A restriction theorem for semisimple Lie groups of rank one

Abstract:

Let ${\mathfrak {g}_{\mathbf {R}}} = {\mathfrak {f}_{\mathbf {R}}} + {\mathfrak {p}_{\mathbf {R}}}$ be a Cartan decomposition of a real semisimple Lie algebra ${\mathfrak {g}_{\mathbf {R}}}$ and let $\mathfrak {g} = \mathfrak {f} + \mathfrak {p}$ be the corresponding complexification. Also let ${\mathfrak {a}_{\mathbf {R}}}$ be a maximal abelian subspace of ${\mathfrak {p}_{\mathbf {R}}}$ and let $\mathfrak {a}$ be the complex subspace of $\mathfrak {p}$ generated by ${\mathfrak {a}_{\mathbf {R}}}$. We assume $\dim {\mathfrak {a}_{\mathbf {R}}} = 1$. Now let $G$ be the adjoint group of $\mathfrak {g}$ and let $K$ be the analytic subgroup of $G$ with Lie algebra ${\text {ad}}_\mathfrak {g}(\mathfrak {f})$. If $S^\prime (\mathfrak {g})$ denotes the ring of all polynomial functions on $\mathfrak {g}$ then clearly $S^\prime (\mathfrak {g})$ is a $G$-module and a fortiori a $K$-module. In this paper, we determine the image of the subring $S^\prime {(\mathfrak {g})^K}$ of $K$-invariants in $S^\prime (\mathfrak {g})$ under the restriction map $f \mapsto f{|_{\mathfrak {f} + \mathfrak {a}}}(f \in S^\prime {(\mathfrak {g})^K})$.