Boundary value problems for various boundaries of Hermitian symmetric spaces (Q1968731)

Let \(G\) be a connected simple Lie group with finite center and \(K\) a maximal compact subgroup of \(G\). We suppose that \(X= G/K\) is a Hermitian symmetric space of rank greater than one. Each boundary component of \(X\) is of the form \(G/P_\Theta\), where \(P_\Theta= M_\Theta A_\Theta N_\Theta\) is a parabolic subgroup of \(G\). Then, a principal series representation of \(G\) is attached to \(G/P_\Theta\) as an induced representation from a holomorphic discrete series of \(M_\Theta\) and a character of \(A_\Theta\), and the Poisson transform gives an intertwining operator from the principal series to a homogeneous line bundle over \(G/K\). Our concern is to determine the image of the Poisson transform and to obtain a system of differential equations that the image satisfies. In this paper Shimeno gives an explicit construction of the differential operators, which is a new example on this subject. For the principal series with a generic parameter the image is the space of functions on a homogeneous line bundle over \(G/K\) that are joint eigenfunctions of invariant differential operators and in the kernel of a \(K\)-covariant differential operator \({\mathcal H}^\Theta_{\pm}\) (the Hua operator) of order \(\dim A_\Theta+ 1\). The author also gives a characterization of the degenerate series representation. These results are a natural generalization if we recall the case that \(P_\Theta\) is minimal or the boundary is of Shilov type. In the author's proof the partial Poisson transforms, which are studied in his previous paper [J. Funct. Anal. 121, 330-388 (1994; Zbl 0830.43018)], play an essential role.