A retract theorem for nilpotent Lie groups

TL;DR

This paper proves a retract theorem for nilpotent Lie groups, establishing a correspondence between certain operator fields and Schwartz functions, with implications for the structure of ideals in group algebras.

Contribution

It introduces a retract theorem for nilpotent Lie groups that links invariant submanifolds with Schwartz functions, advancing understanding of the group's harmonic analysis.

Findings

Existence of Schwartz functions matching operator fields on invariant submanifolds.

Characterization of proper G-prime ideals via G-orbits in the dual space.

Application to the structure of ideals in L^1(G).

Abstract

Let $G= \exp(\g)$ be a connected, simply connected nilpotent Lie group. We show that for every $G$-invariant smooth sub-manifold $M$ of $g^*$, there exists an open relatively compact subset $\mathcal{M}$ of $M$ such that for any smooth adapted field of operators $(F(l))_{l\in M}$ supported in $G\cdot \mathcal{M}$ there exists a Schwartz function $f$ on $G$ such that $\pi_l(f)= \op_{F(l)}$ for all $l\in M$. This retract theorem can then be used to show that for every Lie group $\G$ of automorphisms of $G$ containing the inner automorphisms of $G$ with locally closed $\G$-orbits in $\g^*$, the proper $\G$-prime two-sided closed ideals of $L^1(G)$ are the kernels of $\G$-orbits in $\hat{G}$.