Kavli Affiliate: Toshiyuki Kobayashi
| First 5 Authors: Toshiyuki Kobayashi, , , ,
| Summary:
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.
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