Good properties of algebras of invariants and defect of linear representations.

*(English)*Zbl 0845.14008Let \(X\) 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 \(\text{def}_G (V) = 1\).

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 \(\text{def}_G (V) = 1\).

Reviewer: E.Oeljeklaus (Bremen)

##### MSC:

14L24 | Geometric invariant theory |

14L30 | Group actions on varieties or schemes (quotients) |

14M10 | Complete intersections |