The Generalized Cayley Map from an Algebraic Group to Its Lie Algebra

Each infinitesimally faithful representation of a reductive complex connected algebraic group G induces a dominant morphism Φ from the group to its Lie algebra g by orthogonal projection in the endomorphism ring of the representation space. The map Φ 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 Spin(V ) the map Φ essentially coincides with the classical Cayley transform. In general, properties of Φ are established and these properties are applied to deal with a separation of variables (Richardson) problem for reductive algebraic groups: Find Harm(G) so that A(G) = A(G) ⊗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) : Q(g)] of the degrees of extension fields. Among other results, Φ 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. Table of contents