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

Abstract

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 ( 2 n ) , the symplectic group of order n , into the Siegel disk D n of symmetric complex n × n matrices. Let ρ ( v ) = L ( v ) + β ( v ) I , the first order differential operator on D n associated to the element v in the Lie algebra G of Sp ( 2 n ) . We denote L ( v ) a vector field, β ( v ) a function on D n and β ( v ) I is the operator of multiplication by β ( v ) . We show the existence of a basis ( e k ) in the Lie algebra G and of constants ( a k ) such that the operator ∑ k a k ρ ( e k ) 2 is equal to the multiplication by a constant. The constants ( a k ) can be taken equal to 1 for n 2 + 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 D n of the form ∑ k a k ρ ( e k ) L ( e k ) ¯ where L ( e k ) ¯ is the complex conjugate of L ( e k ) . In particular the Kählerian Laplacian on D n is expressed as ∑ k a k L ( e k ) L ( e k ) ¯ . The imaginary part of the vector field ∑ k a k β ( e k ) L ( e k ) ¯ 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] .