  We consider the asymmetric simple exclusion process (ASEP) on a finite
lattice with periodic boundary conditions, conditioned to carry an atypically
low current. For an infinite discrete set of currents, parametrized by the
driving strength $s_K$, $K \geq 1$, we prove duality relations which arise from
the quantum algebra $U_q[\mathfrak{gl}(2)]$ symmetry of the generator of the
process with reflecting boundary conditions. Using these duality relations we
prove on microscopic level a travelling-wave property of the conditioned
process for a family of shock-antishock measures for $N>K$ particles: If the
initial measure is a member of this family with $K$ microscopic shocks at
positions $(x_1,\dots,x_K)$, then the measure at any time $t>0$ of the process
with driving strength $s_K$ is a convex combination of such measures with
shocks at positions $(y_1,\dots,y_K)$. which can be expressed in terms of
$K$-particle transition probabilities of the conditioned ASEP with driving
strength $s_N$.
