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

A binary operation on irreducible components of Lusztig's nilpotent  varieties {II}: applications and conjectures for representations of $GL_n$  over a non-archimedean local field