  Using a transversality argument, we demonstrate the positivity of certain
coefficients in the equivariant cohomology and K-theory of a generalized flag
manifold. This strengthens earlier equivariant positivity theorems (of Graham
and Anderson-Griffeth-Miller) by further constraining the roots which can
appear in these coefficients.
  As an application, we deduce that structure constants for comultiplication in
the equivariant K-theory of an infinite flag manifold exhibit an unusual
positivity property, establishing conjectures of Lam-Lee-Shimozono. Along the
way, we present alternative formulas for the back stable Grothendieck
polynomials defined by those authors, as well as a new method for computing the
coproduct coefficients.
