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Abstract: Abstract Let \(\Gamma \) be a discrete subgroup of \(\text {PU}(1,n)\) . In this work, we look at the induced action of \(\Gamma \) on the projective space \(\mathbb {P}(\wedge ^{k+1}\mathbb {C}^{n+1})\) by the Plücker embedding, where \(\wedge ^{k+1}\) denotes the exterior power. We define a limit set for this action called the k-Chen-Greenberg limit set, which extends the classical definition of the Chen-Greenberg limit set \(L(\Gamma )\) , and we show several of its properties. We prove that its Kulkarni limit set is the union taken over all \(p\in L(\Gamma )\) of the projective subspace generated by all k-planes that contain p or are contained in \(p^{\perp }\) via the Plücker embedding. We also prove a duality between both limit sets. PubDate: 2023-12-02

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Abstract: The article “Generalised Gelfand-Graev Represenations in Bad Characteristic',” written by Meinolf Geck, was originally published Online First without Open Access. PubDate: 2023-12-01

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Abstract: Abstract For complex parallelisable manifolds Γ\G, with G a solvable or semisimple complex Lie group, the Frölicher spectral sequence degenerates at the second page. In the solvable case, the de Rham cohomology carries a pure Hodge structure. In contrast, in the semisimple case, purity depends on the lattice, but there is always a direct summand of the de Rham cohomology which does carry a pure Hodge structure and is independent of the lattice. PubDate: 2023-12-01

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Abstract: Abstract We study the simple Bershadsky–Polyakov algebra ð’²k = ð’²k(sl3, fθ) at positive integer levels and classify their irreducible modules. In this way, we confirm the conjecture from [9]. Next, we study the case k = 1. We discover that this vertex algebra has a Kazama–Suzuki-type dual isomorphic to the simple affine vertex superalgebra Lk′ (osp(1 2)) for k′ = –5=4. Using the free-field realization of Lk′ (osp(1 2)) from [3], we get a free-field realization of ð’²k and their highest weight modules. In a sequel, we plan to study fusion rules for ð’²k. PubDate: 2023-12-01

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Abstract: Abstract We find complete expanding Kähler-Ricci solitons of two types. The first are of cohomogeneity one under the action of the (2m - 1)-dimensional Heisenberg group or a certain quotient of it, for any m ≥ 2. The second type, which generalizes the first, reside on ℂ*-bundles over compact Kähler Ricci-flat manifolds, admit an S1 action by isometries, and have two ends. Both types have the local structure stemming from an ansatz first described in [20]. We give curvature bounds for the first type, and asymptotics for both types. PubDate: 2023-12-01

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Abstract: Abstract In this work, we study the integrability of quotients of quasi-Poisson manifolds. Our approach allows us to put several classical results about the integrability of Poisson quotients in a common framework. By categorifying one of the already known methods of reducing symplectic groupoids we also describe double symplectic groupoids, which integrate the recently introduced Poisson groupoid structures on gauge groupoids. PubDate: 2023-12-01

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Abstract: Abstract Over a field of positive characteristic, a semisimple algebraic group G may have some nonreduced parabolic subgroup P. In this paper, we study the Schubert and Bott–Samelson–Demazure–Hansen (BSDH) varieties of G/P, with P nonreduced, when the base field is perfect. It is shown that, in general, the Schubert and BSDH varieties of such a G/P are not normal, and the projection of the BSDH variety onto the Schubert variety has nonreduced fibers at closed points. When the base field is finite, the generalized convolution morphisms between BSDH varieties (as in [dCHL18]) are also studied. It is shown that the decomposition theorem holds for such morphisms, and the pushforward of intersection complexes by such morphisms are Frobenius semisimple. PubDate: 2023-12-01

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Abstract: Abstract We study the interplay between the following types of special non-Kähler Hermitian metrics on compact complex manifolds (locally conformally Kähler, k-Gauduchon, balanced, and locally conformally balanced) and prove that a locally conformally Kähler compact nilmanifold carrying a balanced or a left-invariant k-Gauduchon metric is necessarily a torus. Combined with the main result in [FV16], this leads to the fact that a compact complex 2-step nilmanifold endowed with whichever two of the following types of metrics—balanced, pluriclosed and locally conformally Kähler—is a torus. Moreover, we construct a family of compact nilmanifolds in any dimension carrying both balanced and locally conformally balanced metrics and finally we show a compact complex nilmanifold does not support a left-invariant locally conformally hyper-Kähler structure. PubDate: 2023-12-01

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Abstract: Abstract Let k be an algebraically closed field of characteristic p > 0. We compute the Weyl filtration multiplicities in indecomposable tilting modules and the decomposition numbers for the general linear group over k in terms of cap diagrams under the assumption that p is bigger than the greatest hook length in the partitions involved. Then we introduce and study the rational Schur functor from a category of GLn-modules to the category of modules for the walled Brauer algebra. As a corollary, we obtain the decomposition numbers for the walled Brauer algebra when p is bigger than the greatest hook length in the partitions involved. This is a sequel to an earlier paper on the symplectic group and the Brauer algebra. PubDate: 2023-12-01

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Abstract: Abstract The aim of this paper is to define a pair of symplectic Dirac operators (D+, D–) in an algebraic setting motivated by the analogy with the algebraic orthogonal Dirac operators in representation theory. We work in the settings of ℤ/2-graded quadratic Lie algebras ð”¤ = ð”¨ + ð” and of graded affine Hecke algebras ℍ. In these contexts, we show analogues of the Parthasarathy’s formula for [D+, D–] and certain generalisations of the Casimir inequality. PubDate: 2023-12-01

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Abstract: Abstract We define the equivariant Cox ring of a normal variety with algebraic group action. We study algebraic and geometric aspects of this object and show how it is related to the ordinary Cox ring. Then we specialize to the case of normal rational varieties of complexity one under the action of a connected reductive group G. We show that the G-equivariant Cox ring is then a finitely generated integral normal G-algebra. Under a mild additional condition, we give a presentation of its subalgebra of U-invariants, where U is the unipotent part of a Borel subgroup of G. The ordinary Cox ring is also finitely generated and inherits a canonical structure of U-algebra. Using a work of Hausen and Herppich, we prove that the subalgebra of U-invariants of the Cox ring is a finitely generated Cox ring of a variety of complexity one under the action of a torus. As an application, we provide a criterion of combinatorial nature for the Cox ring of an almost homogeneous G-variety of complexity one to have log terminal singularities. Finally, we prove that for a normal rational G-variety of complexity one satisfying a mild additional condition (e.g., complete or almost homogeneous), the iteration of Cox rings is finite. PubDate: 2023-12-01

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Abstract: Abstract To each complex semisimple Lie algebra \( \mathfrak{g} \) and regular element a ∈ \( \mathfrak{g} \) reg, one associates a Mishchenko–Fomenko subalgebra \( \mathcal{F} \) a ⊆ ℂ[ \( \mathfrak{g} \) ]. This subalgebra amounts to a completely integrable system on the Poisson variety \( \mathfrak{g} \) , and as such has a bifurcation diagram Σa ⊆ Spec( \( \mathcal{F} \) a). We prove that Σa has codimension one in Spec( \( \mathcal{F} \) a) if a ∈ \( \mathfrak{g} \) reg is not nilpotent, and that it has codimension one or two if a ∈ \( \mathfrak{g} \) reg is nilpotent. In the nilpotent case, we show each of the possible codimensions to be achievable. Our results significantly sharpen existing estimates of the codimension of Σa. PubDate: 2023-12-01

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Abstract: Abstract To a compact Lie group G one can associate a space E(2;G) akin to the poset of cosets of abelian subgroups of a discrete group. The space E(2;G) was introduced by Adem, F. Cohen and Torres-Giese, and subsequently studied by Adem and Gómez, and other authors. In this short note, we prove that G is abelian if and only if πi(E(2;G)) = 0 for i = 1; 2; 4. This is a Lie group analogue of the fact that the poset of cosets of abelian subgroups of a discrete group is simply connected if and only if the group is abelian. PubDate: 2023-12-01

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Abstract: Abstract In this paper we shall prove that the ℤ-subalgebra generated by the divided powers of the Drinfeld generators \( {x}_r^{\pm } \) (r ∈ ℤ) of the Kac–Moody algebra of type \( {\textrm{A}}_2^{(2)} \) is an integral form (strictly smaller than Mitzman’s; see [Mi]) of the enveloping algebra, we shall exhibit a basis generalizing the one provided in [G] for the untwisted affine Kac–Moody algebras and we shall determine explicitly the commutation relations. Moreover, we prove that both in the untwisted and in the twisted case the positive (respectively negative) imaginary part of the integral form is an algebra of polynomials over ℤ. PubDate: 2023-12-01

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Abstract: Abstract Suppose G is a finite cyclic group and M a closed smooth G-manifold. We will show that there is a nonsingular real algebraic G-variety X that is equivariantly diffeomorphic to M so that all G-vector bundles over X are strongly algebraic. PubDate: 2023-12-01

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Abstract: Abstract Let G be a compact connected Lie group and let K be a closed subgroup of G. In this paper, we study whether the functional ð”¤ ↦ ⋋1 (G/K, ð”¤) diam (G/K, ð”¤)2 is bounded among G-invariant metrics ð”¤ on G/K. Eldredge, Gordina, and Saloff-Coste conjectured in 2018 that this assertion holds when K is trivial; the only particular cases known so far are when G is abelian, SU(2), and SO(3). In this article, we prove the existence of the mentioned upper bound for every compact homogeneous space G/K having multiplicity-free isotropy representation. PubDate: 2023-12-01

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Abstract: Abstract Let G be a reductive group. We prove that a family of polynomial actions of G on ℂ2, holomorphically parametrized by an open Riemann surface, is linearizable. As an application, we show that a particular class of reductive group actions on ℂ3 is linearizable. The main step of our proof is to establish a certain restrictive Oka property for groups of equivariant algebraic automorphisms of ℂ2. PubDate: 2023-12-01

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Abstract: Abstract After proving that every Schubert variety in the full flag variety of a complex reductive group G is a general Hessenberg variety, we show that not all such Schubert varieties are adjoint Hessenberg varieties. In fact, in types A and C, we provide pattern avoidance criteria implying that the proportion of Schubert varieties that are adjoint Hessenberg varieties approaches zero as the rank of G increases. We show also that in type A, some Schubert varieties are not isomorphic to any adjoint Hessenberg variety. PubDate: 2023-11-18

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Abstract: Abstract Following Lusztig and Vogan, we study the Bruhat G-order on the set \( \mathcal{D} \) of rank 1 local systems on B-orbits in an Hermitian symmetric variety G/L. Supposing Φ irreducible, we obtain a complete combinatorial characterization of the order for Φ of type A, B, D, E and a partial characterization for Φ of type C. PubDate: 2023-09-23