Spectral synthesis for coadjoint orbits of nilpotent Lie groups

TL;DR

This paper characterizes the primary ideals in the group algebra of a connected nilpotent Lie group by linking them to invariant polynomials associated with coadjoint orbits, advancing the understanding of the algebraic structure of such groups.

Contribution

It introduces a novel method to identify primary ideals in $L^1(G)$ using invariant polynomials related to coadjoint orbits of nilpotent Lie groups.

Findings

Primary ideals correspond to invariant polynomials on coadjoint orbits.

The method applies to all connected nilpotent Lie groups.

Provides a new algebraic description of the primitive ideal space.

Abstract

We determine the space of primary ideals in the group algebra $L^1(G)$ of a connected nilpotent Lie group by identifying for every $\pi\in\hat G $ the family ${\mathcal I}^\pi $ of primary ideals with hull $\{\pi\}$ with the family of invariant polynomials of a certain finite dimensional subspace ${\mathcal P}_Q^\pi $ of the space of polynomials ${\mathcal P}(G) $ on $G $.