Abstract: In this talk I will first present the classical Cayley Tranformation for matrices. Then the generalization to reductive groups, due to B. Kostant and me:
Each infinitesimally faithful representation of a reductive complex connected algebraic group $G$ induces a dominant morphism $\Phi$ from the group to its Lie algebra $\g$ by orthogonal projection in the endomorphism ring of the representation space. The map $\Phi$ identifies the field $Q(G)$ of rational functions on $G$ with an algebraic extension of the field $Q(\g)$ of rational functions on $\g$. For the spin representation of $\on{Spin}(V)$ the map $\Phi$ essentially coincides with the classical Cayley transform. In general, properties of $\Phi$ are established and these properties are applied to deal with a separation of variables (Richardson) problem for reductive algebraic groups: Find $\on{Harm}(G)$ so that for the coordinate ring $A(G)$ of $G$ we have $A(G) = A(G)^G\otimes \on{Harm}(G)$. As a consequence of a partial solution to this problem and a complete solution for $SL(n)$ one has in general the equality $[Q(G):Q(\g)] = [Q(G)^G:Q(\g)^G]$ of the degrees of extension fields. Among other results, $\Phi$ yields (for the complex case) a generalization, involving generic regular orbits, of the result of Richardson showing that the Cayley map, when $G$ is semisimple, defines an isomorphism from the variety of unipotent elements in $G$ to the variety of nilpotent elements in $\g$. In addition if $G$ is semisimple the Cayley map establishes a diffeomorphism between the real submanifold of hyperbolic elements in $G$ and the space of infinitesimal hyperbolic elements in $\g$. Some examples are computed in detail. In particular the classical Cayley tranformation for orthogonal matrices, multiplied by a regular function (vanishing on the poles) equal the Cayley map fort the spin representation.