Topics in Representation Theory: More About the Exponential Map 1 More About the Exponential Map

We defined the exponential map rather abstractly, using the definition of vector fields as derivations. For many purposes it is useful to think of the tangent space Tm(M) of vectors at a point m ∈ M not in terms of derivations, but as the set of possible velocity vectors γ′(0) for smooth curves γ(t) in M, such that γ(0) = m. A vector field then corresponds to a set of integral curves. A smooth map f:M → N takes these integral curves γ(t) to differentiable curves f(γ(t)) on N, the derivative map df = f ∗ takes the original vector field to the vector field of velocity vectors for the curves on N. In this section we’ll derive two very useful related formulas about the ex-ponential map. These formulas often appear in surprising places througout mathematics. For instance, the first one we will consider appears crucially in the Atiyah-Singer index theorem. In this section we will be assuming that our groups are matrix groups and that we can use the power series formula for the exponential. For a much more detailed exposition of this material including careful attention to the analytical details, see [1]. In calculus the differential of the exponential function satisfies deax = aeaxdx with a a constant. Formulas from calculus like this often have simple general-izations to the case where x is a vector and a a matrix, often it is just a matter of paying careful attention to the order of the symbols. The generalization of this formula turns out to be not so trivial. Theorem 1. The differential of the exponential map exp: X ∈ g → G is a map dexp = exp ∗ : g → Texp(X)G given by exp∗(X)Y = (lexp(X)) ∗ ◦