A proof of the polynomial conjecture for nilpotent Lie groups monomial representations

Abstract

Let G = exp ⁡ ( g ) G = \operatorname {exp}(\mathfrak {g}) be a connected and simply connected real nilpotent Lie group with Lie algebra g \mathfrak g , H = exp ⁡ ( h ) H = \operatorname {exp}(\mathfrak {h}) an analytic subgroup of G G with Lie algebra h \mathfrak h , χ \chi a unitary character of H H and τ = ind H G χ \tau = \text {ind}_H^G \chi the monomial representation of G G induced by χ \chi . Let D τ ( G / H ) D_{\tau }(G/H) be the algebra of the G G -invariant differential operators on the fiber bundle over the base space G / H G/H associated to the data ( H , χ ) (H,\chi ) . We prove the polynomial conjecture due to Corwin-Greenleaf stating that if τ \tau is of finite multiplicities, the algebra D τ ( G / H ) D_{\tau }(G/H) is isomorphic to the algebra C [ Γ τ ] H {{\mathbb C}[{\Gamma }_{\tau }]}^H of the H H -invariant polynomial functions on the affine subspace Γ τ = { ℓ ∈ g ∗ ; ℓ | h = − − 1 d χ } {\Gamma }_{\tau } = \{\ell \in {\mathfrak g}^*; {\ell }|_{\mathfrak h} = -\sqrt {-1}d\chi \} of g ∗ {\mathfrak g}^* . In this case, we show that any non-zero element of D τ ( G / H ) D_{\tau }(G/H) admits a fundamental tempered solution.