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giancarlo travaglini Abstract. We determine the best possible estimate for the determinant of the Jacobian of a holomorphic mapping of a bounded symmetric domain into a ball. 1. This note is concerned with the following analogue of the classical Schwarz lemma. Suppose D — ÍI^-D ^ is a bounded symmetric domain in C " with irreducible components D, realized as a circular starlike bounded domain with center 0, in accordance with Harish-Chandra's imbedding (here and in the sequel the reference for the theory of bounded symmetric domains is [5]). Let F: D- » Bn be a holomorphic mapping of D into the unit ball of C". Let J(F) denote the Jacobian matrix of F. To estimate det(/(F)(z)) we can suppose F(0) = 0 and evaluate the Jacobian at the origin (we write /(F)(0) =/(F)). Let l ^ and n ^ denote the rank and the dimension of D ^ respectively. We prove that |det(/(F))|<«-/2-n(V/,)V2' ß and that this estimate cannot be improved. The above inequality was proved by Carathéodory [2] for the polydisc and by Kubota [7] for the classical Cartan domains. Related results may be found in [3] under more general hypotheses, but in our case these results do not give sharp estimates. Let us refer also to Korányi [4] for a different, and more classical, extension of the Schwarz lemma to bounded symmetric domains. 2. Let D ^ = G^/Kp, where G ^ is the connected component of the group of holomorphic automorphisms of D ^ and K ^ is the subgroup of GM which leaves 0 fixed (A"M is a connected compact group of unitary transformations). Let g £ and ï £ be the complexifications of the Lie algebras of G ^ and Kß respectively. Under the symmetry au of gM we have the decomposition (L. = f „ + p ^ into eigenspaces of o ^ for the eigenvalues +1 and-1 respectively. We choose a Cartan subalgebra b in f: then fj £ is a Cartan subalgebra in g£. To every noncompact root aß (a root of g ^ which is not a root of f£) we associate in the standard way the element E £ of g£. The canonical realization of D is in the complex vector space p ~ which is the subalgebra of g ^ spanned by the E_a (a positive noncompact). Let A ^ denote the Harish-Chandra Received by the editors June 1, 1982

Year: 2013

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