On valuations invariant under a reductive group (Q1318063)

Let \(G\) be a reductive group defined over an algebraically closed field \(k\) and let \(X\) be a \(G\)-variety. In this paper we study \(G\)-invariant valuations \(v\) of the field \(K\) of rational functions on \(X\). These objects are fundamental for the theory of equivariant completions of \(X\). Let \(B\) be a Borel subgroup and \(U\) the unipotent radical of \(B\). It is proved that \(v\) is uniquely determined by its restriction to \(K^ U\). Then we study the set of invariant valuations having some fixed restriction \(v_ 0\) to \(K^ B\). If \(v_ 0\) is geometric (i.e., induced by a prime divisor) then this set is a polyhedron in some vector space. In characteristic zero we prove that this polyhedron is a simplicial cone and in fact the fundamental domain of the finite reflection group \(W_ X\). Thus, the classification of invariant valuations is almost reduced to the classification of valuations of \(K^ B\).