Investigating $kG$-modules using nilpotent operators
Seminar Room 1, Newton Institute
This is a report of on-going work with Jon Carlson, Julia Pevtsova, and Andrei Suslin. Our object of study is the representation theory of $kG$ where $G$ is a finite group scheme. Following Quillen's early work, first invariants involve cohomology and cohomological support varieties. These have interpretations in terms of 1-parameter subgroups and $\pi$-points. Finer invariants arise from considering the Jordan type of nilpotent operators, leading to local Jordan types, generalized support varieties, and algebraic vector bundles on projective varieties.