Identities for vector fields in the infinitesimal representation of the symplectic group into the Siegel disk of complex symmetric matrices

We discuss the notion of Ornstein–Uhlenbeck operator on a complex manifold endowed with a Kählerian metric. We give the example of the Siegel disk. We consider the infinitesimal holomorphic representation of Sp(2n), the symplectic group of order n, into the Siegel disk Dn of symmetric complex n×n matrices. Let ρ(v)=L(v)+β(v)I, the first order differential operator on Dn associated to the element v in the Lie algebra G of Sp(2n). We denote L(v) a vector field, β(v) a function on Dn and β(v)I is the operator of multiplication by β(v). We show the existence of a basis (ek) in the Lie algebra G and of constants (ak) such that the operator ∑kakρ(ek)2 is equal to the multiplication by a constant. The constants (ak) can be taken equal to 1 for n2+n of them and to −1 for the others. Varying the coefficients in the modular factor of the representation, we obtain Ornstein–Uhlenbeck type operators on Dn of the form where is the complex conjugate of L(ek). In particular the Kählerian Laplacian on Dn is expressed as . The imaginary part of the vector field is divergence free for the measure of the holomorphic representation. This extends some of the identities obtained for the Poincaré disk in H. Airault and H. Ouerdiane (2011, 2009) [4,3].