International Congress of Mathematicians

Some Results Related to Group Actions in Several Complex Variables 745A group action on a set can be restricted on various cases. When the set isa topological space and the group is a topological group, the action is continuous,then one gets a topological transformation group; when the space is a metric space,the transformation preserves the metric, then one gets a motion group; when theset is a differentiable manifold and the group is a Lie group, the action is differentiable,then one gets a Lie transformation group; when the set is a vector space, thetransformation preserves the vector space structure, then one gets a linear transformationgroup; when the set is an algebraic variety (or a scheme), the group is analgebraic group, and the action is algebraic, one gets an algebraic transformationgroup; when the set is a complex space, the transformation is holomorphic, andthe action is real analytic, then one gets a (real) holomorphic transformation group(note that in this case, if the action is continuous then it is also real analytic); ifthe set is a complex space, the group is a complex Lie group, and the the action isholomorphic, then one gets a complex (holomorphic) transformation group.In this talk, we're mainly concerned with the last case. We consider a complexLie group Gc with a real form GR acting holomorphically on a complex manifold(also called holomorphic Gc- manifold) and a G^-invariant domain. It's knownthat a complex reductive Lie group has a unique maximal compact subgroup up toconjugate as its real form, but it also has many noncompact real forms.A group action on a set can be regarded as a representation of the group on thewhole group of transformations. An effective group action means the representationis faithful, so it corresponds to a (closed) subgroup of the whole transformationgroup.Actually, many domains in several complex variables such as Hartogs, circular,Reinhardt and tube domains can be formulated in the setting of group actions.Examples, a) Hartogs and circular domains: consider the Hartogs actionof C* with the real form S 1 on C: C x C" -) C n given by (t, (z t ,-- -,z n j) -•(tzi, z 2 , • • •, z n ), then Hartogs domain is ^-invariant domain; consider the circularaction of C* with the real form S 1 on C": C'xC -I C" given by (t, (zi, • • •, z n j) -t(tzi,tz 2 , • • • ,tz n ), then circular domain is SMnvariant domain.b) Reinhardt domains: consider the Reinhardt action of (C*) n on C" given byvv^i, ' ' ', in), \Zi, - - -, z n )) y \%iZi, - - -, t n z n ),then Reinhardt domain is (S' 1 )"-invariant domain. One can similarly defines matrixReinhardt domainsc) tube domains: consider the action of R" on C" given by (r, z) —¥ r + z, thenR"-invariant domain is tube domain.d) future tube: let AT 4 be the Alinkowski space with the Lorentz metric:x-y = xnyn-xiyi-x 2 y 2 -X3y3, where a; = (xo,xi,x 2 ,xz),y = (yo,yi,y 2 ,yz) £ -R 4 ;let V + and V~ = —Y r+ be the future and past light cones in R 4 respectively, i.e.Y r± = {y £ M : y 2 > 0, ±yn > 0}, the corresponding tube domains r 1 * 1 = T v =R 4 + iY r± in C 4 are called future and past tubes; let L be the Lorentz group, i.e.L = 0(1,3), L has four connected components, denote the identity component ofL by Lfi, which is called the restricted Lorentz group, i.e. Lfi = 30+(1,3); letL(C) be the complex Lorentz group, i.e.L = 0(1,3,C) = 0(4, C),L(C) has two