  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.
