Pseudo unitary conformal groups and Clifford algebras for standard pseudo hermitian spaces

Abstract.

This paper, self-contained, deals with pseudo-unitary spin geometry. First, we present pseudo-unitary conformal structures over a 2n-dimensional complex manifold V and the corresponding projective quadrics $\ifmmode\expandafter\tilde\else\expandafter\~\fi{H}_{{p,q}}$ for standard pseudo-hermitian spaces H _p,q . Then we develop a geometrical presentation of a compactification for pseudo-hermitian standard spaces in order to construct the pseudo-unitary conformal group of H _p,q . We study the topology of the projective quadrics $\ifmmode\expandafter\tilde\else\expandafter\~\fi{H}_{{p,q}}$ and the “generators” of such projective quadrics. Then we define the space S of spinors canonically associated with the pseudo-hermitian scalar product of signature (2ⁿ⁻¹, 2ⁿ⁻¹). The spinorial group Spin U(p,q) is imbedded into SU(2ⁿ⁻¹, 2ⁿ⁻¹). At last, we study the natural imbeddings of the projective quadrics $\ifmmode\expandafter\tilde\else\expandafter\~\fi{H}_{{p,q}}.$