  We investigate the action of discretized Virasoro generators, built out of
generators of the lattice Temperley-Lieb algebra ("Koo-Saleur
generators"[arXiv:hep-th/9312156]), in the critical XXZ quantum spin chain. We
explore the structure of the continuum-limit Virasoro modules at generic
central charge for the XXZ vertex model, paralleling [arXiv:2007.11539] for the
loop model. We find again indecomposable modules, but this time not logarithmic
ones. The limit of the Temperley-Lieb modules $\mathcal{W}_{j,1}$ for $j\neq 0$
contains pairs of "conjugate states" with conformal weights
$(h_{r,s},h_{r,-s})$ and $(h_{r,-s},h_{r,s})$ that give rise to dual
structures: Verma or co-Verma modules. The limit of
$\mathcal{W}_{0,\mathfrak{q}^{\pm2}}$ contains diagonal fields
$(h_{r,1},h_{r,1})$ and gives rise to either only Verma or only co-Verma
modules, depending on the sign of the exponent in $\mathfrak{q}^{\pm 2}$. In
order to obtain matrix elements of Koo-Saleur generators at large system size
$N$ we use Bethe ansatz and Quantum Inverse Scattering methods, computing the
form factors for relevant combinations of three neighbouring spin operators.
Relations between form factors ensure that the above duality exists already at
the lattice level. We also study in which sense Koo-Saleur generators converge
to Virasoro generators. We consider convergence in the weak sense,
investigating whether the commutator of limits is the same as the limit of the
commutator? We find that it coincides only up to the central term. As a side
result we compute the ground-state expectation value of two neighbouring
Temperley-Lieb generators in the XXZ spin chain.
