Berezin Kernels and Analysis on Makarevich Spaces

Following ideas of van Dijk and Hille we study the link which exists between maximal degenerate representations and Berezin kernels. We consider the conformal group ${\rm Conf}(V)$ of a simple real Jordan algebra $V$. The maximal degenerate representations $\pi_s$ ($s\in {\mathbb C}$) we shall study are induced by a character of a maximal parabolic subgroup $\bar P$ of ${\rm Conf}(V)$. These representations $\pi_s$ can be realized on a space $I_s$ of smooth functions on $V$. There is an invariant bilinear form ${\mathfrak B}_s$ on the space $I_s$. The problem we consider is to diagonalize this bilinear form ${\mathfrak B}_s$, with respect to the action of a symmetric subgroup $G$ of the conformal group ${\rm Conf}(V)$. This bilinear form can be written as an integral involving the Berezin kernel $B_{\nu}$, 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 $B_{\nu}$. From these, one deduces that the Berezin kernel satisfies a remarkable Bernstein identity : $$D(\nu)B_{\nu} =b(\nu)B_{\nu +1},$$ where $D(\nu)$ is an invariant differential operator on $G/K$ and $b(\nu)$ 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 $I_s$. 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 ${\rm Conf}(V)$.