Automorphism groups of ind-varieties of generalized flags

TL;DR

This paper determines the automorphism groups of ind-varieties of generalized flags, revealing they are larger than classical ind-groups and describing them via Mackey groups, with explicit matrix forms and detailed analysis of the Sato grassmannian.

Contribution

It provides a comprehensive computation of automorphism groups of ind-varieties of generalized flags, extending understanding beyond classical ind-groups and introducing Mackey groups.

Findings

Automorphism groups are larger than classical ind-groups.

Explicit matrix forms of automorphism groups are provided.

The automorphism group of the Sato grassmannian is characterized.

Abstract

We 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 $Sp(\infty)$. We 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 Japanese $GL(\infty)$.