Quotients of group completions by spherical subgroups. (Q1398158)

The author is concerned with describing projective embeddings of \(G/H\), where \(G\) is a semi-simple algebraic group and \(H\) is a spherical subgroup. The author exploits knowledge of \(G\times G\)-equivariant embeddings \(X\) of \(G\), and in fact a main thrust of the article is to determine a condition under which a projective embedding \(Y\) of \(G/H\) is a quotient of \(X\) by \(H\) (or rather a quotient by \(H\) of an open subset of \(X\), denoted \(X^{ss}({\mathcal L})\), which is determined by a \(G\times H\)-linearized line bundle over \(X\)). The condition for having \(Y=X^{ss}({\mathcal L})//H\) requires the existence of a \(G\times G\)-equivariant embedding \(X_{{\mathcal O}}\) together with an equivariant surjective rational mapping \(\phi: X_{{\mathcal O}}\rightarrow G/H \cup {\mathcal O}\) for each \(G\) orbit \({\mathcal O}\) of codimension one in \(Y\). The author points out that the quotient \(X^{ss}({\mathcal L})//H\) is not a space of orbits in general. However, when the index of \(H\) in its normalizer is finite, together with other conditions, it is shown that there is a surjective, \(G\)-equivariant mapping \(\pi:X^{ss}({\mathcal L})\rightarrow X^{ss}({\mathcal L})//H\) whose fibers are the orbits of \(\{1\}\times H\).