  In the first part of the paper we defined and studied a binary operation on
the set of irreducible components of Lusztig's nilpotent varieties of a quiver.
For type $A$ we conjecture, following Geiss and Schr\"oer, that this operation
is compatible with taking the socle of parabolic induction of representations
of general linear groups over a local non-archimedean field, at least when one
of the irreducible components is rigid. We verify this conjecture in special
cases.
