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The Capelli identity and unitary representations. (English) Zbl 0758.22008
Let \(\Omega=G/K\) be an irreducible Hermitian symmetric space of tube type of rank \(n\), \(G/P\) its Shilov boundary where \(P=LN\) is a maximal parabolic subgroup of \(G\). Let \(\varphi\) be the Jordan norm on the Lie algebra \(\mathfrak n\) of \(N\). If \(\overline P\) is the opposite parabolic and \(t\in \mathbb{R}\), consider the induced representation \(I(G)=Ind^ G_{\overline P}(\nu^ t\oplus 1)\), where \(\nu^{-2}\) is the positive character of \(L\) associated with \(\varphi\). Using a canonical isomorphism from \(\mathfrak n\) to \(\mathfrak n^*\) one can associate to \(\varphi\) a differential operator \(D\). On \(I(m)\), \(m\in\mathbb{Z}\), one can define the invariant Hermitian form \((f_ 1,f_ 2)_ m=\langle D^ mf_ 1,f_ 2\rangle\), where \(\langle , \rangle\) is the Hermitian pairing between \(I(m)\) and \(I(-m)\). The author finds an explicit formula for the signature of this form for each \(K\)-type. He also obtains information on \(I(m)/\text{Ker }D^ m\).

MSC:
22E45 Representations of Lie and linear algebraic groups over real fields: analytic methods
43A85 Harmonic analysis on homogeneous spaces
22D30 Induced representations for locally compact groups
11E39 Bilinear and Hermitian forms
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References:
[1] B. Boe , Homomorphisms between Generalized Verma Modules , Trans. Amer. Math. Soc. 288 (1985), 791-799. · Zbl 0568.17004 · doi:10.2307/1999964
[2] J. Faraut and A. Koranyi , Function Spaces and Reproducing Kernels on Bounded Symmetric Domains , J. Funct. Anal. 88 (1990), 64-89. · Zbl 0718.32026 · doi:10.1016/0022-1236(90)90119-6
[3] A. Guillemonat , On Some Semi-spherical Representations of a Hermitian Symmetric Pair of Tubular Type , Math. Ann. 246(1980), 93-116. · Zbl 0499.22016 · doi:10.1007/BF01420161 · eudml:163330
[4] H. Jakobsen and M. Vergne , Wave and Dirac Operators and Representations of the Conformal Group , J. Funct. Anal. 24(1977), 52-106. · Zbl 0361.22012 · doi:10.1016/0022-1236(77)90005-2
[5] M. Kashiwara and M. Vergne , Functions on the Shilov Boundary of the Generalized Half Plane, Non-Commutative Harmonic Analysis , Lect. Notes Math. 728, Springer, 1979, pp. 136-176. · Zbl 0416.22006
[6] A. Koranyi and J. Wolf , Realization of Hermitian Symmetric Spaces as Generalized Halfplanes , Ann. of Math. 81 (1965), 265-268. · Zbl 0137.27402 · doi:10.2307/1970616
[7] B. Kostant and S. Rallis , Orbits and Representations Associated to Symmetric Spaces , Amer. J. Math. 93 (1971), 753-809. · Zbl 0224.22013 · doi:10.2307/2373470
[8] B. Kostant and S. Sahi , The Capelli Identity, Tube Domains and the Generalized Laplace Transform , Adv. Math. 87 (1991), 71-92. · Zbl 0748.22008 · doi:10.1016/0001-8708(91)90062-C
[9] S. Kudla and S. Rallis , Degenerate Principal Series and Invariant Distributions , Israel J. Math. 69 (1990), 25-45. · Zbl 0708.22005 · doi:10.1007/BF02764727
[10] C. Moore , Compactifications of Symmetric Domains II. The Cartant Domains , Amer. J. Math. 86 (1964), 358-378. · Zbl 0156.03202 · doi:10.2307/2373040
[11] H. Rossi and M. Vergne , Analytic Continuation of the Holomorphic Discrete Series of a Semi-simple Lie Group , Acta Math. 136 (1976), 1-59. · Zbl 0356.32020 · doi:10.1007/BF02392042
[12] B. Speh , Degenerate Series Representations of the Universal Covering Group of SU(2, 2 ), J. Funct. Anal. 33 (1979), 95-118. · Zbl 0415.22012 · doi:10.1016/0022-1236(79)90019-3
[13] D. Vogan and G. Zuckerman , Unitary Representations with Non-Zero Cohomology , Comp. Math. 53 (1984), 51-90. · Zbl 0692.22008 · numdam:CM_1984__53_1_51_0 · eudml:89677
[14] N. Wallach , The Analytic Continuation of the Discrete Series II , Trans. Amer. Math. Soc. 251 (1979), 19-37. · Zbl 0419.22018 · doi:10.2307/1998681
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