This talk is based on a series of joint works with M. Eastwood, A. Isaev, A. Medvedev, and G. Schmalz.
Due to the fundamental theorems of Vinberg, Gindikin and Pyatetsky-Shapiro, all bounded homogeneous domains in Cn
are equivalent to Siegel domains of either I or II kind. Although this result does not imply a complete classification of bounded homogeneous domains, it narrows down the problem to study only very special domains. R. Penney introduced a class of unbounded rational homogeneous N-P domains with a Levi-nondegenerate boundary, that he calles nil-balls. on which solvable algebraic group acts transitively and polynomially by holomorphic transformations, such that its nil-radical acts simply transitively on the Levi-nondegenerate boundary.
Building on the classification of affinely homogeneous tube hypersurfaces in C4 we define homogeneous domains lying on the either side of the hypersurface. We observe that two pairs of these domains do not have bounded realisation and are not equivalent to nil-balls. Accidentally, their holomorphic automorphism groups are larger than their transitive affine subgroups and have equal dimension 10. We generalise the equation of one of such hypersurfaces to a family of affinely homogeneous hypersurfaces in Cn+1 for every n>3, explicitly determine the holomorphic automorphism groups of the corresponding homogeneous domains and show that they are not the nil-balls. The dimension of the automorphism group appears to be n2 – 2n + 7, which we conjecture is maximal in its class.
Links:
[1] http://www.mpim-bonn.mpg.de/de/taxonomy/term/39
[2] http://www.mpim-bonn.mpg.de/de/node/158