By I. I Piatetskii-Shapiro
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Extra info for Automorphic Functions and the Geometry of Classical Domains (Mathematics and Its Applications)
Theorem 1. :». We denote the algebras of the Lie groups ® and ®o by G and Go, respectively, and by j an endomorphism induced in G by an endomorphism of the complex structure. Then the triple (G, Go,}) is aj-algebra. In order to prove this theorem it is sufficient to verify conditions 0:, [3, and'}' of the definition of j-algebras. Condition 0: is the well-known condition for a manifold to have an invariant complex structure, while [3 follows from results of Koszul  in the following way. Let ds 2 be the 48 THE GEOMETRY OF CLASSICAL DOMAINS Bergman metric in f0.
It is also clear that K is a bounded domain in P. We should also note that (27) implies that the transformation E + p is nondegenerate. , (1) the form Liu, v) is complex linear with respect to the first argument 40 THE GEOMETRY OF CLASSICAL DOMAINS and real Imear with respect to the second argument, and (2) the difference Lp(u, v) - LP( v, u) is purely imaginary. The first of these statements is clear, while the second can be proved in the following manner. We set U1 = (E+p)-1 U, V 1 = (E+p) 1V.
A condition for membership (of a point obtained) in S can be written in the following manner: 1m (A(t)z + a(O, t)) - Re LtCb(t), bet)) E V (9) for any tEf». The initial point (z, 0, t) belongs to S if and only if y = 1m Z E V. z, 0, t), where A is an arbitrary positive real number, belong to S, along with the point (z, 0, t). Substituting AZ for Z in (9) and passing to the limit as A --? 00, we find that for any t E f» 1m (A(t)z)EV , then ImzE V (10) (V is the closure of V). From this it clearly follows that ACt) is a real matrix for any t.
Automorphic Functions and the Geometry of Classical Domains (Mathematics and Its Applications) by I. I Piatetskii-Shapiro