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

CR-manifolds of dimension 5: A Lie algebra approach

“…Denote by C := (t, x) ∈ 2 × n : t 1 t 2 = x|x , t 1 + t 2 > 0 the future light cone in (n + 2)-dimensional space time and by T := C ⊕ i n+2 the corresponding tube manifold over C . Then T is a simply connected, 2-nondegenerate homogeneous CR-manifold with hol (T , a) ∼ = so (2, n+2) for all a ∈ T , see [11] for the last statement. Then the hypersurface M , given in U := {(w, z 1 , z 2 ) ∈ ¼ × ¼ n × ¼ : |z 2 | < 1} by the equation…”

“…One of the main results of [11] states (even for higher dimensional examples of this type) that all three Lie algebras above coincide, see Proposition 4.3 in [11].…”

“…The group Aut(D) acts transitively on each R p,0 and can be identified with Aut(R p,0 ) this way for every p > 0, see [11]. The Cayley transformation γ gives a CR-isomorphism from T p,0 to the dense domain {z ∈ R p,0 : det(1 1 n − zz) = 0} in R p,0 .…”

“…From [11] we recall the explicit form of this Lie algebra and present a root decomposition in terms of vector fields. This will enable us in the next section to show that a certain local equation defines a CR-manifold locally equivalent to T .…”

“…R has the remarkable property that every CR-equivalence between domains U , V in R extends to a CR-automorphism of R, compare [11]. In particular, Aut(U ) ∼ = {g ∈ G : g(U ) = U } and the groups Aut(U ), Aut(V ) are conjugate in Aut(R) for any pair of CR-equivalent domains U , V ⊂ R. Via the Cayley transformation R contains a copy of the CR-manifold T and hence may be considered as a CR-extension of T .…”