Bott canonical basis?

TL;DR

This paper proposes a method to construct canonical bases for unitary representations using torus actions on Bott-Samelson manifolds, contingent on a conjectural cohomology vanishing.

Contribution

It introduces a new construction of canonical bases based on geometric actions, dependent on specific algebraic choices and a conjecture about cohomology.

Findings

Construction depends on maximal torus, Borel subgroup, and reduced expression.

Relies on a conjectural vanishing of higher cohomology.

Provides a geometric approach to canonical bases.

Abstract

Expanding an idea of Raoul Bott, we propose a construction of canonical bases for unitary representations that comes from big torus actions on families of Bott-Samelson manifolds. The construction depends only on the choices of a maximal torus, a Borel subgroup ,and a reduced expression for the longest element of the Weyl group. It relies on a conjectural vanishing of higher cohomology of sheaves of holomorphic sections of certain line bundles on the total spaces of the families, hence the question mark in the title.