On enhanced general linear groups: nilpotent orbits and support variety for Weyl module

Associated with a reductive algebraic group $G$ and its rational representation $(\rho ,M)$ over an algebraically closed filed $\mathbf{k}$, the authors define the enhanced reductive algebraic group $\underset{\_}{G}:=G{⋉}_{\rho }M$, which is a product variety $G\times M$ and endowed with an enhanced cross product in ^[5]. If $\underset{\_}{G}=GL(V){⋉}_{\eta }V$ with the natural representation $(\eta ,V)$ of $\text{GL}(V)$, it is called an enhanced general linear algebraic group. And the authors give a precise classification of finite nilpotent orbits via a finite set of so-called enhanced partitions of $n=\dim V$ for the enhanced group $\underset{\_}{G}=\text{GL}(V){⋉}_{\eta }V$ in [6, Theorem 3.5]. We will give another way to prove this classification theorem in this paper. Then we focus on the support variety of the Weyl module for $\underset{\_}{G}=\text{GL}(V){⋉}_{\eta }V$ in characteristic $p$, and obtain that it coinsides with the closure of an enhanced nilpotent orbit under some mild condition.