Abstract
Let G be a semisimple Lie group with a Cartan decomposition G = K exp(\( \mathfrak{p} \)). For every finite-dimensional representation of G, there is an inner product on the representation space such that with respect to it, K acts as unitary operators, and exp(\( \mathfrak{p} \)) acts as positive definite Hermitian operators. In this paper, we observe that similar phenomena appear for infinite-dimensional representations with real infinitesimal characters.
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References
M. Flensted-Jensen, Discrete series for semisimple symmetric spaces, Annals of Mathematics. Second Series 111 (1980), 253–311.
M. Harris and J.-S. Li, A Lefschetz property for subvarieties of Shimura varieties, Journal of Algebraic Geometry 7 (1998), 77–122.
G. Hochschild, Complexification of real analytic groups, Transactions of the American Mathematical Society 125 (1966), 406–413.
M. Kashiwara and W. Schmid, Quasi-equivariant D-modules, equivariant derived category, and representations of reductive Lie groups, in Lie Theory and Geometry, in Honor of Bertram Kostant, Progress in Mathematics, Vol. 123, Birkhäuser Boston, Boston, 1994, pp. 457–488.
A. Knapp and D. Vogan, Cohomological Induction and Unitary Representations, Princeton University Press, Princeton, NJ, 1995.
J.-S. Li, Theta liftings for unitary representations with nonzero cohomology, Duke Mathematical Journal, 61 (1990), 913–937.
J.-S. Li, On the discrete spectrum of (G 2, PGS p6 ), Inventiones Mathematicae 130 (1997), 189–207.
H. Moscovici, The signature with local coefficients of locally symmetrix spaces, The Tohoku Mathematical Journal. Second Series 37 (1985), 513–522.
S. Salamanca-Riba and D. Vogan, On the classification of unitary reperesentations of reductive Lie groups, Annals of Mathematics. Second Series 148 (1998), 1067–1133.
B. Sun, Matrix coefficients of cohomologically induced representations, Compositio Mathematica 143 (2007), 201–221.
D. Vogan, Branching to a maximal compact subgroup, in Harmonic Analysis, Group Representations, Automorphic Forms and Invariant Theory, in Lecture Notes Series, Institute for Mathematical Sciences, National University of Singapore, Vol. 12, World Scientific Publishing, Hackensack, NJ, 2007, pp. 321–401.
D. Vogan, Classification of the irreducible representations of semisimple Lie groups, Proceedings of the National Academy of Sciences of the United States of America 74 (1977), 2649–2650.
D. Vogan, Representations of Real Reductive Lie Groups, Birkhäuser, Boston, MA, 1981.
D. Vogan and G. Zuckerman, Unitary representations with non-zero cohomology, Compositio Mathematica 53 (1984), 51–90.
N. Wallach, Real Reductive Groups I, Academic Press Inc., Boston, MA, 1988.
G. Warner, Harmonic Analysis on Semi-Simple Lie Groups II, Springer-Verlag, Berlin, 1972.
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Supported by Tianyuan Mathematics Foundation of NSFC (Grant No. 10626050).
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Sun, B. Positivity of matrix coefficients of representations with real infinitesimal characters. Isr. J. Math. 170, 395–410 (2009). https://doi.org/10.1007/s11856-009-0034-9
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DOI: https://doi.org/10.1007/s11856-009-0034-9