Global sections of line bundles on a wonderful compactification of the general linear group

TL;DR

This paper studies the structure of global sections of line bundles on a compactification of the general linear group, revealing a canonical decomposition into simple modules related to flag varieties.

Contribution

It provides a canonical decomposition of global sections of line bundles on the compactification into simple modules, connecting to flag varieties.

Findings

Decomposition of global sections into simple G-modules

Connection to flag varieties and their line bundles

Canonical nature of the decomposition

Abstract

In a previous paper we have constructed a compactification $KGl_n$ of the general linear group $Gl_n$, which in many respects is analogous to the so called wonderful compactification of adjoint semisimple algebraic groups as studied by De Concini and Procesi. In particular there is an action of $G=Gl_n\times Gl_n$ on this compactification. In this paper we show how the space of global section of an arbitrary $G$-linearized line bundle on $KGl_n$ decomposes canonically into a direct sum of simple $G$-modules which are themselves given as the spaces of global sections of line bundles on the product of two copies of the full flag manifold parametrizing flags in an $n$-dimensional vector space.