A ug 2 00 4 TOWARDS A CLASSIFICATION OF HOMOGENEOUS TUBE DOMAINS IN C 4

We classify the tube domains in C 4 with affinely homogeneous base whose boundary contains a non-degenerate affinely homogeneous hypersurface. It follows that these domains are holo-morphically homogeneous and amongst them there are four new examples of unbounded homogeneous domains (that do not have bounded realisations). These domains lie to either side of a pair of Levi-indefinite hypersurface. Using the geometry of these two hypersurfaces, we find the automorphism groups of the domains. The study of holomorphically homogeneous domains in complex space goes back tó E. Cartan [C] who determined all bounded symmetric domains in C n as well as all bounded homogeneous domains in C 2 and C 3. Due to the fundamental theorem of Vinberg, Gindikin, and Pyatetskii-Shapiro, every bounded homogeneous domains can be re-alised as a Siegel domain of the second kind (see [P-S]). Although this result does not immediately imply a complete classification of bounded homogeneous domains, it reduces the classification problem to that for domains of a very special form. A generalisation of the above theorem to the case of unbounded domains for the class of rational homogeneous domains was obtained by Penney in his remarkable paper [P2], where the role of models is played by so-called Siegel domains of type N-P. Nevertheless, the classification problem for the unbounded case is far from fully understood. In [L2], Loboda shows that any holomorphically homogeneous non-spherical tube hypersurface in C 2 has an affinely homogeneous base. The generalisation of this result to higher dimensions is an open problem. Loboda's result motivates our study. We shall consider tubes over