Over a (not necessarily
symmetrizable) Kac-Moody algebra, we define translation functors in both the
dominant and the antidominant directions, and prove an adjoint-like property
relating the two translation functors. Using this property, we show that for any x and
y in the Weyl group, the numbers dimExtn(M(x ⋅ λ),L(y ⋅ λ)) relating Verma
modules and irreducible modules do not depend on the choice of dominant
integral weight λ. We then define operators of coherent continuation and
polynomials analogous to the Kazhdan-Lusztig polynomials and study some of their
properties.
