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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



On a degenerate principal series of representations of $ {\rm U}(2, 2)$

Author: Yang Hua
Journal: Trans. Amer. Math. Soc. 238 (1978), 229-252
MSC: Primary 22E45; Secondary 43A30
MathSciNet review: 0466417
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Abstract: A degenerate principal series of representations $ T(\rho ,m; \cdot ),(\rho ,m) \in {\mathbf{R}} \times {\mathbf{Z}}$, of $ U(2,2)$, is realized on the Hilbert space of all square integrable functions on the space X of $ 2 \times 2$ Hermitian matrices. Using Fourier analysis, gamma functions, and Mellin analysis, we spectrally analyze the operator equation $ AT(\rho ,m;g) = T(\rho ,m;g)A$ for all $ g \in \mathfrak{G} = U(2,2)$ on an invariant subspace of $ {L^2}(X)$, and obtain the first main result: For $ \rho \ne 0$ or m odd, $ T(\rho ,m; \cdot )$ is irreducible. Then we define certain integral transforms on $ {L^2}(X)$ the analytic continuation of which leads to the second main result: $ T(0,2n; \cdot )$ is reducible.

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Keywords: $ U(2,2)$, $ SU(2)$, unitary representation, irreducible representation, reducible representation, degenerate principal series, Fourier transform, intertwining operator, Mellin transform, distribution, gamma function, analytic continuation, singular integral
Article copyright: © Copyright 1978 American Mathematical Society