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

(1)给出了Krull整环R上群G作用下单位群unit（R）中G的1-上圈R_chi是秩1的自由R^G-模的一个判别准则，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模的一个局部特征标判别准则. (2)设G是特征为零的代数闭域K上的单位元为代数环面的仿射代数群，X是具有平凡单位的仿射阶乘G簇.考虑$Cal O（X）$的一个有限维生成$G$-子模$V^^*$，它有一个由$G ^0 $的权向量组成的$K$-基$Omega$，使得$Omega$在标准态射$V反斜杠{0}下不退化到Bbb P（V）$。设作用$（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 mapsto x + w in V$诱导$Wdslash G_w cong Vdsla ...更多信息 Sh G$.研究了$Xdslash G$，$Xdslash G_w$和$Vdslash G$的Weil因子.我们的结果是有用的，在确定某些表示的非半单约化代数群的不变量理论。(3)令$varrho：G到GL（V）$是特征为零的代数闭域K$上可对角化代数群G$的有限维有理表示。利用（2）定义的varrho$的$（W，w）$，我们证明了存在一个余自由表示$widetilde{G_w} hookrightarrow GL（W）$，使得$varrho（G_w）子集widetilde{G_w}$和$Wdslash G_w到Wdslash widetilde{G_w}$整除非分歧等价于$Vdslash G$的Gorensteinness.研究了使得Vdslash G是完全交的超曲面，并根据G的特征给出了Vdslash G是超曲面的一个判别准则. (4)我们将定义一个称为半对合的线性映射作为对合的推广，并证明任何幂零线性自同态都是对合和半对合的乘积。我们还将给出关于两个对合乘积的Djocovic定理的一个新证明。(5)研究了交换环R上具有正交基的秩为n的二次模V上的正交群O_n（V），该正交群满足有理整数环Bbb Z所满足的条件。群$O_n（V）$将被证明是一个有限群，并且$O_n（V）$中的任何元素都是三个2阶元素的乘积。少

(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