On the Cayley Degree of an Algebraic Group

A connected linear algebraic group G is called a Cayley group if the Lie algebra of G endowed with the adjoint G-action and the group variety of G endowed with the conjugation G-action are birationally G-isomorphic. In particular, the classical Cayley map X 7→ (In − X)(In + X) −1 between the special orthogonal group SOn and its Lie algebra son, shows that SOn is a Cayley group. In an earlier paper we classified the simple Cayley groups defined over an algebraically closed field of characteristic zero. Here we consider a new numerical invariant of G, the Cayley degree, which “measures” how far G is from being Cayley, and prove upper bounds on Cayley degrees of some groups.