The Berezin form for rank one para-Hermitian symmetric spaces

Recently one of the authors presented a general scheme of quantization tin the spirit of Berezin) for para-Hermitian symmetric spaces. One of the main notions of this theory is the Berezin form. It is an invariant Hermitian form, which, generally speaking, is not positive-definite. The rank one para-Hermitian spaces are exhausted by the spaces X = SL(n, R)/GL(n - 1, R), up to coverings. In this paper we decompose the Berezin form beta(mu, nu), mu is an element of R, nu = 0, 1 for these spaces X into irreducible components. For values mu < -n+1/2 the decomposition contains the same irreducible components as L-2(X), but the Plancherel measure is not necessarily positive. For mu > -n+1/2 a finite number of irreducible components, belonging to the (non-unitary) complementary series, is added, We also pay attention to the behaviour of beta(mu, nu) as mu--> -infinity, which has a meaning in the context of quantization. It turns out that the correspondence principle holds on the discrete spectrum for n even. (C) Elsevier, Paris