Let $k$ be an algebraically closed field with characteristic $p>3$  and $X$ be the restricted Cartan type Lie algebras $W$, $S$ or $H$ over $k$. In this talk, we will introduce the Weyl groups of $X$ with respect to a maximal torus and investigate some applications of these Weyl groups. Firstly, we give a complete description of semisimple orbits of $X$ by the Weyl group action, which has been obtained in a joint work with Hao Chang. Secondly, for each Weyl group $W$, there exists a natural representation $V$. The $W$-invariants of $S(V)$ and $\wedge(V)\otimes S(V)$ will be considered. Thirdly, we determine the block basis for the coinvariants of each Weyl group and their generalizations.

会议密码：2021