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 ). 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  for the polydisc and by Kubota  for the classical Cartan domains. Related results may be found in  under more general hypotheses, but in our case these results do not give sharp estimates. Let us refer also to Korányi  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
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