## A Foundation of Group-theoretic Analysis on Manifolds. Colloquium di dipartimento. Dipartimento di Matematica, Università di Roma “Tor Vergata” (online), 18 February 2021.

Symmetry of geometry is inherited by symmetry of function spaces, called the regular representation. From this viewpoint, the classical theory of expansions such as Fourier series or spherical harmonics may be interpreted as "analysis and synthesis" of the regular representation.

In this talk, we address some fundamental questions about the regular representation on manifolds $X$ acted algebraically by reductive Lie groups $G$ such as $GL(n, \textbf{R})$.

A. Does the group $G$ "control well" the space of function on $X$?
B. What can we say about "spectrum" for $L^2(X)$?

We highlight "multiplicity" for A and "temperdness" for B, and explain some geometric ideas of the solution.