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Hermitian symmetric spaces and Kähler rigidity. (English) Zbl 1138.32012

The paper concerns irreducible bounded symmetric domains \({\mathcal D}\) in \(\mathbb C^n\), especially those not of tube type, and the goal is to extend to the general case results of M. Burger and A. Iozzi [Ann. Sci. Éc. Norm. Supér. (4) 37, No. 1, 77–103 (2004; Zbl 1061.32016)] for the case of rectangular matrices. In fact the first result characterises the non tube type irreducible domains in terms of the existence of certain nonconstant ‘Hermitian triple products’.
For a subset of the Cartesian product \(\overline{\mathcal D}^3\) of 3 copies of the closure \(\overline{\mathcal D}\) of \(\mathcal D\) (which includes \({\mathcal D}^3\) and intersects \(\check{S}^3\) in a dense set, where \(\check{S}\) means the Shilov boundary of \({\mathcal D}\)), the Hermitian triple product is given in terms of the Bergmann kernel \(k_{\mathcal D}\) by \( \langle \langle x, y, z \rangle \rangle = k_{\mathcal D}(x,y) k_{\mathcal D}(y,z) k_{\mathcal D}(z,x)\) and is considered modulo multiplication by nonzero reals \(\mathbb R^\times\) (so that it has values in the quotient \(\mathbb R^\times \backslash \mathbb C^\times\)). If G denotes the connected complex Lie group associated to the complexification of the Lie algebra of Aut\(({\mathcal D})\), there is a parabolic subgroup Q of G so that \(\check{S}\) embeds naturally in G\(/\)Q. The Hermitian triple product \(\langle \langle \cdot, \cdot, \cdot \rangle \rangle\) extends to a G-invariant rational function on G\(/\)Q (with values in a suitable quotient of \((\mathbb C^2)^\times\)).
The argument \(\beta(x,y,z)\) of \( \langle \langle x, y, z \rangle \rangle\) (suitably defined) is called the Bergmann cocycle, and for \(x, y, z \in \mathcal D\), the integral of the Kähler form \(\omega_{\text{Berg}}\) over the geodesic triangle \(\Delta(x,y,z)\) is \(\int_{\Delta(x,y,z)} \omega_{\text{Berg}} = \beta(x,y,z)\). On \(\overline{\mathcal D}^3\), \(\beta\) is a closed cocycle. When \(\mathcal D\) is of tube type, \(\beta( \check{S}^3)\) is discrete, while it is an interval in the non tube case.
A continuous \(G\)-invariant homogeneous cocycle may be defined on \(G\) via
\((g_1, g_2, g_3) \mapsto \beta( g_10, g_20, g_30)/(2 \pi)\) (\(0 \in {\mathcal D}\) the origin) and the corresponding class \(\kappa_{G, B} \in \text{H}^2_{\text{c}}(G, \mathbb R)\) in continuous cohomology is integral. \(\kappa_{G, B}\) is called the Kähler class and gives rise to a bounded class \(\kappa_{G, B}^b \in \text{H}^2_{\text{cb}}(G, \mathbb R)\).
Let \(G\) be the connected component of the identity in Aut\(({\mathcal D})\), where \(\mathcal D\) is not of tube type, and \(H\) a locally compact second countable group. The bounded Kähler class is applied to characterise continuous homomorphisms \(\rho \colon H \to G\) with Zariski dense image up to conjugation. The pullback \(\rho^*(\kappa_{G, B}^b)\) is a determining invariant. Let \(\Gamma\) be a finitely generated group. If \(\text{H}^2_{\text{b}}(\Gamma, \mathbb R)\) is finite dimensional, or if the comparison map \(\text{H}^2_{\text{b}}(\Gamma, \mathbb R) \to \text{H}^2(\Gamma, \mathbb R)\) is injective, it is shown that there are, up to \(G\)-conjugation, only finitely many representations \(\Gamma \to G\) with Zariski dense image.

MSC:

32M15 Hermitian symmetric spaces, bounded symmetric domains, Jordan algebras (complex-analytic aspects)
46M20 Methods of algebraic topology in functional analysis (cohomology, sheaf and bundle theory, etc.)

Citations:

Zbl 1061.32016
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