Geometric conditions for \(\square\)-irreducibility of certain representations of the general linear group over a non-Archimedean local field. (English) Zbl 1400.20047
Adv. Math. 339, 113-190 (2018); corrigendum ibid. 450, Article ID 109767, 2 p. (2024).
From the authors’ abstract: “Let \(\pi\) be an irreducible, complex, smooth representation of \(\mathrm{GL}_n\) over a local non-Archimedean (skew) field.” Assume that no two segments in the Zelevinsky parameters of \(\pi\) have the same beginning or the same end. One says \(\pi\) is \(\square\)-irreducible if the parabolic induction of \(\pi \otimes \pi\) to \(\text{GL}_{2 n}\) is irreducible …‘we give a geometric necessary and sufficient criterion for the irreducibility. The latter irreducibility property is the \(p\)-adic analogue of a special case of the notion of “real representations” introduced by B. Leclerc [Transform. Groups 8, No. 1, 95–104 (2003; Zbl 1044.17009)] and studied recently by S.-J. Kang et al. [Compos. Math. 151, No. 2, 377–396 (2015; Zbl 1366.17014)] (in the context of KLR or quantum affine algebras). Our criterion is in terms of singularities of Schubert varieties of type \(A\) and admits a simple combinatorial description. It is also equivalent to a condition studied by C. Geiß et al. [Invent. Math. 165, No. 3, 589–632 (2006; Zbl 1167.16009); Adv. Math. 228, No. 1, 329–433 (2011; Zbl 1232.17035)]. The proofs involve a lot of combinatorics with multisegments.
Reviewer: Wilberd van der Kallen (Utrecht)
MSC:
| 20G25 | Linear algebraic groups over local fields and their integers |
| 22E50 | Representations of Lie and linear algebraic groups over local fields |
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