New familiy of unbounded homogeneous tube domains in C^n

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.