Berezin kernels and analysis on Makarevich spaces (Q855596)

Let \(V\) be a simple real Jordan algebra of dimension \(n\). Let \({\widetilde G}=\text{ Conf}(V)\) be the conformal group. The stabilizer of the point \(0\) is a parabolic subgroup \(\overline P\) of \(\widetilde G\). The manifold \({\mathcal M}={\widetilde G}/{\overline P}\) is the conformal compactification of \(V\). Makarevich symmetric spaces (the term was introduced by Bertram) of tube type are reductive symmetric spaces which can be realized as open symmetric orbits in \({\mathcal M}\). The main goal of the paper under review is to compare two approaches to Berezin kernels for Makarevich spaces: a) using the Kantor cross ratio for \(V\); b) restricting maximal degenerate series representations of \(\widetilde G\). The authors concentrate on such spaces carrying an invariant Riemannian metric. These spaces are obtained in the following way (a list of these spaces is included). Let \(\alpha\) be a Euclidean involution of \(V\). Let \(V_0\) and \(V_1\) be \(+1\) and \(-1\) eigenspaces of \(\alpha\). The space \(V_0\) is a Euclidean Jordan algebra, let \(r_0=\text{ rank} \;V_0\). The interior \(\Omega_0\) of the set elements \(x^2\), \(x\in V_0\), is a symmetric cone of \(V_0\). Let \(G\) be a connected subgroup of \(\widetilde G\) consisting of \(g\in\widetilde G\) such that \((-\alpha)\circ g=g\circ(-\alpha)\), and \(K\subset G\) the stabilizer of the identity element \(e\) of \(V\). The group \(K\) is a maximal compact subgroup of \(G\). The Riemannian symmetric space \({\mathcal X}=G/K\) is a real tube domain \(\Omega_0+V_1\). The Berezin kernel \(B_\nu(x,y)\), \(\nu\in \mathbb C\), on \(\mathcal X\) is constructed by means of the Kantor cross ratio. An important theorem on Berezin kernels is formulated: it claims that for \(\text{ Re} \;\nu>(n/r_0)-1\) the Berezin function \(\psi_\nu(x)=B_\nu(x,e)\) is integrable, and gives an explicit expression of its spherical Fourier transform: a fraction whose nominator and denominator are products of \(2r_0\) Gammas of linear functions of \(\nu\). For Hermitian symmetric spaces, this remarkable formula was found by Berezin in 1978 (classical spaces) and by Unterberger and Upmeier in 1994 (all spaces). For tube domains, it was done by Zhang, and van Dijk and Pevzner in 2001. On the other hand, the Berezin kernel \(B_\nu(x,y)\) can be obtained from the kernel of an operator intertwining maximal degenerate series representations \(\pi_s\), \(s\in\mathbb C\), of \(\widetilde G\). Here the parameters \(\nu\) and \(s\) are connected by a linear relation. The restriction \(T_s\) of representations \(\pi_s\) to \(G\) and to the orbit \(\mathcal X\) are called canonical representations of \(G\). (For Hermitian symmetric spaces, canonical representations were introduced by Berezin and Vershik, Gelfand and Graev in the seventies. The idea that in the general case canonical representations should be defined as restrictions of maximal degenerate series representations was put forward by the reviewer in the nineties.) If \(\nu>(n/r_0)-1\), then the kernel \(B_\nu(x,y)\) is of positive type, and the canonical representation \(T_s\) is unitary with respect to this inner product, it decomposes multiplicity free as a direct integral of spherical principal series representations, with the explicitly written Plancherel formula. Similar results are obtained for the Riemannian compact dual.