By I. I Piatetskii-Shapiro
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Extra resources for Automorphic Functions and the Geometry of Classical Domains (Mathematics and Its Applications)
Theorem 1. Any linear transformation of a Siegel domain of genus 2 has theform z ~ Az+a +2iF(Bu, b)+ iF(b, b), } u ~ (7) Bu+b, where a is any real n-dimensional vector, b is any m-dimensional complex vector, A is a linear transformation of the cone V onto itself, and B is a complex linear transformation such that AF(u, v) = F(Bu, Bv) for any complex u and v. Proof Let (8) be an affine transformation of S onto itself. We will use the fact that the skeleton is invariant under transformation (8). The point (0,0) is clearly contained in the skeleton.
Note that the form (5) is analytic with respect to t for all fixed c 1 (t) and cit) and takes real B 34 THE GEOMETRY OF CLASSICAL DOMAINS values only. As a result, it is independent of t. The law of composition in the group ~, as we can easily show, is defined by the formula (Cl' a l ) x (C2' a 2 ) = (Cl + C2 , a 1 + a 2 + Q(c l , c2 )). (6) The group of linear transformations of Siegel domains of genuses 1 and 2 plays an important role in the study of these domains. Quasilinear transformations play an analogous part in the study of Siegel domains of genus 3.
That [jro, g] = 0 when g E G'. It follows from the Jacobi identity and the fact that G' is a subalgebra that w([[jr o,g1J,g2])+W([g1, [jr o,g2J]) = w([jr o, [g1,g2J]) = O. This equation and the fact that adzj 1'0 commutes with the operator j shows that the operator ad G , jro is skew Hermitian. It remains to note that skew Hermitian operators are always semisimple. The semisimplicity of the operator adzjro follows from the fact that, as we can show with no difficulty, the operator ¢(z) = [jro, z] --tz, is skew Hermitian.