The Algebra of K-invariant Vector Fields on a Symmetric Space G/k

When G is a complex reductive algebraic group and G/K is a reductive symmetric space, the decomposition of C[G/K] as a K-module was obtained (in a non-constructive way) by Richardson, generalizing the celebrated result of Kostant-Rallis for the linearized problem (the harmonic decomposition of the isotropy representation). To obtain a constructive version of Richardson’s results, this paper studies the infinite dimensional Lie algebra X(G/K)K of K-invariant regular algebraic vector fields using the geometry of G/K and the K-spherical representations of G. Assume G is semisimple and simply-connected and let J be the algebra of K biinvariant functions on G. An explicit set of free generators for the localization X(G/K)ψ is constructed for a suitable ψ ∈ J . A commutator formula is obtained for K-invariant vector fields in terms of the corresponding K-covariant maps from G to the isotropy representation of G/K. Vector fields on G/K whose horizontal lifts to G are tangent to the Cartan embedding of G/K into G are called flat. When G is simple and simply connected, it is shown that every element of X(G/K)K is flat if and only if K is semisimple. The gradients of the fundamental characters of G are shown to generate all conjugation-invariant vector fields on G. These results are applied in the case of the adjoint representation of G = SL(2,C) to construct a conjugation invariant differential operator whose kernel furnishes a harmonic decomposition of C[G].