Time: 2:30, Saturday, June 4 2022

Abstract: The studying of some topics in algebraic geometry such as GIT contructions of moduli space of curves, moduli space of vector bundles, ... leads us to consider the set of (semi)-stable vectors of a representation of algebraic groups. More concretely, let $G$ be a linear algebraic group acting on the vector space $V$ via the representation $rho: G rightarrow GL(V)$ defined over a field $k$, and let $v in V(k)$ be any nonzero vector. Then $v$ is called semistable (resp., stable) if $0
otin overline{G.v}$ (resp., the orbit $G.v$ is closed and the stabilizer $G_{v}$ is finite). The Hilbert-Mumford Theorem gave an useful criterion for semistable vectors, namely, if $G$ is reductive and $0 in overline{G.v}$, then there exists a cocharacter $lambda in X_{*}(G)_{overline{k}}$ such that $lim_{al to 0} lambda(al).v=0$. Furthermore, in 1978, G. Kempf and G. Rousseau improved this remarkable result by showing that there exists a so-called optimal cocharacter $lambda_{v}$ satisfying $lim_{al to 0}lambda_{v}(al).v=0$ and $lambda_{v}$ takes $v$ outside $G.v$ fastest in some sense. In this talk, I present some further extensions of the Hilbert-Mumford Theorem, and a couple of recent applications of invariant theory in completely reducible subgroups, and $p$-adic representations.