ABSTRACT

In this chapter the basic notion of complex Lie groups is introduced, and some of the essential tools that will be used in the remaining chapters are developed ([2], [8], [22]).

Complex Analytic Manifolds We first recall the definition of an analytic function of n variables. Let U ⊂ Cn be an open set. A function f : U→ C is called complex analytic (or holomorphic) on U if, given any (a1, · · · , an) ∈ U , there exist a positive number η and a power series∑

s1,···,sn≥0 cs1,···,sn(x1 − a1)s1 · · · (xn − an)sn , cs1,···,sn ∈ C

around (a1, · · · , an) such that the series converges absolutely and uniformly to the sum f(x1, · · · , xn) for all n-tuples (x1, · · · , xn) with |xi − ai| < η, 1 ≤ i ≤ n.