Geometric conditions for $\square$-irreducibility of certain representations of the general linear group over a non-archimedean local field

Abstract

Let $\pi$ be an irreducible, complex, smooth representation of $G{L}_n$ over a local non-archimedean (skew) field. Assuming $\pi$ has regular Zelevinsky parameters, we give a geometric necessary and sufficient criterion for the irreducibility of the parabolic induction of $\pi \otimes \pi$ to $G{L}_{2n}$. The latter irreducibility property is the $p$-adic analogue of a special case of the notion of "real representations" introduced by Leclerc and studied recently by Kang-Kashiwara-Kim-Oh (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 Geiss-Leclerc-Schröer.

Added references to arXiv:1710.06115 and arXiv:1705.06517

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Cited by in corpus (13)

• Advances in R-matrices and their applications (after Maulik-Okounkov, Kang-Kashiwara-Kim-Oh,...)
• Conjectures and results about parabolic induction of representations of $G{L}_n(F)$
• Decomposition rules for the ring of representations of non-Archimedean $G{L}_n$
• Modulo $p$ representations of reductive $p$-adic groups: functorial properties
• Robinson-Schensted-Knuth correspondence in the representation theory of the general linear group over a non-archimedean local field
• A tightness property of relatively smooth permutations
• Conjectures about certain parabolic Kazhdan--Lusztig polynomials
• Models of representations and Langlands functoriality
• Quantum affine algebras and Grassmannians
• Reality determining subgraphs and strongly real modules
• Some combinatorial results on smooth permutations
• Proof of a conjecture of Kudla and Rallis on quotients of degenerate principal series
• On a determinant formula for some real regular representations