Proper Actions and Representation Theory

TL;DR

This paper explores recent advances in proper actions, linking geometric criteria with representation theory, and introduces new quantitative methods to analyze properness in group actions.

Contribution

It presents novel approaches to quantify proper actions using sharpness and dynamical volume estimates, with applications to temperedness criteria in representation theory.

Findings

Established geometric criteria for properness in reductive groups.

Linked properness of actions to discrete decomposability of unitary representations.

Introduced quantitative measures like sharpness and volume estimates to analyze properness.

Abstract

This exposition presents recent developments on proper actions, highlighting their connections to representation theory. It begins with geometric aspects, including criteria for the properness of homogeneous spaces in the setting of reductive groups. We then explore the interplay between the properness of group actions and the discrete decomposability of unitary representations realized on function spaces. Furthermore, two contrasting new approaches to quantifying proper actions are examined: one based on the notion of sharpness, which measures how strongly a given action satisfies properness; and another based on dynamical volume estimates, which measure deviations from properness. The latter quantitative estimates have proven especially fruitful in establishing temperedness criterion for regular unitary representations on $G$-spaces. Throughout, key concepts are illustrated with concrete geometric and representation-theoretic examples.