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

Abstract

Each infinitesimally faithful representation of a reductive complex connected algebraic group Ginduces 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 Gwith 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 for the coordinate ring A(G) of G we have A(G) = A(G)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) ^G : Q(g) ^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, whenGis semisimple, defines an isomorphism from the variety of unipotent elements inGto 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.