On some geometrical consequences of physical symmetries (Q1086686)

The orbits of the natural operation of GL(n, \({\mathbb{C}})\times GL(m, {\mathbb{C}})\) on \({\mathbb{C}}^{n\times m}\) are the submanifolds \(O_ k^{(n,m)}\) of complex \(n\times m\)-matrices of rank k. For \(n=4\), \(k=2\), GL(n, \({\mathbb{C}})\) is restricted to SU(2,2) or the Poincaré-group, GL(m, \({\mathbb{C}})\) to SU(m, \({\mathbb{C}})\). The decomposition of \(O_ k^{(n,m)}\) into invariant submanifolds in this case is explicitly described by means of calculations in special coordinates. Representations of SU(2,2)\(\times SU(m)\) in the resulting submanifolds and in the Hilbert space of functions over these manifolds are also considered.