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Small unitary representations of classical groups. (English) Zbl 0635.22014

Group representations, ergodic theory, operator algebras, and mathematical physics, Proc. Conf. Hon. G. W. Mackey, Berkeley/Calif. 1984, Publ., Math. Sci. Res. Inst. 6, 121-150 (1987).
[For the entire collection see Zbl 0602.00003.]
Let X be an n-dimensional real vector space, with dual \(X^*\); on \(X\oplus X^*_ 0=W\), there is a natural symplectic form. Let \(Sp=Sp(W)\) be the symplectic group defined by this form; let P be the parabolic subgroup of elements taking X to X; write \(P=MN\) (Levi decomposition), where M is the subgroup taking X to X and \(X^*\) to \(X^*\); it turns out that \(M\cong GL(X)\), and that \(N\cong S^ 2(X)\) \((=\) the second symmetric tensor product of X) naturally. Hence, N is Abelian and \(N^{{\hat{\;}}}\cong S^ 2(X^*).\)
Let \(\tau\) be a unitary representation of Sp or some cover of Sp, and consider \(\tau |_ N\). It is then described by a projection-valued measure on \(S^ 2(X^*)\). Each element of \(S^ 2(X^*)\), being a bilinear form of X, has a rank, and it turns out \(\tau |_ N\) can be regarded as living on the bilinear forms of a fixed rank r. When r is sufficiently small, the author gives a complete description of all \(\tau\) of the given rank. He also shows that \(\tau\) gives a representation of Mp, the 2-fold cover of Sp; in all “small” cases, and that the representation factors through Sp iff r is even.
Reviewer: L.Corwin

MSC:

22E45 Representations of Lie and linear algebraic groups over real fields: analytic methods

Citations:

Zbl 0602.00003