By Jacques Faraut

Offers self contained exposition of the geometry of symmetric cones, and develops research on those cones and at the advanced tube domain names linked to them.

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**Extra info for Analysis on Symmetric Cones **

**Sample text**

8 and cf. Sect. 7 of Chap. 2). Let F(xo, Xl, X2) be an irreducible homogeneous polynomial defining C. , X2), X2 - 1) is the ideal generated by the polynomials F(xo, Xl, X2) and X2 - 1. The mapping f(p) = (xo(p) : XI(P) : 1), where pES, extends by continuity to a desingularization of the curve C. As a special case, if C is nonsingular then 8 is the Riemann surface of C and f = id. If, on the other hand, C has some singularities then f resolves them, for it is an isomorphism over the nonsingular points.

Further F satisfies equation (2). By Riemann's removable singularity theorem, the coefficients Ci have meromorphic continuations on S2' We return to the construction of the Riemann surface of the algebraic function F. Note that this function is holomorphic on U and is given by the projection map (p, z) 1---+ z. 1. Again, by Riemann's removable singularity theorem and by the Proposition of Sect. 10, we obtain the required mapping f: Sl --+ S2, U eSt, and a meromorphic function F on Sl. But according to Proposition 2 the decomposition of Sl into connected components defines a factorization of P, whence Sl (and U) are connected.

Definition. A mapping of Riemann surfaces f: 8 1 mal, or Galois, if its automorphism group --t 8 2 is said to be nor- Aut f ~f {g E Aut 8 1 I fog = J} acts transitively on the fibres f- 1 (p), p E 8 2 . Corollary 2. 1)jF. 11. The Riemann Surface of an Algebraic Function Proposition 1. Let be an irreducible polynomial. Then there exists a finite mapping of Riemann surfaces f: 8 1 --t 8 2 , of degree n, and a meromorphic function F E M(81 ) that satisfies the equation Fn + r(Cl) F n - l + ... + r(en) = O.