. Soc., Providence, RI, 1998, 1-31 (and references therein).
  12. Kobayashi T., Discretely decomposable restrictions of unitary representations of reductive Lie groups - examples and conjectures, in Analysis on Homogeneous Spaces and Representation Theory of Lie Groups, Editor T. Kobayashi, Advanced Studies in Pure Mathematics, Vol. 26, Kinokuniya, Tokyo, 2000, 99-127.
  13. Kobayashi T., Discrete series representations for the orbit spaces arising from two involutions of real reductive groups, J. Funct. Anal. 152 (1998), 100-135.
  14. Kobayashi T., Discrete decomposability of the restriction of Aq(λ) with respect to reductive subgroups. III. Restriction of Harish-Chandra modules and associated varieties, Invent. Math. 131 (1997), 229-256.
  15. Kobayashi T., The restriction of Aq(λ) to reductive subgroups, Proc. Japan Acad. Ser. A Math. Sci. 69 (1993), no. 7, 262-267.
  16. Kobayashi T., Discrete decomposability of the restriction of Aq(λ) with respect to reductive subgroups and its applications, Invent. Math. 117 (1994), 181-205.
  17. Kobayashi T., Ørsted B., Conformal geometry and branching laws for unitary representations attached to minimal nilpotent orbits, C. R. Acad. Sci. Paris 326 (1998), 925-930.
  18. Loke H., Restrictions of quaternionic representations, J. Funct. Anal. 172 (2000), 377-403.
  19. Martens S., The characters of the holomorphic discrete series, Proc. Nat. Acad. Sci. USA 72 (1976), 3275-3276.
  20. Rohlfs J., On the cuspidal cohomology of the Bianchi modular groups, Math. Z. 188 (1985), 253-269.
  21. Sahi S., Stein E., Analysis in matrix space and Speh's representations. Invent. Math. 101 (1990), 379-393.
  22. Schwermer J., On Euler products and residual cohomology classes for Siegel modular varieties, Forum Math. 7 (1995), 1-28.
  23. Speh B., Unitary representations of GL(n,R) with nontrivial (g,K)-cohomology, Invent. Math. 71 (1983) 443-465.
  24. Vargas J., Restriction of some discrete series representations, Algebras Groups Geom. 18 (2001), 85-100.
  25. Vargas J., Restriction of holomorphic discrete series to real forms, Rend. Sem. Mat. Univ. Politec. Torino, to appear.
  26. Vogan D. Jr., Representations of real reductive Lie groups, Birkhäuser, Boston, 1981.
  27. Wolf J., Representations that remain irreducible on parabolic subgroups, in Differential Geometrical Methods in Mathematical Physics IV (Proceedings, Aix-en-Provènce and Salamanca, 1979), Springer Lecture Notes in Mathematics, Vol. 836, Springer, Berlin, 1980, 129-144.
  28. Zhang G., Berezin transform of holomorphic discrete series on real bounded symmetric domains, Trans. Amer. Math. Soc. 353 (2001), 3769-3787.

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