Geometric invariant theory for reductive groups over non-algebraically closed fields

Online seminar Quadratic forms, Linear algebraic groups and Beyond, 8:30-9:30

https://uottawa-ca.zoom.us/j/99397061432?pwd=LzNBS1UrSVNJQ1ZiU28yWHAyNlV...

Let $G$ be a reductive linear algebraic group acting on an affine variety $X$ over a field $k$. If $k$ is algebraically closed then the Hilbert-Mumford Theorem gives a powerful tool for understanding the structure of $G(k)$-orbits in $X$: an orbit is closed if and only if it is closed under taking limits along cocharacters. Moreover, if an orbit is not closed then one can reach a closed orbit by taking a limit along a cocharacter, and by work of Hesselink/Kempf/Rousseau one can choose this cocharacter in a canonical way.

Now suppose $k$ is arbitrary, and let $x\in X(k)$. We say the orbit $G(k)\cdot x$ is cocharacter-closed over $k$ if it is closed under taking limits of $k$-defined cocharacters. There is a version of the Hilbert-Mumford Theorem which holds in this more general setting. I will discuss the notion of cocharacter-closure and its interactions with the theory of spherical buildings and the theory of $G$-complete reducibility.

This talk is based on joint work with Michael Bate, Sebastian Herpel, Gerhard R\"ohrle and Rudolf Tange.

References:

Bate M.E., Herpel S., Martin B., Röhrle G. Cocharacter-closure and the rational Hilbert-Mumford Theorem. Math. Zeit. 287 (2017), 39–72.
Open access link: https://link.springer.com/article/10.1007/s00209-016-1816-5

Bate M.E., Martin B., Röhrle G., Tange R. Closed orbits and uniform S-instability in geometric invariant theory. Trans. Amer. Math. Soc. 365 (2013), 3643–3673.
Arxiv: https://arxiv.org/abs/0904.4853

Bate M.E., Martin B., Röhrle G., Tange R. Complete reducibility and separable field extensions.C. R. Math Acad. Sci. Paris 348 (2010), no. 9–10, 495-497.
Arxiv: https://arxiv.org/abs/1002.4319