AN ATTEMPT OF UNIFIED INVARIANT THEORY OF ALGEBRAIC GROUPS AND RELATED TOPICS

(1)给出了在Krull整环$R$上的群$G$作用下，单位群$单位$(R)$中的$G$的$1$-上循环$X$的一个1阶自由$R^G$-模的一个判据，从而建立了代数环面的有限中心扩张的相对不变量模是自由的局部特征标的一个判据。(2)设$G$是特征为零的代数闭域$K$上单位为平凡单位的仿射代数群和仿射阶乘$G$-簇.考虑一个有限维生成$Cal O(X)$的$G$-子模$V^*$，具有一个由$G^0$的权重向量组成的$K$-基$Omega$，使得$Omega$在标准态射$Vback slash{0}到BBB P(V)$下不退化.假设作用$(X，G)$是稳定的，并考虑$V$的$G$-子模$W$使得$G$在$V/W$上是对角的，并且对于某些$Win V$，$W ni x映射到V$中的x+w诱导$Wdslash G_w cong Vdsla…更多的是他的钱。我们研究了$Xdslash G$、$Xdslash G_w$和$Vdslash G$的Weil因子。我们的结果对于确定不变理论中非半单还原代数群的某些表示是有用的。(3)设$varrho：G to GL(V)$是特征为零的代数闭域$K上的可对角化代数群$G$的有限维有理表示。利用定义的$varrho$(2)的$(W，w)$，我们证明了存在一个无余码表示$widetilde{G_w}hookright tarrow GL(W)$使得$varrho(G_W)子集{G_w}$Wdslash G_w到Wdslash widetilde{G_w}$除法去分支等价于$Vdslash G$的Gorenstein性.我们研究了$RHO$‘S使得$Vdslash G$是完全交的问题，给出了$Vdslash G$是超曲面的一个判据。(4)我们将定义一个称为半对合的线性映射为对合的推广，并证明了任何幂零的线性自同态都是对合和半对合的乘积。我们还将给出两个对合积的Djocovi‘c定理的一个新的证明。(5)我们研究了秩为$n$的二次模$V$上的正交群$O_n(V)$在交换环$R上满足有理整数环$BBB Z$的某些条件。群$O_n(V)$将被证明为有限群，并且$O_n(V)$中的任何元素都是2阶三个元素的乘积。

(1) We give a criterion for $R_chi$ to be a free $R^G$-module of rank one for a $1$-cocycle $chi$ of $G$ in the unit group $unit(R)$ under the action of a group $G$ on a Krull domain $R$, and consequently establish a criterion for module of relative invariants of a finite central extension of algebraic tori to be free, in terms of local characters.(2) Let $G$ be an affine algeraic group whose identity component is an algebraic torus and $X$ an affine factorial $G$-variety with trivial units, over an algebraically closed field $K$ of characteristic zero. Consider a finite dimensional generating $G$-submodule $V^^*$ of $Cal O(X)$ having a $K$-basis $Omega$ consisting of weight vectors of $G^0$ such that $Omega$ does not degenerate under the canonical morphism $Vbackslash {0} to Bbb P (V)$. Suppose that the action $(X, G)$ is stable and consider the $G$-submodule $W$ of $V$ such that $G$ is diagonal on $V/W$ and, for some $win V$, $W ni x mapsto x + w in V$ induces $Wdslash G_w cong Vdsla … More sh G$. We study on the Weil divisors of $Xdslash G$, $Xdslash G_w$ and $Vdslash G$. Our results are useful in determining certain representations of non-semisimple reductive algebraic groups in invariant theory.(3) Let $varrho : G to GL(V)$ be a finite dimensional rational representation of a diagonalizable algebraic group $G$ over an algebraically closed field $K$ of characteristic zero. Using $(W, w)$ of $varrho$ defined (2), we show the existence of a cofree representation $widetilde{G_w} hookrightarrow GL(W)$ such that $varrho(G_w) subseteq widetilde{G_w}$ and $Wdslash G_w to Wdslash widetilde{G_w}$ is divisorially unramified is equivalent to the Gorensteinness of $Vdslash G$. We study on $rho$'s such that $Vdslash G$ are complete intersections and give a criterion for $Vdslash G$ to be a hypersurface in terms of characters of $G$.(4) We shall define a linear map called a semiinvolution as a generalization of an involution, and show that any nilpotent linear endmorphism is a product of an involution and a semiinvolution. Also we shall give a new proof for Djocovi'c's theorem on a product of two involutions.(5) We study the orthogonal group $O_n(V)$ on a quadratic module $V$ of rank $n$ with an orthogonal basis over a commutative ring $R$ satisfying some condition which holds for the ring of rational integers $Bbb Z$. The group $O_n(V)$ will be shown to be a finite group and that any element in $O_n(V)$ is a product of three elements of order 2. Less