Socle formulas for rigid irreducible components of Lusztig's nilpotent varieties in type $A$

Source abstract

Let $F$ be a non-archimedean local field. For any irreducible component $C$ of a nilpotent variety of type $A$ let $π(C)$ be the irreducible representation of a general linear group over $F$ attached to it. We show that if the multisegment of $C$ is multiplicity free, $C$ is rigid if and only if $π(C)$ is $\square$-irreducible. In this case, $$π(C*D)=soc\bigl(π(C)\timesπ(D)\bigr),\qquad π(D*C)=soc\bigl(π(D)\timesπ(C)\bigr)$$ for every irreducible component $D$, where $*$ is the binary operation defined by Aizenbud and Lapid, in accordance with a conjecture of Lapid and Mínguez. We also give a computer-assisted example in $A_7$ of a non-simple rigid module $z\ne0$ which does not admit a proper rigid submodule $y\ne0$ such that $Ext_Π^1(z,z/y)=0$, answering negatively a question of Aizenbud and Lapid.