In this paper we continue previous work on counting open string
states between D-branes by considereing open strings between D-branes
with nonzero Higgs vevs, and in particular, nilpotent Higgs
vevs, as arise, for example, when studying D-branes in orbifolds.
Ordinarily Higgs vevs can be interpreted as moving the D-brane,
but nilpotent Higgs vevs have zero eigenvalues, and so their
interpretation is more interesting - for example, they often
correspond to nonreduced schemes, which furnishes an important link
in understanding old results relating classical D-brane moduli
spaces in orbifolds to Hilbert schemes, resolutions of quotient
spaces, and the McKay correspondence. We give a sheaf-theoretic
description of D-branes with Higgs vevs, including nilpotent Higgs
vevs, and check that description by noting that Ext groups between
the sheaves modelling the D-branes, do in fact correctly count
open string states. In particular, our analysis expands the types
of sheaves which admit on-shell physical interpretations, which is
an important step for making derived categories useful for physics.