A filtration of a representation whose successive quotients are isomorphic to Demazure modules is called an excellent filtration. In this paper we study graded multiplicities in excellent filtrations of fusion products for the current algebra \( {\mathfrak{sl}}_2\left[t\right] \). We give a combinatorial formula for the polynomials encoding these multiplicities in terms of two-dimensional lattice paths

We prove that the Yangian associated to an untwisted symmetric affine Kac–Moody Lie algebra is isomorphic to the Drinfeld double of a shuffle algebra. The latter is constructed in [YZ14] as an algebraic formalism of cohomological Hall algebras. As a consequence, we obtain the Poincare–Birkhoff–Witt (PBW) theorem for this class of affine Yangians. Another independent proof of the PBW theorem is given

Let G be a connected reductive algebraic group defined over a finite field with q elements. In the 1980’s, Kawanaka introduced generalised Gelfand–Graev representations of the finite group \( G\left({\mathbbm{F}}_q\right) \), assuming that q is a power of a good prime for G. These representations have turned out to be extremely useful in various contexts. Here we investigate to what extent Kawanaka’s

A Tits polygon is a bipartite graph in which the neighborhood of each vertex is endowed with an “opposition relation” satisfying certain axioms. Moufang polygons are precisely the Tits polygons in which these opposition relations are all trivial. Every Tits polygon has a distinguished set of circuits. A Tits quadrangle is a Tits polygon in which these circuits all have length 8. There is a standard

We continue our program on classiffication of holomorphic vertex operator algebras of central charge 24. In this article, we show that there exists a unique strongly regular holomorphic VOA of central charge 24, up to isomorphism, if its weight one Lie algebra has the type C4,10, D7,3A3,1G2,1, A5,6C2,3A1,2, A3,1C7,2, D5,4C3,2A\( {A}_{1,1}^2 \), or E6,4C2,1A2,1. As a consequence, we have verified that

We call a flag variety admissible if its automorphism group is the projective general linear group. (This holds in most cases.) Let K be a field of characteristic 0, containing all roots of unity. Let the K-variety X be a form of an admissible flag variety. We prove that X is either ruled, or the automorphism group of X is bounded, meaning that there exists a constant C ∈ ℕ such that if G is a finite

This paper is a continuation of the theory of cyclic elements in semisimple Lie algebras, developed by Elashvili, Kac and Vinberg. Its main result is the classification of semisimple cyclic elements in semisimple Lie algebras. The importance of this classification stems from the fact that each such element gives rise to an integrable hierarchy of Hamiltonian PDE of Drinfeld–Sokolov type.

We consider the set of affine alcoves associated with a root system R as a topological space and define a certain category S of sheaves of \( {\mathcal{Z}}_k \)-modules on this space. Here \( {\mathcal{Z}}_k \) is the structure algebra of the root system over a field k. To any wall reection s we associate a wall crossing functor on S. In the companion article [FL] we prove that S encodes the simple

In this paper, we study the Gelfand–Cetlin systems and polytopes of the co-adjoint SO(n)-orbits. We describe the face structure of Gelfand–Cetlin polytopes and iterated bundle structure of Gelfand–Cetlin fibers in terms of combinatorics on the ladder diagrams. Using this description, we classify all Lagrangian fibers.

The minimal model \( \mathfrak{osp}\left(1|2\right) \) vertex operator superalgebras are the simple quotients of affine vertex operator superalgebras constructed from the affine Lie super algebra \( \hat{\mathfrak{osp}}\left(1\left|2\right.\right) \) at certain rational values of the level k. We classify all isomorphism classes of ℤ2-graded simple relaxed highest weight modules over the minimal model

This erratum concerns the Kähler potential F defined in the second displayed equation of [M, Sect. 4.2], which is incorrectly claimed to be proper.

We relate the category of sheaves on alcoves that was constructed in [FL1] to the representation theory of reductive algebraic groups. In particular, we show that its indecomposable projective objects encode the irreducible rational characters of a connected, semisimple and simply connected reductive algebraic group for characteristics above the Coxeter number.

We prove a version of a theorem of Auslander for finite group coactions on noetherian graded down-up algebras.

An indecomposable Lie group with Riemannian bi-invariant metric is always simple and hence Einstein. Indefinite bi-invariant metrics are not necessarily Einstein, not even on simple Lie groups. We study the question of whether a semi-Riemannian bi-invariant metric is conformal to an Einstein metric. We obtain results for all three cases in the structure theorem by Medina and Revoy for indecomposable

We introduce rectangular elements in the symmetric group. In the framework of PBW degenerations, we show that in type A the degenerate Schubert variety associated with a rectangular element is indeed a Schubert variety in a partial ag variety of the same type with larger rank. Moreover, the degenerate Demazure module associated with a rectangular element is isomorphic to the Demazure module for this

We prove that commutative algebras in braided tensor categories do not admit faithful Hopf algebra actions unless they come from group actions. We also show that a group action allows us to see the algebra as the regular algebra in the representation category of the acting group.

Let V be an n-dimensional algebraic representation over an algebraically closed field K of a group G. For m > 0, we study the invariant rings K[Vm]G for the diagonal action of G on Vm. In characteristic zero, a theorem of Weyl tells us that we can obtain all the invariants in K[Vm]G by the process of polarization and restitution from K[Vn]G. In particular, this means that if K[Vn]G is generated in

Let X be a complete toric variety equipped with the action of a torus T, and G a reductive algebraic group, defined over an algebraically closed field K. We introduce the notion of a compatible ∑-filtered algebra associated to X, generalizing the notion of a compatible ∑-filtered vector space due to Klyachko, where ∑ denotes the fan of X. We combine Klyachko's classification of T-equivariant vector

We define two algebra automorphisms Ͳ0 and Ͳ1 of the q-Onsager algebra \( {\mathcal{B}}_c \), which provide an analog of G. Lusztig's braid group action for quantum groups. These automorphisms are used to define root vectors which give rise to a PBW basis for \( {\mathcal{B}}_c \). We show that the root vectors satisfy q-analogs of Onsager's original commutation relations. The paper is much inspired

We classify the effective and transitive actions of a Lie group G on an n-dimensional non-degenerate hyperboloid (also called real pseudo-hyperbolic space), under the assumption that G is a closed, connected Lie subgroup of SO0(n–r; r+1), the connected component of the indefinite special orthogonal group. Assuming additionally that G acts completely reducibly on ℝ n +1 , we also obtain that any G-homogeneous

Let \( \mathfrak{g} \) be a complex semisimple Lie algebra. For a regular element x in \( \mathfrak{g} \) and a Hessenberg space H ⊆ \( \mathfrak{g} \), we consider a regular Hessenberg variety X(x, H) in the ag variety associated with \( \mathfrak{g} \). We take a Hessenberg space so that X(x, H) is irreducible, and show that the higher cohomology groups of the structure sheaf of X(x, H) vanish. We

We study non-standard Verma type modules over the Kac-Moody queer Lie superalgebra 𝔮(n)(2). We give a sufficient condition under which such modules are irreducible. We also give a classification of all irreducible diagonal ℤ-graded modules over certain Heisenberg Lie superalgebras contained in 𝔮(n)(2).

We study the Fulton-MacPherson operational Chow rings of good moduli spaces of properly stable, smooth, Artin stacks. Such spaces are étale locally isomorphic to geometric invariant theory quotients of affine schemes, and are therefore natural extensions of GIT quotients. Our main result is that, with ℚ-coefficients, every operational class can be represented by a topologically strong cycle on the

We consider Lie groups G endowed with a pair of anticommuting left-invariant abelian complex structures (J1, J2) and a left-invariant (possibly indefinite) metric g such that (G, J1, J2, g) results to be a hyperkähler manifold. We give a classification of their Lie algebras up to dimension 12 and study some of their geometric properties. In particular, we show that all such groups are locally symmetric

Let X be an algebraic variety isomorphic to the complement of a closed subvariety of dimension at most n − 3 in \( {\mathbbm{A}}_{\mathrm{k}}^n \). We find some conditions under which an isomorphism of two closed subvarieties of X can be extended to an automorphism of X. We also study the similar problem for subvarieties of affine quadrics and SL(n, k).

This note is motivated by the problem of understanding Springer’s remarkable representation of the Weyl group W of a semisimple complex linear algebraic group G on the cohomology algebra of an arbitrary Springer variety in the ag variety of G from the viewpoint of torus actions and localization. Continuing the work [CK] which gave a sufficient condition for a group \( \mathcal{W} \) acting on the fixed

We prove results about automatic continuity and openness of abstract surjective group homomorphisms \( K\overset{\varphi }{\to }G, \) where G and K belong to a certain class К of topological groups, and where the kernel of φ satisies a certain topological countability condition. Our results apply in particular to the case where G is a semisimple Lie group or a semisimple compact group, and where К

We suggest two explicit descriptions of the Poisson q-W algebras which are Poisson algebras of regular functions on certain algebraic group analogues of the Slodowy transversal slices to adjoint orbits in a complex semisimple Lie algebra \( \mathfrak{g} \). To obtain the first description we introduce certain projection operators which are analogous to the quasi-classical versions of the so-called

Extending results of Rais–Tauvel, Macedo–Savage, and Arakawa–Premet, we prove that under mild restrictions on the Lie algebra \( \mathfrak{q} \) having the polynomial ring of symmetric invariants, the m-th Takiff algebra of \( \mathfrak{q} \), \( \mathfrak{q} \)⟨m⟩, also has a polynomial ring of symmetric invariants.

Let G be a connected semisimple group over an algebraically closed field k of characteristic 0. Let Y = G/H be a spherical homogeneous space of G, and let Y′ be a spherical embedding of Y. Let k0 be a subfield of k. Let G0 be a k0-model (k0-form) of G. We show that if G0 is an inner form of a split group and if the subgroup H of G is spherically closed, then Y admits a G0-equivariant k0-model. If we

We use Boij–Söderberg theory to give two lower bounds for the dimension of the cohomology of a finite CW-complex in terms of the toral rank and certain Betti numbers of the space. One of our bounds turns out to be particularly effective for c-symplectic spaces, proving the toral rank conjecture for c-symplectic spaces of formal dimension ≤ 8.

Let G be a linearly reductive group acting on a vector space V, and f a semi-invariant polynomial on V. In this paper we study systematically decompositions of the Bernstein–Sato polynomial of f in parallel with some representation-theoretic properties of the action of G on V. We provide a technique based on a multiplicity one property, that we use to compute the Bernstein–Sato polynomials of several

We prove, in the case of hyperbolic 3-space, a couple of conjectures raised by J. J. Seidel in On the volume of a hyperbolic simplex, Stud. Sci. Math. Hung. 21 (1986), 243–249. Seidel’s first conjecture states that the volume of an ideal tetrahedron in hyperbolic 3-space is determined by (the permanent and the determinant of) a certain Gram matrix G of its vertices; Seidel’s fourth conjecture claims

We provide a new proof of the super duality equivalence between infinite-rank parabolic BGG categories of general linear Lie (super) algebras conjectured by Cheng and Wang and first proved by Cheng and Lam. We do this by establishing a new uniqueness theorem for tensor product categorifications motivated by work of Brundan, Losev, and Webster. Moreover we show that these BGG categories have Koszul

Let P be a parabolic subgroup in G = SLn(k), for k an algebraically closed field. We show that there is a G-stable closed subvariety of an affine Schubert variety in an affine partial flag variety which is a natural compactification of the cotangent bundle T*G/P. Restricting this identification to the conormal variety N*X(w) of a Schubert divisor X(w) in G/P, we show that there is a compactification

Suppose G is a cyclic group and M a closed smooth G-manifold with exactly one isotropy type. We will show that there is a nonsingular real algebraic G-variety X such that X is equivariantly diffeomorphic to M and all G-vector bundles over X are strongly algebraic.

We present a new link between the Invariant Theory of infinitesimal singular Riemannian foliations and Jordan algebras. This, together with an inhomogeneous version of Weyl's First Fundamental Theorems, provides a characterization of the recently discovered Clifford foliations in terms of basic polynomials. This link also yields new structural results about infinitesimal foliations, such as the existence

In this paper we will show that the pull-back of any regular differential form defined on the smooth locus of a GIT quotient of dimension at most four to any resolution yields a regular differential form.

We study criteria for deciding when the normal subgroup generated by a single special polynomial automorphism of 𝔸n is as large as possible, namely, equal to the normal closure of the special linear group in the special automorphism group. In particular, we investigate m-triangular automorphisms, i.e., those that can be expressed as a product of affine automorphisms and m triangular automorphisms

Let K be the field of fractions of a complete discrete valuation ring \( \mathcal{A} \) with residue field k, and let G be a connected reductive algebraic group over K. Suppose \( \mathcal{P} \) is a parahoric group scheme attached to G. In particular, \( \mathcal{P} \) is a smooth affine \( \mathcal{A} \)-group scheme having generic fiber \( \mathcal{P} \)K = G; the group scheme \( \mathcal{P} \)

We study singular hyperkähler quotients of the cotangent bundle of a complex semisimple Lie group as stratified spaces whose strata are hyperkähler. We focus on one particular case where the stratification satisfies the frontier condition and the partial order on the set of strata can be described explicitly by Lie theoretic data.

In this paper we discuss the highest weight \( {\mathfrak{k}}_r \)-finite representations of the pair (𝔤r, \( {\mathfrak{k}}_r \)) consisting of 𝔤r, a real form of a complex basic Lie superalgebra of classical type 𝔤 (𝔤 ≠ A(n, n)), and the maximal compact subalgebra \( {\mathfrak{k}}_r \) of 𝔤r,0, together with their geometric global realizations. These representations occur, as in the ordinary