Harish–Chandra modules over invariant subalgebras in a skew-group ring

Mazorchuk, Volodymyr; Vishnyakova, Elizaveta

doi:10.4310/ajm.2021.v25.n3.a6

Cited by 5 publications

(1 citation statement)

References 17 publications

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“…With the aid of KLRW algebras, in [KTW + 18], the authors provide a bijection between the set of simple Gelfand-Tsetlin gl(n, C)-modules with a fixed character and the zero weight space of an sl(n, C)-crystal. Furthermore, the properties of the singular Gelfand-Tsetlin modules have been studied with combinatorial tools, [FGR16,FGRZ18,FRZ19,RZ18], as well as with geometric methods, [EMV,MV,Vis18].…”

Section: Introductionmentioning

confidence: 99%

Bounds of Gelfand-Tsetlin multiplicities and tableaux realizations of Verma modules

Futorny¹,

Grantcharov

Ramirez

et al. 2020

Journal of Algebra

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We introduce the notion of essential support of a simple Gelfand-Tsetlin gl n -module as an attempt towards understanding the character formula of such module. This support detects the weights in the module having maximal possible Gelfand-Tsetlin multiplicities. Using combinatorial tools we describe the essential supports of the simple socles of the universal tableaux modules. We also prove that every simple Verma module appears as the socle of a universal tableaux module. As a consequence, we prove the Strong Futorny-Ovsienko Conjecture on the sharpness of the upper bounds of the Gelfand-Tsetlin multiplicities. We also give a very explicit description of the support and essential support of the simple singular Verma module M(−ρ). MSC 2010 Classification: 16G99, 17B10. tools, [FGR16, FGRZ18, FRZ19, RZ18], as well as with geometric methods,has a basis of derivative tableaux, and the action of gl(n, C) on this basis was described in [FGRZ18] in terms of BGG differential operators and Postnikov-Stanley polynomials. We conjecture that the module V(T(v)) is universal in the sense that every simple Gelfand-Tsetlin module having m in its support is a subquotient of V(T(v m )). This conjecture was proven for generic v in [FGR15] and for 1-singular v in [FGR16, FGR17b].In the present paper we make a significant step in the understanding of the structure of V(T(v)), in particular its socle. As a generating vector we choose a special vector v ∈ C µ , called a seed (see Definition 2.2), and show that the module V(T(v)) = V(T(v)) has a simple socle V soc whose structure can be described in terms of certain oriented graphs. The simple module V soc is also a Gelfand-Tsetlin module such that V soc = z V soc [v + z] where the sum is taken over a certain set of points of C µ with integral coordinates. The dimensions of the weight spaces V soc [v + z] (called Gelfand-Tsetlin multiplicities) are finite and uniformly bounded as explained in more detail below.Set S µ := S 1 × · · · × S n and consider the free abelian group Z µ 0 consisting of elements in C µ with integer coordinates the last n of which equal zero. Denote by GT the category of all Gelfand-Tsetlin gl(n, C)-modules, and for each equivalence class ζ ∈ C µ /(Z µ 0 #S µ ) denote by GT ζ the full subcategory of GT consisting of modules whose support is contained in ζ. We have a decomposition of GT into a direct sum of componentsGT ζ in the sense that Ext i GT (M, N) = 0 for all i ≥ 0 and for any M and N in different components (see [FO14, Corollary 3.4]).An upper bound for the Gelfand-Tsetlin multiplicities of any simple Gelfand-Tsetlin module was found in [FO14, Theorem 4.12(c)]. To write this bound, fix a seed v and consider the stabilizer S π(v) of v in S µ . Let z ∈ Z µ 0 be such that v + z is in normal form (see Definition 2.1) and let (S π(v) ) z be the stabilizer of z in S π (v) . Set ζ = ζ v = C µ /(Z µ 0 #S µ )v. Then, as shown in [FO14], for any simple module M in GT ζ the upper bound on University of Texas at Arlington,

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Section: Introductionmentioning

confidence: 99%

Bounds of Gelfand-Tsetlin multiplicities and tableaux realizations of Verma modules

Futorny¹,

Grantcharov

Ramirez

et al. 2020

Journal of Algebra

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An extension of U(gln) related to the alternating group and Galois orders

Jauch

2021

Journal of Algebra

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Twisting functors and Gelfand–Tsetlin modules over semisimple Lie algebras

Futorny

Křižka

2022

Commun. Contemp. Math.

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We associate to an arbitrary positive root [Formula: see text] of a complex semisimple finite-dimensional Lie algebra [Formula: see text] a twisting endofunctor [Formula: see text] of the category of [Formula: see text]-modules. We apply this functor to generalized Verma modules in the category [Formula: see text] and construct a family of [Formula: see text]-Gelfand–Tsetlin modules with finite [Formula: see text]-multiplicities, where [Formula: see text] is a commutative [Formula: see text]-subalgebra of the universal enveloping algebra of [Formula: see text] generated by a Cartan subalgebra of [Formula: see text] and by the Casimir element of the [Formula: see text]-subalgebra corresponding to the root [Formula: see text]. This covers classical results of Andersen and Stroppel when [Formula: see text] is a simple root and previous results of the authors in the case when [Formula: see text] is a complex simple Lie algebra and [Formula: see text] is the maximal root of [Formula: see text]. The significance of constructed modules is that they are Gelfand–Tsetlin modules with respect to any commutative [Formula: see text]-subalgebra of the universal enveloping algebra of [Formula: see text] containing [Formula: see text]. Using the Beilinson–Bernstein correspondence we give a geometric realization of these modules together with their explicit description. We also identify a tensor subcategory of the category of [Formula: see text]-Gelfand–Tsetlin modules which contains constructed modules as well as the category [Formula: see text].

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Harish–Chandra modules over invariant subalgebras in a skew-group ring

Cited by 5 publications

References 17 publications

Bounds of Gelfand-Tsetlin multiplicities and tableaux realizations of Verma modules

Bounds of Gelfand-Tsetlin multiplicities and tableaux realizations of Verma modules

An extension of U(gln) related to the alternating group and Galois orders

Twisting functors and Gelfand–Tsetlin modules over semisimple Lie algebras

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