Abstract: [Abe--Matsumura 2015] obtain a positivity result for a ring with basis arising from the equivariant cohomology of weighted grassmannians, but state that they do not have geometric or representation theoretic interpretations of their parameters.In joint work with William Graham, we obtain a strengthened positivity theorem for any weighted flag variety, define the basis as classes of \emph{weighted Schubert varieties}, and interpret the structure constants via \emph{weighted roots}.
In the case of weighted grassmannians, the positivity result of [AM] can be recovered from our main theorem, providing a geometric interpretation of their parameters. Along the way, we use Lie theory to generalize the definition of weighted flag varieties, prove that our main theorem also holds for equivariant Chow groups, and describe a weighted Chevalley formula.
This talk strives to illuminate the theory for the special case of weighted projective space.