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

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(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