|
SIGMA 4 (2008), 017, 19 pages arXiv:0802.0974
https://doi.org/10.3842/SIGMA.2008.017
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
Abstract
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.
pdf (432 kb)
ps (406 kb)
tex (329 kb)
References
- 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.
- 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.
- Gross B., Prasad D., On the decomposition
of a representation of SOn when restricted to SOn-1,
Canadian J. Math. 44 (1992), 974-1002.
- 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.
- 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.
- Hecht H., Schmid W., A proof of Blatner's conjecture, Invent. Math. 31 (1976), 129-154.
- 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.
- Jacobsen H.P., Vergne M., Restriction and expansions of
holomorphic representations, J. Funct. Anal. 34 (1979),
29-53.
- Kim H., The residual spectrum of Sp4, Compositio Math. 99 (1995), 129-151.
- Knapp A.W., Vogan D.A. Jr.,
Cohomological induction and unitary representations, Princeton
Mathematical Series, Vol. 45,
Princeton University Press, Princeton, NJ, 1995.
- 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).
- 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.
- Kobayashi T., Discrete series representations for the orbit spaces arising
from two involutions of real reductive groups, J. Funct.
Anal. 152 (1998), 100-135.
- 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.
- Kobayashi T., The restriction of Aq(λ) to reductive
subgroups, Proc. Japan Acad. Ser. A Math. Sci. 69 (1993), no. 7, 262-267.
- Kobayashi T., Discrete decomposability of the restriction of
Aq(λ) with respect to reductive subgroups and its
applications, Invent. Math. 117 (1994), 181-205.
- 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.
- Loke H., Restrictions of quaternionic representations, J. Funct. Anal. 172 (2000), 377-403.
- Martens S., The characters of the holomorphic discrete series, Proc.
Nat. Acad. Sci. USA 72 (1976), 3275-3276.
- Rohlfs J., On the cuspidal cohomology of the
Bianchi modular groups, Math. Z. 188 (1985), 253-269.
- Sahi S., Stein E., Analysis in matrix space and Speh's representations.
Invent. Math. 101 (1990), 379-393.
- Schwermer J., On Euler products and residual
cohomology classes for Siegel modular varieties, Forum Math. 7 (1995), 1-28.
- Speh B., Unitary representations of GL(n,R)
with nontrivial (g,K)-cohomology, Invent. Math. 71 (1983)
443-465.
- Vargas J., Restriction of some discrete series representations,
Algebras Groups Geom. 18 (2001), 85-100.
- Vargas J., Restriction of holomorphic discrete series to real forms,
Rend. Sem. Mat. Univ. Politec. Torino, to appear.
- Vogan D. Jr., Representations of real reductive
Lie groups, Birkhäuser, Boston, 1981.
- 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.
- Zhang G., Berezin transform of holomorphic discrete series on real bounded symmetric domains,
Trans. Amer. Math. Soc. 353 (2001), 3769-3787.
|
|