Automorphism groups of ind-varieties of generalized flags (Q6594718)

Given a homogenous space \(X\), it is a natural problem to compute its automorphism group \(\mathsf{Aut}(X)\). In the case when \(X\) is a complex flag variety, that is, \(X = G/P\) for a connected reductive complex algebraic group and a parabolic subgroup \(P \subset G\), the automorphism group of \(X\) is well known. The authors compute the group of automorphisms of an arbitrary ind-variety of (possibly isotropic) generalized flags. Such an ind-variety is a homogeneous ind-space for one of the ind-groups \(SL(\infty)\), \(O(\infty)\) or \(Sp(\infty)\). They show that the respective automorphism groups are much larger than \(SL(\infty)\), \(O(\infty)\) or \(Sp(\infty)\), and present the answer in terms of Mackey groups. The latter are groups of automorphisms of nondegenerate pairings of (in general infinite-dimensional) vector spaces. An explicit matrix form of the automorphism group of an arbitrary ind-variety of generalized flags is also given. The case of the Sato Grassmannian is considered in detail, and its automorphism group is the projectivization of the connected component of unity in the group known as Japanese \(GL(\infty)\).