Berezin kernels and analysis on Makarevich spaces

We consider the conformal group Conf(V) of a simple real Jordan algebra V. The maximal degenerate representations πs (s ε ℂ) we shall study are induced by a character of a maximal parabolic subgroup P¯ of Conf(V). These representations πs can be realized on a space Is of smooth functions on V. There is an invariant bilinear form ℬs on the space Is. The problem we consider is to diagonalize this bilinear form ℬs, with respect to the action of a symmetric subgroup G of the conformal group Conf(V). This bilinear form can be written as an integral involving the Berezin kernel Bv an invariant kernel on the Riemannian symmetric space G/K, which is a Makarevich symmetric space in the sense of Bertram. Then we can use results by van Dijk and Pevzner who computed the spherical Fourier transform of Bv. From these, one deduces that the Berezin kernel satisfies a remarkable Bernstein identity: D(ν)Bν=b(ν)Bν+1, where D(ν) is an invariant differential operator on G/K and b(ν) is a polynomial. By using this identity we compute a Hua type integral which gives the normalizing factor for an intertwining operator from I−s to Is. Furthermore, we obtain the diagonalization of the invariant bilinear form with respect to the action of the maximal compact group U of the conformal group Conf(V).