Principal Nilpotent Research Articles

Admissible representations of simple affine vertex algebras

We provide an explicit combinatorial description of highest weights of simple highest weight modules over the simple affine vertex algebra Lκ(sln+1) with n∈N of admissible level κ. For admissible simple highest weight modules corresponding to the principal, subregular and maximal parabolic nilpotent orbits we give a realization using the Gelfand–Tsetlin theory, which also allows us to obtain a realization of certain classes of simple admissible sl2-induced modules in these orbits. In particular, simple admissible sl2-induced modules are fully classified for the principal nilpotent orbit.

Generators of Supersymmetric Classical W-Algebras

Let \({\mathfrak {g}}\) be a Lie superalgebra of type \(\mathfrak {sl}\) or \(\mathfrak {osp}\) with an odd principal nilpotent element f. We consider a matrix \({\mathcal {A}}_{{\mathfrak {g}},f}\) determined by \({\mathfrak {g}}\) and f and find a generating set of the supersymmetric classical W-algebra \({\mathcal {W}}(\bar{{\mathfrak {g}}},f)\) using the row determinant of \({\mathcal {A}}_{{\mathfrak {g}},f}\).

Unipotent representations attached to the principal nilpotent orbit

In this paper, we construct and classify the special unipotent representations of a real reductive group attached to the principal nilpotent orbit. We give formulas for theK\mathbf {K}-types, associated varieties, and Langlands parameters of all such representations.

Semi-infinite cohomology and the linkage principle for [formula omitted]-algebras

Let g be a simple Lie algebra, and let Wκ be the affine W-algebra associated to a principal nilpotent element of g and level κ. We explain a duality between the categories of smooth W modules at levels κ+κc and −κ+κc, where κc is the critical level. Their pairing amounts to a construction of semi-infinite cohomology for the W-algebra.As an application, we determine all homomorphisms between the Verma modules for W, verifying a conjecture from the conformal field theory literature of de Vos–van Driel. Along the way, we determine the linkage principle for Category O of the W-algebra.

Supersymmetric W-algebras

We develop a general theory of $W$-algebras in the context of supersymmetric vertex algebras. We describe the structure of $W$-algebras associated with odd nilpotent elements of Lie superalgebras in terms of their free generating sets. As an application, we produce explicit free generators of the $W$-algebra associated with the odd principal nilpotent element of the Lie superalgebra $\mathfrak{gl}(n+1|n).$

Permutation orbifolds of Virasoro vertex algebras and W-algebras

We study permutation orbifolds of the 2-fold and 3-fold tensor product for the Virasoro vertex algebra Vc of central charge c. In particular, we show that for all but finitely many central charges (Vc⊗3)Z3 is a W-algebra of type (2,4,5,63,7,83,93,102). We also study orbifolds of their simple quotients and obtain new realizations of certain rational affine W-algebras associated to a principal nilpotent element. Further analysis of permutation orbifolds of the celebrated (2,5)-minimal vertex algebra L−225 is presented.

Representations of principal W-algebra for the superalgebra Q(n) and the super Yangian YQ(1)

We classify irreducible representations of finite W-algebra for the queer Lie superalgebra Q(n) associated with the principal nilpotent coadjoint orbits. We use this classification and our previous results to obtain a classification of irreducible finite-dimensional representations of the super Yangian YQ(1).

Whittaker coinvariants for GL(m|n)

Let Wm|n be the (finite) W-algebra attached to the principal nilpotent orbit in the general linear Lie superalgebra glm|n(C). In this paper we study the Whittaker coinvariants functor, which is an exact functor from category O for glm|n(C) to a certain category of finite-dimensional modules over Wm|n. We show that this functor has properties similar to Soergel's functor V in the setting of category O for a semisimple Lie algebra. We also use it to compute the center of Wm|n explicitly, and deduce consequences for the classification of blocks of O up to Morita/derived equivalence.

Confluent hypergeometric systems associated with principal nilpotent p-tuples

Kimura and Takano showed that taking limits of regular elements of [Formula: see text] corresponds to the process of confluence of Aomoto–Gel’fand systems. We introduce a hypergeometric system associated with a principal nilpotent [Formula: see text]-tuple, and, by using the principal nilpotent [Formula: see text]-tuple, we directly deform a hypergeometric system of Gauss type into that of Airy type. Moreover, we explicitly describe the deformation.

Weight Representations of Admissible Affine Vertex Algebras

For an admissible affine vertex algebra $V_k(\mathfrak{g})$ of type $A$, we describe a new family of relaxed highest weight representations of $V_k(\mathfrak{g})$. They are simple quotients of representations of the affine Kac-Moody algebra $\widehat{\mathfrak{g}}$ induced from the following $\mathfrak{g}$-modules: 1) generic Gelfand-Tsetlin modules in the principal nilpotent orbit, in particular all such modules induced from $\mathfrak{sl}_2$; 2) all Gelfand-Tsetlin modules in the principal nilpotent orbit which are induced from $\mathfrak{sl}_3$; 3) all simple Gelfand-Tsetlin modules over $\mathfrak{sl}_3$. This in particular gives the classification of all simple positive energy weight representations of $V_k(\mathfrak{g})$ with finite dimensional weight spaces for $\mathfrak{g}=\mathfrak{sl}_3$.

Explicit generators in rectangular affine $$\mathcal {W}$$ W -algebras of type A

We produce in an explicit form free generators of the affine W-algebra of type A associated with a nilpotent matrix whose Jordan blocks are of the same size. This includes the principal nilpotent case and we thus recover the quantum Miura transformation of Fateev and Lukyanov.

Total positivity, Schubert positivity, and geometric Satake

Let G be a simple, simply-connected complex algebraic group, and let X⊂G∨ be the centralizer of a principal nilpotent. Ginzburg and Peterson independently related the ring of functions on X with the homology ring of the affine Grassmannian GrG. Peterson furthermore connected X to the quantum cohomology rings of partial flag varieties G/P.In this paper we study three notions of positivity for X: (1) Schubert positivity arising via Peterson's work, (2) Lusztig's total positivity and (3) Mirković–Vilonen positivity obtained from the MV-cycles in GrG. The first main theorem establishes that these three notions of positivity coincide. Our second main theorem proves a parametrization of the totally nonnegative part of X, confirming a conjecture of the second author.In type A the parametrization and relationship with Schubert positivity were proved earlier by the second author. Here we tackle the general type case and also introduce a crucial new connection with the affine Grassmannian and geometric Satake correspondence.

Classical affine W-algebras associated to Lie superalgebras

In this paper, we prove classical affine W-algebras associated to Lie superalgebras (W-superalgebras), which can be constructed in two different ways: via affine classical Hamiltonian reductions and via taking quasi-classical limits of quantum affine W-superalgebras. Also, we show that a classical finite W-superalgebra can be obtained by a Zhu algebra of a classical affine W-superalgebra. Using the definition by Hamiltonian reductions, we find free generators of a classical W-superalgebra associated to a minimal nilpotent. Moreover, we compute generators of the classical W-algebra associated to spo(2|3) and its principal nilpotent. In the last part of this paper, we introduce a generalization of classical affine W-superalgebras called classical affine fractional W-superalgebras. We show these have Poisson vertex algebra structures and find generators of a fractional W-superalgebra associated to a minimal nilpotent.

Rationality of W-algebras: principal nilpotent cases

We prove the rationality of all the minimal series principal W-algebras discovered by Frenkel, Kac and Wakimoto in 1992, thereby giving a new family of rational and C_2-cofinite vertex operator algebras. A key ingredient in our proof is the study of Zhu's algebra of simple W-algebras via the quantized Drinfeld-Sokolov reduction. We show that the functor of taking Zhu's algebra commutes with the reduction functor. Using this general fact we determine the maximal spectrums of the associated graded of Zhu's algebra of all the admissible affine vertex algebras as well.

Hitchin's conjecture for simply-laced Lie algebras implies that for any simple Lie algebra

Let g be any simple Lie algebra over C. Recall that there exists an embedding of sl2 into g, called a principal TDS, passing through a principal nilpotent element of g and uniquely determined up to conjugation. Moreover, ∧(g⁎)g is freely generated (in the super-graded sense) by primitive elements ω1,…,ωℓ, where ℓ is the rank of g. N. Hitchin conjectured that for any primitive element ω∈∧d(g⁎)g, there exists an irreducible sl2-submodule Vω⊂g of dimension d such that ω is non-zero on the line ∧d(Vω). We prove that the validity of this conjecture for simple simply-laced Lie algebras implies its validity for any simple Lie algebra.Let G be a connected, simply-connected, simple, simply-laced algebraic group and let σ be a diagram automorphism of G with fixed subgroup K. Then, we show that the restriction map R(G)→R(K) is surjective, where R denotes the representation ring over Z. As a corollary, we show that the restriction map in the singular cohomology H⁎(G)→H⁎(K) is surjective. Our proof of the reduction of Hitchin's conjecture to the simply-laced case relies on this cohomological surjectivity.

PrincipalW-algebras for GL(m|n)

We consider the (finite) $W$-algebra $W_{m|n}$ attached to the principal nilpotent orbit in the general linear Lie superalgebra $\mathfrak{gl}_{m|n}(\mathbb C)$. Our main result gives an explicit description of $W_{m|n}$ as a certain truncation of a shifted version of the Yangian $Y(\mathfrak{gl}_{1|1})$. We also show that $W_{m|n}$ admits a triangular decomposition and construct its irreducible representations.

Cyclic elements in semisimple lie algebras

We develop a theory of cyclic elements in semisimple Lie algebras. This notion was introduced by Kostant, who associated a cyclic element with the principal nilpotent and proved that it is regular semisimple. In particular, we classfiy all nilpotents giving rise to semisimple and regular semisimple cyclic elements. As an application, we obtain an explicit construction of all regular elements in Weyl groups.

A Brylinski filtration for affine Kac–Moody algebras

Braverman and Finkelberg have recently proposed a conjectural analogue of the geometric Satake isomorphism for untwisted affine Kac–Moody groups. As part of their model, they conjecture that (at dominant weights) Lusztig's q-analog of weight multiplicity is equal to the Poincare series of the principal nilpotent filtration of the weight space, as occurs in the finite-dimensional case. We show that the conjectured equality holds for all affine Kac–Moody algebras if the principal nilpotent filtration is replaced by the principal Heisenberg filtration. The main body of the proof is a Lie algebra cohomology vanishing result. We also give an example to show that the Poincare series of the principal nilpotent filtration is not always equal to the q-analog of weight multiplicity. Finally, we give some partial results for indefinite Kac–Moody algebras.

A Finite Analog of the AGT Relation I: Finite W-Algebras and Quasimaps’ Spaces

Recently Alday, Gaiotto and Tachikawa [2] proposed a conjecture relating 4-dimensional super-symmetric gauge theory for a gauge group G with certain 2-dimensional conformal field theory. This conjecture implies the existence of certain structures on the (equivariant) intersection cohomology of the Uhlenbeck partial compactification of the moduli space of framed G-bundles on $${\mathbb{P}^2}$$ . More precisely, it predicts the existence of an action of the corresponding W-algebra on the above cohomology, satisfying certain properties. We propose a “finite analog” of the (above corollary of the) AGT conjecture. Namely, we replace the Uhlenbeck space with the space of based quasi-maps from $${\mathbb{P}^1}$$ to any partial flag variety G/P of G and conjecture that its equivariant intersection cohomology carries an action of the finite W-algebra $${U(\mathfrak{g},e)}$$ associated with the principal nilpotent element in the Lie algebra of the Levi subgroup of P; this action is expected to satisfy some list of natural properties. This conjecture generalizes the main result of [5] when P is the Borel subgroup. We prove our conjecture for G = GL(N), using the works of Brundan and Kleshchev interpreting the algebra $${U(\mathfrak{g},e)}$$ in terms of certain shifted Yangians.

On some exotic finite subgroups of E 8 and Springer’s regular elements of the Weyl group

A proof (by Serre and by Cohen, Griess and Lisser) verified, in the special case of E8, a conjecture of mine, that the finite projective group L2(61) embeds in \( {E_8}\left( \mathbb{C} \right) \). Subsequently, Griess and Ryba have shown (using computers) that L2(49) and, in addition, (established by Serre without computers) L2(41) also embed in \( {E_8}\left( \mathbb{C} \right) \). That is, if K = 30, 24, 20 and k ∈ K then L2(2k + 1) embeds in \( {E_8}\left( \mathbb{C} \right) \). In this paper we show that the “Borel” subgroup B(k) of L2(2k + 1), k ∈ K, has a uniform construction. The theorem uses a result of T. Springer on the existence in \( {E_8}\left( \mathbb{C} \right) \) of three regular elements of the Weyl group, having orders k ∈ K, and associated to the regular, subregular and subsubregular nilpotent elements. Springer’s result generalizes (in the E8 case) a 1959 general result of mine relating the principal nilpotent element with the Coxeter element.