The Generalized Dolbeault Complex in Two Clifford Variables

The Penrose transform is used to construct a complex starting with the Dirac operator in two Clifford variables. The corresponding relative BGG complex and its direct image is computed for cohomology with values in line bundles induced by representations in singular infinitesimal character. The limit of the induced spectral sequence is computed in cases connected with the Dirac operator in two Clifford variables. The work presented here is a part of the research project MSM 0021620839 and was supported also by the grants GAUK 447/2004 and GA ČR 201/05/2117.