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

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