Non-compact Hermitian Symmetric Spaces
As with symmetric spaces in general, each compact Hermitian symmetric space H/K has a noncompact dual H*/K obtained by replacing H with the Lie group H* in G whose Lie algebra is
However, whereas the natural map from H/K to G/P is an isomorphism, the natural map from H*/K to G/P is only an injection. In fact its image lies in the exponential image of and the corresponding domain in is bounded (this is the Harish-Chandra embedding theorem). The biholomorphism group of H*/K is equal to its isometry group H*.
A bounded domain Ω in a complex vector space (i.e., Ω is an open subset whose closure is compact with respect to the standard topology) is said to be a bounded symmetric domain if for every x in Ω, there is an involutive biholomorphism σx of Ω for which x is an isolated fixed point. Given such a domain Ω, the Bergman kernel defines a metric on Ω, the Bergman metric, for which every biholomorphism is an isometry. This realizes Ω as a Hermitian symmetric space of noncompact type.
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