A Bott-Borel-Weil theorem for diagonal ind-groups

TL;DR

This paper extends the classical Bott-Borel-Weil theorem to diagonal ind-groups, providing a method to compute cohomology of line bundles on their homogeneous ind-varieties, overcoming challenges in defining an appropriate Weyl group analog.

Contribution

It introduces a new framework for computing cohomology on ind-varieties associated with diagonal ind-groups, including defining an analog of the Weyl group for these infinite-dimensional structures.

Findings

Established a cohomology computation theorem for line bundles on $G/B$ for diagonal ind-groups.

Defined an analog of the Weyl group $W_B$ compatible with cohomology actions.

Extended classical representation theory results to infinite-dimensional ind-group settings.

Abstract

We establish a theorem computing the cohomology groups of line bundles on homogeneous ind-varieties $G/B$ for diagonal ind-groups $G$. The main difficulty in proving this analog of the classical Bott-Borel-Weil theorem is in defining an appropriate analog $W_B$ of the Weyl group so that the action of $W_B$ on weights of $G$ is compatible with the analog of the Demazure "action" of the Weyl group on the cohomology of line bundles.