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Good properties of algebras of invariants and defect of linear representations. (English) Zbl 0845.14008
Let $$X$$ be an irreducible affine algebraic variety and $$G$$ a connected reductive group over an algebraically closed field $$k$$. Assume that $$G$$ operates on $$X$$ as a regular transformation group. Fix a Borel subgroup $$B$$ in $$G$$, a maximal torus $$T \subset B$$ and the unipotent radical $$U$$ of $$B$$. Let $$\Gamma$$ denote the character group of $$T$$. The author is interested in the structure of algebras of invariants, namely the algebra $$k[X]^U$$ of covariants, the algebra $$k[X \times X^*]^G$$ of the doubled $$G$$-action and, for homogeneous $$X = G/H$$, $$H$$ connected, the algebra $$k[{\mathfrak h}^\perp]^H$$ of invariants of the coisotropy representation of $$H$$.
Let $$c_G (X)$$ denote the complexity of the $$G$$-variety $$X$$, i.e. the minimal codimension of a $$B$$-orbit in $$X$$. If $$X$$ is spherical (i.e. $$c_G (X) = 0)$$ and factorial without non-constant invertible regular functions it is known that the above algebras are polynomial, e.g. the related orbit spaces $$X//U$$, $$X \times X^*//G$$ and $${\mathfrak h}^\perp//H$$ are affine spaces. The author studies the situation $$c_G (X) = 1$$ and is convinced that in this case the orbit spaces will turn out to be complete intersections. He proves the following result:
Let $$X$$ be unirational without non-constant invertible regular functions and $$c_G (X) = 1$$. If $$k = k[X]^G$$, then $$X//U$$ is a complete intersection. If $$k \neq k [X]^G$$ and if the semigroup generated by those dominant weights $$\lambda \in \Gamma$$ which occur non-trivially in the weight decomposition $$k[X]^U=\bigoplus_\lambda k [X]^U_\lambda$$, is contained in an open half space of $$Q \otimes_Z \Gamma$$, then $$X//U$$ is an affine space (theorem 1.6).
For the proof it is sufficient to study toric actions, since the $$T$$-variety $$X//U$$ inherits the required properties of the $$G$$-variety $$X$$.
Finally the author studies finite dimensional linear representations $$G \to GL (V)$$. Let $$\pi : V \to V//G$$ be the quotient morphism and define $$\text{def}_G (V) : = \dim \pi^{-1} (\pi (0)) - (\dim V - \dim V//G)$$. The conjecture that $$V//G$$ is an affine space if $$\text{def}_G(V) = 0$$ seems still to be open. The author presents several examples to confirm his conjecture that $$V//G$$ is a complete intersection when $$V$$ is a self dual $$G$$-module and with $$\text{def}_G (V) = 1$$.

##### MSC:
 14L24 Geometric invariant theory 14L30 Group actions on varieties or schemes (quotients) 14M10 Complete intersections
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