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Pseudo-Hermitian symmetric spaces and Siegel domains over nondegenerate cones. (English) Zbl 0751.32017

fermitian symmetric spaces have been the subject of intensive study for more than five decades. They form a special instance of homogeneous Kähler manifolds with transitive reductive group of automorphism and they contain the subclass of symmetric Siegel domains. Recently the reviewer and Z.-D. Guan have investigated homogeneous pseudo-Kähler manifolds which are compact or admit a reductive transitive group of holomorphic isometrics. The present paper investigates in detail a special subclass of such manifolds, namely simple irreducible pseudo- hermitian symmetric spaces of \(K_ \varepsilon\)-type and determines their relation with Siegel domains over a nondegenerate cone.
More precisely, the following theorems are proven:
Theorem 3.6. Let \(G\) be the adjoint group of a real simple Lie algebra \({\mathfrak g}\) of hermitian type of real rank \(r\), and \(H_ k(0\leq k\leq r)\) be the centralizer in \(G\) of the element \(iE_ k\in g\) (cf. \((3.10)_ k)\). Then the coset space \(M_ k=G/H_ k(0\leq k\leq r)\) is a simply connected simple irreducible pseudo-hermitian symmetric space of \(K_ \varepsilon\)-type. Conversely every simply connected simple irreducible pseudo-hermitian symmetric space of \(K_ \varepsilon\)-type is obtained in this manner. Furthermore, if the restricted root system of \({\mathfrak g}\) is of type \(C_ r\), then we have the isomorphism \(M_ k\simeq M_{r- k}(0\leq k\leq(r/2))\) as pseudo-hermitian symmetric spaces.
Theorem 5.3. Let \(G\) be a simple Lie group of hermitian type or real rank \(r\). Let \(M_ k=G/H_ k(0\leq k\leq r)\) be a (simply connected) simple irreducible pseudo-hermitian symmetric space of \(K_ \varepsilon\)-type constructed in §3 and realized as an open subset of \(M^*\), the compact dual of the hermitian symmetric space \(M_ 0\) (cf. Proposition 3.7). Then the intersection of the Cayley image \(c(M_ k)\) with \(\xi(\overline g_{-1})\) is holomorphically equivalent to the affine homogeneous Siegel domain \(D(V_{r-k,k},F)\) in \(\overline g_{-1}\), where \(V_{r-k,k}\) is the nondegenerate cone given in (4.25) and \(F\) is the \(V_{r,0}\)- hermitian form given in (5.2). More precisely we have \(\xi^{-1}(c(M_ k)\cap\xi(\overline g_{-1}))=D(V_{r-k,k},F)\), \(0\leq k\leq r\). If the restricted root system of \({\mathfrak g}\)=Lie \(G\) is of type \(C_ r\), then the Siegel domain \(D(V_{r-k,k},F)\) is reduced to the tube domain \(D(V_{r-k,k})\).
It is remarkable that, contrary to the hermitian symmetric case, not the whole part of a simple irreducible pseudo-hermitian (non-hermitian) symmetric space \(M\) of \(K_ \varepsilon\)-type but only an open dense subset of \(M\) is realized as an affine homogeneous Siegel domain over a non-degnerate cone.

MSC:

32M15 Hermitian symmetric spaces, bounded symmetric domains, Jordan algebras (complex-analytic aspects)
53C50 Global differential geometry of Lorentz manifolds, manifolds with indefinite metrics
22E46 Semisimple Lie groups and their representations
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