Symmetries in CR complexity theory

We introduce the Hermitian-invariant group $\Gamma_f$ of a proper rational map $f$ between the unit ball in complex Euclidean space and a generalized ball in a space of typically higher dimension. We use properties of the groups to define the crucial new concepts of essential map and the source rank of a map. We prove that every finite subgroup of the source automorphism group is the Hermitian-invariant group of some rational proper map between balls. We prove that $\Gamma_f$ is non-compact if and only if $f$ is a totally geodesic embedding. We show that $\Gamma_f$ contains an $n$-torus if and only if $f$ is equivalent to a monomial map. We show that $\Gamma_f$ contains a maximal compact subgroup if and only if $f$ is equivalent to the juxtaposition of tensor powers. We also establish a monotonicity result; the group, after intersecting with the unitary group, does not decrease when a tensor product operation is applied to a polynomial proper map. We give a necessary condition for $\Gamma_f$ (when the target is a generalized ball) to contain automorphisms that move the origin.