Construction of the Hermitian Symmetric Spaces

Abstract

In this chapter we start with an irreducible oiLa of non-compact type, g = t + p, and assume only that t is not semisimple. We shall prove that there exists a unique corresponding Riemannian symmetric space, that it is actually Hermitian symmetric, and it can be realized as a bounded domain. Along the way we are also going to do quite a lot more. We shall give a description of the compact Hermitian symmetric space corresponding to the dual oiLa gu = t + ip (which is also unique, although there may be more than one Riemannian symmetric space corresponding to gu; an example being ℂP 1, the complex projective line, and P 2, the real projective plane). We shall also get a full generalization of the situation in one complex dimension where the complex plane ℂ is imbedded into the Riemann sphere ℂP 1, and the unit disc U into the lower hemisphere, by stereographic projection. This example, where the initial G is SU(1, 1) (or its quotient modulo {±I}), the group of fractional linear transformations of U, should be constantly borne in mind during the following discussion.KeywordsSymmetric SpaceSimple RootRiemann SphereFractional Linear TransformationHermitian Symmetric SpaceThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.