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

  • Erez Lapid
  • Alberto M\'ınguez
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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 □-irreducible. In this case, π(C*D)=soc(π(C)×π(D)), π(D*C)=soc(π(D)×π(C)) 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 A7 of a non-simple rigid module z≠0 which does not admit a proper rigid submodule y≠0 such that ExtΠ1(z,z/y)=0, answering negatively a question of Aizenbud and Lapid.

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Cite

BibTeXRIS
BibTeX
@misc{lapid2026socle,
  title = {{Socle formulas for rigid irreducible components of Lusztig's nilpotent varieties in type $A$}},
  author = {Erez Lapid and Alberto M\'ınguez},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.26764},
  eprint = {2610.26764},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.26764},
}
RIS
TY  - UNPB
TI  - Socle formulas for rigid irreducible components of Lusztig's nilpotent varieties in type A
AU  - Erez Lapid
AU  - Alberto M\'ınguez
PY  - 2026
DA  - 2026/10/19
PB  - arXiv
DO  - 10.48550/arxiv.2610.26764
UR  - https://arxiv.org/abs/2610.26764
AB  - 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 □-irreducible. In this case, π(C*D)=soc(π(C)×π(D)), π(D*C)=soc(π(D)×π(C)) 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≠0 which does not admit a proper rigid submodule y≠0 such that Ext_Π^1(z,z/y)=0, answering negatively a question of Aizenbud and Lapid.
ER  -