Spherical subgroups and double coset varieties (Q2892866)

Let \(G\) be a connected reductive algebraic group over an algebraically closed field of characteristic zero and let \(T \subset G\) be a maximal torus. If \(F,H \subset G\) are reductive subgroups, then \(G\) admits a categorical quotient \(F \backslash \!\! \backslash G /\!\!/ H\) with respect to the action of \(F \times H\) given by \((f,h) \circ g = fgh^{-1}\), which is called a double coset variety: the paper under review studies the variety \(F \backslash \!\! \backslash G /\!\!/ H\) in the case where \(G\) is a classical group, \(F = T\) and \(H\) is a connected spherical reductive subgroup. Under these assumptions, the author shows that the double coset variety \(T \backslash \!\! \backslash G /\!\!/ H\) is an affine space if and only if \(\pi(e)\) is a regular point, where \(e \in G\) is the identity element and \(\pi : G \to T \backslash \!\! \backslash G /\!\!/ H\) is the quotient morphism: in particular, if \(T \backslash \!\! \backslash G /\!\!/ H\) is non-singular, then it is necessarily an affine space. In the case of a classical group \(G\), the author lists all the connected spherical reductive subgroups \(H\) such that \(T \backslash \!\! \backslash G /\!\!/ H\) is an affine space. The proofs rely on the classification of connected spherical reductive subgroups of a simple group \(G\) given by \textit{M.~Krämer} [Compos. Math. 38, 129--153 (1979; Zbl 0402.22006)].NEWLINENEWLINEDenote \(\Lambda_+(G/H)\) the weight semigroup of \(G/H\), i.e. the semigroup of the highest weights of the simple \(G\)-modules having non-trivial \(H\)-invariant vectors. If \(G\) is a simple classical group and if \(H\subset G\) is a connected spherical reductive subgroup such that \(T \backslash \!\! \backslash G /\!\!/ H\) is an affine space, it follows by the list that \(\Lambda_+(G/H)\) is generated by a single element. This is conjectured to be true for every simple algebraic group \(G\). If the conjecture is true, then \(T \backslash \!\! \backslash G /\!\!/ H\) is never an affine space if \(G\) is an exceptional group and if \(H\subset G\) is a connected spherical reductive subgroup.