Modular Invariants for Some Finite Pseudo-Reflection Groups

In this paper, we determine the modular invariants of finite modular pseudo-reflection subgroups of the finite general linear group ${\text{GL}}_n(q)$ acting on the tensor product of the symmetric algebra $S^{\bullet }(V)$ and the exterior algebra ${\wedge }^{\bullet }(V)$ of the natural ${\text{GL}}_n(q)$-module $V$. We are particularly interested in the case where $G$ is a subgroup of the parabolic subgroups of ${\text{GL}}_n(q)$ which is a generalization of Weyl group of Cartan type Lie algebra.