Let g be a Kac-Moody algebra
defined by a not necessarily symmetrizable generalized Cartan matrix. We use
operators of coherent continuation to define modules UαL(w ⋅λ) with α a simple root
of g and w in the Weyl group W of g, and then use these modules to study the
integers dimExtn(M(x ⋅ λ),L(y ⋅ λ)) for x and y in W, where λ is a dominant
integral weight, M(μ) denotes the Verma module over g of highest weight μ and L(μ)
denotes its irreducible quotient. In particular, we show that in the presence of a
parity conjecture and a weak assumption on the behavior of the modules UαL(w ⋅λ),
both of which hold in the case of a finite dimensional g, we may compute the
dimensions by induction on the length of x, recovering the coefficients of
“twisted” versions of the Kazhdan-Lusztig polynomials, where the twist comes
from the fact that we start at the top of the orbit W ⋅ λ, rather than at the
bottom.
