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Equidimensional varieties and associated cones. (English) Zbl 0808.14039

Let \(G\) be a complex reductive algebraic group acting on a finite- dimensional complex vector space \(V\). Let \(R\) be the ring of regular functions on \(V\), and \(R^ G\) the subring of \(G\)-invariant functions. The action (or the representation) is said to be co-free if \(R\) is a free \(R^ G\)-module. If the representation is co-free, then it is co-regular (i.e., \(R^ G\) is a polynomial ring), and equidimensional (i.e., all fibers of \(\text{Spec} R \to \text{Spec} R^ G\) have the same dimension \((=\dim (\text{Spec} R) - \dim (\text{Spec} R^ G))\). Conversely, we have, a co-regular, equidimensional representation \(V\) is co-free. Popov conjectured that every equidimensional representation of a connected reductive group \(G\) is co-free. This conjecture has been proved to be true for simple groups, and tori (by Schwarz and Wehlau), and for irreducible representations of semi-simple groups (by Littelmann). In this paper, the author proves that a normal, conical, equidimensional \(G\)-variety is co-free.
This paper makes a valuable contribution to invariant theory.

MSC:

14L30 Group actions on varieties or schemes (quotients)
14L24 Geometric invariant theory
20G05 Representation theory for linear algebraic groups
14L35 Classical groups (algebro-geometric aspects)
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