Symbols and orbits for 3-step nilpotent Lie groups (Q1061875)

Let \(f\in {\mathcal S}^*(G)\) where G is a 3-step nilpotent Lie group with 1-dimensional center. Assume that the Kirillov symbol \(\sigma (f)=(f\circ \exp)^{\wedge}\in {\mathcal S}^*({\mathfrak g}^*)\) is in \(C^{\infty}({\mathfrak g}^*)\), where \({\mathfrak g}\) denotes the Lie algebra of G. The author shows that left convolution by f is a bounded operator on \(L^ 2(G)\) if certain estimates on \(\sigma\) (f) and its derivatives hold. An important ingredient is a structure theorem, proved in the paper: \({\mathfrak g}\) is isomorphic to a Lie subalgebra of an explicitly described 3-step nilpotent Lie algebra \({\mathcal S}_ n{\mathcal H}_ n\) for some \(n\in {\mathbb{N}}\). \({\mathcal S}_ n{\mathcal H}_ n\) is a semidirect product of the abelian Lie algebra \({\mathcal S}_ n\) of \(n\times n\) symmetric matrices with the \((2n+1)\)-dimensional Heisenberg algebra \({\mathcal H}_ n\).