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A restriction theorem for semisimple Lie groups of rank one

1983; American Mathematical Society; Volume: 279; Issue: 2 Linguagem: Inglês

10.1090/s0002-9947-1983-0709574-9

1088-6850

Juan Tirao,

Homotopy and Cohomology in Algebraic Topology

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