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 $\X(G/K)^K_{\psi}$ is constructed for a suitable $\psi \in \J$. 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 \emph{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 $\C[G]$.