On the holomorphic structure of G-orbits in compact hermitian symmetric spaces

Abstract

For every irreducible bounded symmetric domain D we are interested in the Cauchy-Riemann structure of the orbits under the real Lie group G = Aut(D) in the compact hermitian dual Z of D. For orbits of this type we solve the CR-equivalence problem, compute explicitly their CR-automorphism groups and determine maximal holomorphic extendibility for continuous CR-functions and infinitesimal CR-transformations. (2000): 17C50, 32M15, 32V10, 32V25, 46J15 Mathematics Subject Classification

On local CR-transformations of Levi-degenerate group orbits in compact Hermitian symmetric spaces

“…Clearly, M is invariant unter the subgroup Aff(H) ⊂ S, and we claim that actually M is an Aff(H)-orbit in E. This follows from the well known fact that in the irreducible Hermitian symmetric space Z of rank r the number of S-orbits is r+2 2 (compare e.g. [12]), which by Lemma 5.4 is also the number of Aff(H)-orbits in E. By the above discussion M is a 2-nondegenerate CR-manifold, by homogeneity this therefore also is true for S. Finally, minimality of S follows from Theorem 3.6 in [12].…”

“…Proof. The antiholomorphic transformation τ (z) = z * of H induces the same Lie algebra automorphism of h as s. Therefore the first claim follows from Proposition 4.5 in [12] (stated for the biholomorphically equivalent domain D). Suppose g(δ) = δ.…”

Local tube realizations of CR-manifolds and maximal abelian subalgebras

“…By Proposition 2.11 in [16], g a = g b for a, b ∈ Z only holds if a = b. The group Aut(M ) ∼ = PSU(p, q) is connected and for H := Aut(M ) ∪ Aut(M )τ the homomorphism Ad : H → Aut(g ) is an isomorphism, compare Proposition 4.5 in [16]. In particular, Aut By the above considerations we know that for every non-open G-orbit M = M p,q j,k in Z there is an integer 1 ≤ k ≤ 3 such that M is k-nondegenerate.…”

“…For every a ∈ M denote by g a := {ξ ∈ g : ξ a = 0} the isotropy subalgebra at a. By Proposition 2.11 in [16], g a = g b for a, b ∈ Z only holds if a = b. The group Aut(M ) ∼ = PSU(p, q) is connected and for H := Aut(M ) ∪ Aut(M )τ the homomorphism Ad : H → Aut(g ) is an isomorphism, compare Proposition 4.5 in [16].…”

CR-manifolds of dimension 5: A Lie algebra approach

“…In analogy to our setting in Section 3 with n = 2 we denote for every p, q by T p,q := C p,q ⊕ iV the tube manifold over the cone C p,q . Then T p,q is open precisely if k = 0 and is closed if k = n. In all other cases T p,q is a 2-nondegenerate CR-manifold and Aut(T p,q ) consists of all transformations (3.2) (with GL(2, ) replaced by GL(n, ), compare [10]). In [10] also the following holomorphic extension property has been shown: In case pq = 0 every continuous CRfunction on T p,q has a holomorphic extension to all of E. In case p > 0 every continuous CR-function on T p,0 has a holomorphic extension to H that is continuous up to T p,0 ⊂ H in a certain sense.…”