On a degenerate principal series of representations of 𝑈(2,2)

Hua, Yang

doi:10.1090/S0002-9947-1978-0466417-7

On a degenerate principal series of representations of $\textrm {U}(2, 2)$
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Trans. Amer. Math. Soc. 238 (1978), 229-252 Request permission

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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