Jubilee 2019

In harmonic analysis on noncommutative groups one often encounters subalgebras of $L^1$, characterized by appropriate invariance properties of its elements, which are commutative. For such an algebra $A$, the Gelfand theory, called "spherical" in this context, can present various degrees of similarity with Fourier analysis on abelian groups, depending on the nature of the involved groups.

In the context of Lie groups with polynomial volume growth, investigation on numerous examples has shown that the spherical transform maps Schwartz functions in $A$ bijectively to Schwartz functions on the Gelfand spectrum, appropriately embedded into some $\mathbb{R}^n$ space.

The open question is how general this property is. In this talk we give a short account of the state of the art on this problem.