Proper Actions and Representation Theory
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arXiv
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| Format: | Preprint |
| Published: |
2025
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| _version_ | 1866917431887265792 |
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| author | Kobayashi, Toshiyuki |
| author_facet | Kobayashi, Toshiyuki |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2506_15616 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Proper Actions and Representation Theory Kobayashi, Toshiyuki Representation Theory Differential Geometry 22F30, 22E46, 53C35, 11F72 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. |
| title | Proper Actions and Representation Theory |
| topic | Representation Theory Differential Geometry 22F30, 22E46, 53C35, 11F72 |
| url | https://arxiv.org/abs/2506.15616 |