For x ∈ End(𝕂n) satisfying x2 = 0 let ℱx be the variety of full flags stable under the action of x (Springer fiber over x). The full classification of the components of ℱx according to their smoothness was provided in [4] in terms of both Young tableaux and link patterns. Moreover in [2] the purely combinatorial algorithm to compute the singular locus of a singular component of ℱx is provided. However

Our main goal is to show that the Gelfand–Tsetlin toric degeneration of the type A ag variety can be obtained within a degenerate representation-theoretic framework similar to the theory of PBW degenerations. In fact, we provide such frameworks for all Gröbner degenerations intermediate between the ag variety and the GT toric variety. These degenerations are shown to induce filtrations on the irreducible

For a reductive group scheme G over a semilocal Dedekind ring R with total ring of fractions K, we prove that no nontrivial G-torsor trivializes over K. This generalizes a result of Nisnevich–Tits, who settled the case when R is local. Their result, in turn, is a special case of a conjecture of Grothendieck–Serre that predicts the same over any regular local ring. With a patching technique and weak

In this paper we define and study a critical-level generalization of the Suzuki functor, relating the affine general linear Lie algebra to the rational Cherednik algebra of type A. Our main result states that this functor induces a surjective algebra homomorphism from the centre of the completed universal enveloping algebra at the critical level to the centre of the rational Cherednik algebra at t

We present an efficient method for determining the conditions that a metric on a cohomogeneity one manifold, defined in terms of functions on the regular part, needs to satisfy in order to extend smoothly to the singular orbit.

We consider several vertex operator algebras and superalgebras closely related to \( {V}_{-1}\left(\mathfrak{sl}(n)\right) \), n ≥ 3 : (a) the parafermionic subalgebra K(\( \mathfrak{sl} \)(n); −1) for which we completely describe its inner structure, (b) the vacuum algebra Ω(V−1(\( \mathfrak{sl} \)(n))), and (c) an infinite extension \( \mathcal{U} \) of V−1(\( \mathfrak{sl} \)(n)) obtained from certain

We construct a minimal Lefschetz decomposition of the bounded derived category of coherent sheaves on the isotropic Grassmannian IGr(3, 7). Moreover, we show that IGr(3, 7) admits a full exceptional collection consisting of equivariant vector bundles.

We show that certain homological regularity properties of graded connected algebras, such as being AS-Gorenstein or AS-Cohen–Macaulay, can be tested by passing to associated graded rings. In the spirit of noncommutative algebraic geometry, this can be seen as an analogue of the classical result that, in a flat family of varieties over the affine line, regularity properties of the exceptional fiber

We study the geometry and topology of exotic Springer fibers for orbits corresponding to one-row bipartitions from an explicit, combinatorial point of view. This includes a detailed analysis of the structure of the irreducible components and their intersections as well as the construction of an explicit affine paving. Moreover, we compute the ring structure of cohomology by constructing a CW-complex

We classify all simple bounded highest weight modules of a basic classical Lie superalgebra \( \mathfrak{g} \). In particular, our result leads to the classification of the simple weight modules with finite weight multiplicities over all classical Lie superalgebras. We also obtain some character formulas of strongly typical bounded highest weight modules of \( \mathfrak{g} \).

We prove various characterisations for the Cohen–Macaulay property for the invariant ring k[x1, …, xn]G, where k is a PID.We also show that, except for one case, all the invariant rings ℤ[x1, x2]G and ℤ[x1, x2, x3]G are Cohen–Macaulay. We also provide some sufficient conditions to obtain a polynomial invariant ring over the ring of integers.

We develop a general theory for irreducible homogeneous spaces M = G/H, in relation to the nullity distribution ν of their curvature tensor. We construct natural invariant (different and increasing) distributions associated with the nullity, that give a deep insight of such spaces. In particular, there must exist an order-two transvection, not in the nullity, with null Jacobi operator. This fact was

Rosay and Rudin introduced the notion of tame discrete subsets of the affine complex space and investigated their properties. We generalize this theory to the case of a complex linear algebraic group with trivial character group.

Let X denote a projective variety over an algebraically closed field on which a linear algebraic group acts with finitely many orbits. Then, a conjecture of Soergel and Lunts in the setting of Koszul duality and Langlands' philosophy, postulates that the equivariant derived category of bounded complexes with constructible equivariant cohomology sheaves on X is equivalent to a full subcategory of the

Our aim is to support the choice of two remarkable connections with torsion in a 3-Sasakian manifold, proving that, in contrast to the Levi-Civita connection, the holonomy group in the homogeneous cases reduces to a proper subgroup of the special orthogonal group, of dimension considerably smaller. We realize the computations of the holonomies in a unified way, by using as a main algebraic tool a nonassociative

We provide explicit formulas for integrating multiplicative forms on local Lie groupoids in terms of infinitesimal data. Combined with our previous work [8], which constructs the local Lie groupoid of a Lie algebroid, these formulas produce concrete integrations of several geometric stuctures defined infinitesimally. In particular, we obtain local integrations and non-degenerate realizations of Poisson

In this note we show that if the automorphism group of a normal affine surface S is isomorphic to the automorphism group of a Danielewski surface, then S is isomorphic to the normalization of a Danielewski surface.

We present a full list of all representations of the special linear group SLn over the complex numbers with complete intersection invariant ring of homological dimension greater than or equal to two, completing the classification of Shmelkin. For this task, we combine three techniques. Firstly, the graph method for invariants of SLn developed by the author to compute invariants, covariants and explicit

It is well known that the cohomology of any non-trivial 1-dimensional local system on a nilmanifold vanishes (this result, due to J. Dixmier, was also announced and proved in some particular case by Alaniya). A complex nilmanifold is a quotient of a nilpotent Lie group equipped with a left-invariant complex structure by an action of a discrete, co-compact subgroup. We prove a Dolbeault version of Dixmier’s

Let G = SL(2, ℤ) ⋉ ℤ2 and H = SL(2, ℤ). We prove that the action G ↷ ℝ2 is uniformly non-amenable and that the quasi-regular representation of G on ℓ2(G/H) has a uniform spectral gap. Both results are a consequence of a uniform quantitative form of ping-pong for affine transformations, which we establish here.

A GLd-pseudocharacter is a function from a group Γ to a ring k satisfying polynomial relations that make it “look like” the character of a representation. When k is an algebraically closed field of characteristic 0, Taylor proved that GLd-pseudocharacters of Γ are the same as degree-d characters of Γ with values in k, hence are in bijection with equivalence classes of semisimple representations Γ →

We study the Ricci iteration for homogeneous metrics on spheres and complex projective spaces. Such metrics can be described in terms of modifying the canonical metric on the fibers of a Hopf fibration. When the fibers of the Hopf fibration are circles or spheres of dimension 2 or 7, we observe that the Ricci iteration as well as all ancient Ricci iterations can be completely described using known

Until a couple of years ago, the only known examples of Lie groups admitting left-invariant metrics with negative Ricci curvature were either solvable or semisimple.We use a general construction from a previous article of the second named author to produce a large number of examples with compact Levi factor. Given a compact semisimple real Lie algebra 𝔲 and a real representation π satisfying some

In parts I and II, we determined which faithful irreducible representations V of a simple linear algebraic group G are generically free for Lie(G), i.e., which V have an open subset consisting of vectors whose stabilizer in Lie(G) is zero, with some assumptions on the characteristic of the field. This paper settles the remaining cases, which are of a different nature because Lie(G) has a more complicated

We determine which faithful irreducible representations V of a simple linear algebraic group G are generically free for Lie(G), i.e., which V have an open subset consisting of vectors whose stabilizer in Lie(G) is zero. This relies on bounds on dim V obtained in prior work (part I), which reduce the problem to a finite number of possibilities for G and highest weights for V , but still infinitely many

We produce braided commutative algebras in braided monoidal categories by generalizing Davydov’s full center construction of commutative algebras in centers of monoidal categories. Namely, we build braided commutative algebras in relative monoidal centers \( {\mathcal{Z}}_{\mathrm{\mathcal{B}}}\left(\mathcal{C}\right) \) from algebras in ℬ-central monoidal categories \( \mathcal{C} \), where ℬ is an

We compute the cohomological invariants of ℋg, the moduli stack of smooth hyperelliptic curves, for every odd g.

Let G be a symplectic or special orthogonal group, let H be a connected reductive subgroup of G, and let X be a flag variety of G. We classify all triples (G, H, X) such that the natural action of H on X is spherical. For each of these triples, we determine the restrictions to H of all irreducible representations of G realized in spaces of sections of homogeneous line bundles on X.

We show that every 3-dimensional affine normal quasihomogeneous SL(2)-variety has an equivariant resolution of singularities given by an invariant Hilbert scheme and we present an explicit description of the invariant Hilbert scheme.

In this paper we study the existence of sections of universal bundles on rational homogeneous varieties–called nestings–classifying them completely on rational homogeneous varieties G/P in the case where G is a simple group of classical type and P is a parabolic subgroup of G. In particular we show that, under this hypothesis, nestings do not exist unless there exists a proper algebraic subgroup of

Metrics on Lie groupoids and differentiable stacks have been introduced recently, extending the Riemannian geometry of manifolds and orbifolds to more general singular spaces. Here we continue that theory, studying stacky curves on Riemannian stacks, measuring their length using stacky metrics, and introducing stacky geodesics. Our main results show that the length of stacky curves measure distances

In this paper we will give a short proof of a special case of Zariski’s result about finite generation in connection with Hilbert’s 14th problem using a new idea. Our result is useful for invariant subrings of unipotent or connected semisimple groups. We will also prove an analogue of Miyanishi’s result for the ring of invariants of a \( {\mathbbm{G}}_a \)-action on R[X, Y, Z] for an affine Dedekind

An involutive diffeomorphism σ of a connected smooth manifold M is called dissecting if the complement of its fixed point set is not connected. Dissecting involutions on a complete Riemannian manifold are closely related to constructive quantum field theory through the work of Dimock and Jaffe/Ritter on the construction of reflection positive Hilbert spaces. In this article we classify all pairs (M

Let S be a del Pezzo surface with at worst Du Val singularities such that it is a hypersurface in a weighted projective space ℙ. We prove that the surface S contains a (−KS)-polar cylinder if and only if the automorphism group of the affine variety ℙ \ S contains a unipotent subgroup.

Following a question of Vinberg, a general method to construct monomial bases for finite-dimensional irreducible representations of a reductive Lie algebra \( \mathfrak{g} \) was developed in a series of papers by Feigin, Fourier, and Littelmann. Relying on this method, we construct monomial bases of multiplicity spaces associated with the restriction of the representation to a reductive subalgebra

We show that generalized orbital varieties for Mirković–Vybornov slices can be indexed by semi-standard Young tableaux, and, via the Mirković–Vybornov isomorphism [MV19], can be identified with Mirković–Vilonen cycles, such that the (combinatorial) Lusztig datum of a generalized orbital variety, which it inherits from its tableau, is equal to the (geometric) Lusztig datum of its Mirković–Vilonen cycle

Motivated by work of Fine and Panov, and of Lindsay and Panov, we prove that every closed symplectic complexity one space that is positive (e.g., positive monotone) enjoys topological properties that Fano varieties with a complexity one holomorphic torus action possess. In particular, such spaces are simply connected, have Todd genus equal to one and vanishing odd Betti numbers.

Let λ ∈ P+ be a level-zero dominant integral weight, and w the coset representative of minimal length for a coset in W/Wλ, where Wλ is the stabilizer of λ in a finite Weyl group W. In this paper, we give a module \( {\mathbbm{K}}_w^{-}\left(\uplambda \right) \) over the negative part of a quantum affine algebra whose graded character is identical to the specialization at t = ∞ of the nonsymmetric Macdonald

We construct a family of (2, n)-almost Grassmannian structures of regularity C1, each admitting a one-parameter group of strongly essential automorphisms, and each not flat on any open set containing the higher-order fixed point in its closure. This shows that Theorem 1.3 of [12] does not hold assuming only C1 regularity of the structure (see also [3, Prop. 3.5]).

Let (U, ) be a quasi-split quantum symmetric pair of arbitrary Kac–Moody type, where “quasi-split” means the corresponding Satake diagram contains no black node. We give a presentation of the group with explicit relations. The verification of new relations is reduced to some new q-binomial identities. Consequently, is shown to admit a bar involution under suitable conditions on the parameters.

For a Weyl group W of classical type, we present a formula to calculate the restriction of (graded) Springer representations of W to a maximal parabolic subgroup W′ where the types of W and W′ are in the same series. As a result, we obtain recursive formulas for the Betti numbers of Springer fibers for classical types.

It was proved by the first-named author and Zubkov [13] that given an affine algebraic supergroup \( \mathbbm{G} \) and a closed sub-supergroup ℍ over an arbitrary field of characteristic ≠ 2, the faisceau \( \mathbbm{G}\tilde{/}\mathrm{\mathbb{H}} \) (in the fppf topology) is a superscheme, and is, therefore, the quotient superscheme \( \mathbbm{G}/\mathrm{\mathbb{H}} \), which has some desirable

We establish rank-finiteness for the class of G-crossed braided fusion categories, generalizing the recent result for modular categories and including the important case of braided fusion categories. This necessitates a study of slightly degenerate braided fusion categories and their centers, which are interesting for their own sake.

We classify the simple bounded weight modules of the Lie algebras \( \mathfrak{sl}\left(\infty \right),\kern0.5em \mathfrak{o}\left(\infty \right) \) and \( \mathfrak{sp}\left(\infty \right) \), and compute their annihilators in \( U\left(\mathfrak{sl}\left(\infty \right)\right),\kern0.5em U\left(\mathfrak{o}\left(\infty \right)\right),\kern0.5em U\left(\mathfrak{sp}\left(\infty \right)\right) \),

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.