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Abstract: Abstract In this work, we study the monodromy group of covers φ ∘ ψ of curves \(\mathcal {Y}\xrightarrow {\quad {\psi }}\) \( \mathcal {X} \xrightarrow {\quad \varphi } \mathbb {P}^{1}\) , where ψ is a q-fold cyclic étale cover and φ is a totally ramified p-fold cover, with p and q different prime numbers with p odd. We show that the Galois group \(\mathcal {G}\) of the Galois closure \(\mathcal {Z}\) of φ ∘ ψ is of the form \( \mathcal {G} = \mathbb {Z}_{q}^{s} \rtimes \mathcal {U}\) , where 0 ≤ s ≤ p − 1 and \(\mathcal {U}\) is a simple transitive permutation group of degree p. Since the simple transitive permutation group of prime degree p are known, and we construct examples of such covers with these Galois groups, the result is very different from the previously known case when the cover φ was assumed to be cyclic, in which case the Galois group is of the form \( \mathcal {G} = \mathbb {Z}_{q}^{s} \rtimes \mathbb {Z}_{p}\) . Furthermore, we are able to characterize the subgroups \({\mathscr{H}}\) and \(\mathcal {N}\) of \(\mathcal {G}\) such that \(\mathcal {Y} = \mathcal {Z}/\mathcal {N}\) and \(X = \mathcal {Z}/{\mathscr{H}}\) . PubDate: 2022-09-20

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Abstract: Abstract This paper gives a classification of stable vectors in dual Vinberg representations coming from a graded Lie algebra of type F4 in a way that is independent of the field of definition. Relating these gradings to Moy–Prasad filtrations, we obtain the input for Reeder–Yu’s construction of epipelagic supercuspidal representations. As a corollary, this construction gives new supercuspidal representations of \(F_{4}(\mathbb {Q}_{p})\) when p is small. PubDate: 2022-09-15

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Abstract: Abstract We prove that the Gromov width of any Bott-Samelson variety associated to a reduced expression and equipped with a rational Kähler form equals the symplectic area of a minimal curve. From this, we derive an estimate for the Seshadri constants of ample line bundles on Bott-Samelson varieties. PubDate: 2022-09-13

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Abstract: Abstract We introduce and study the notion of a logarithmic vertex algebra, which is a vertex algebra with logarithmic singularities in the operator product expansion of quantum fields; thus providing a rigorous formulation of the algebraic properties of quantum fields in logarithmic conformal field theory. We develop a framework that allows many results about vertex algebras to be extended to logarithmic vertex algebras, including in particular the Borcherds identity and Kac Existence Theorem. Several examples are investigated in detail, and they exhibit some unexpected new features that are peculiar to the logarithmic case. PubDate: 2022-09-09

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Abstract: Abstract We use the Springer correspondence to give a partial characterization of the irreducible representations which appear in the Tymoczko dot action of the Weyl group on the cohomology ring of a regular semisimple Hessenberg variety. In type A, we apply these techniques to prove that all irreducible summands which appear in the pushforward of the constant sheaf on the universal Hessenberg family have full support. We also observe that the recent results of Brosnan and Chow, which apply the local invariant cycle theorem to the family of regular Hessenberg varieties in type A, extend to arbitrary Lie type. We use this extension to prove that regular Hessenberg varieties, though not necessarily smooth, always have the “Kähler package.” PubDate: 2022-09-06

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Abstract: Abstract Firstly, we prove that every closed subgroup H of type-preserving automorphisms of a locally finite thick affine building Δ of dimension ≥ 2 that acts strongly transitively on Δ is Moufang. If moreover Δ is irreducible and H is topologically simple, we show that H is the subgroup \(\mathbb {G}(k)^{+}\) of the k-rational points \(\mathbb {G}(k)\) of the isotropic simple algebraic group \(\mathbb {G}\) over a non-Archimedean local field k associated with Δ. Secondly, we generalise the proof given in Burger and Mozes (Inst. Hautes Études Sci. Publ. Math., 92, 151–194 (2001) 2000) for the case of bi-regular trees to any locally finite thick affine building Δ, and obtain that any topologically simple, closed, strongly transitive and type-preserving subgroup of Aut(Δ) has the Howe–Moore property. This proof is different than the strategy used so far in the literature and does not rely on the polar decomposition KA+K, where K is a maximal compact subgroup, and the important fact that A+ is an abelian maximal sub-semi-group. PubDate: 2022-09-06

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Abstract: Abstract In this paper we study the category ð’ª over the hyperalgebra of a reductive algebraic group in positive characteristics. For any locally closed subset K of weights, we define a subquotient ð’ª[K] of ð’ª. It has the property that its simple objects are parametrized by elements in K. We then show that ð’ª[K] is equivalent to ð’ª [K +pl γ] for any dominant weight γ if l > 0 is an integer such that K ∩ (K – pl η) = ∅ for all weights η > 0. Hence it is enough to understand the subquotients inside the dominant (or the antidominant) chamber. PubDate: 2022-09-02

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Abstract: Abstract 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 approximation in the style of Harder, we reduce to the case when R is a complete discrete valuation ring. Afterwards, we consider Levi subgroups to reduce to the case when G is semisimple and anisotropic, in which case we take advantage of Bruhat–Tits theory to conclude. Finally, we show that the Grothendieck–Serre conjecture implies that any reductive group over the total ring of fractions of a regular semilocal ring S has at most one reductive S-model. PubDate: 2022-09-01

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Abstract: Abstract Let G be a reductive algebraic group over a field k and let B be a Borel subgroup in G. We demonstrate how a number of results on the cohomology of line bundles on the ag manifold G/B have had interesting consequences in the representation theory for G, and vice versa. Our focus is on the case where the characteristic of k is positive. In this case, both the vanishing behavior of the cohomology modules for a line bundle on G/B and the G-structures of the nonzero cohomology modules are still very much open problems. We give an account of the developments over the years, trying to illustrate what is now known and what is still not known today. PubDate: 2022-09-01

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Abstract: Abstract Let G be one of the ind-groups GL(∞), O(∞), Sp(∞), and let P1, ..., Pℓ be an arbitrary set of ℓ splitting parabolic subgroups of G. We determine all such sets with the property that G acts with finitely many orbits on the ind-variety X1 × × Xℓ where Xi = G/Pi. In the case of a finite-dimensional classical linear algebraic group G, the analogous problem has been solved in a sequence of papers of Littelmann, Magyar–Weyman–Zelevinsky and Matsuki. An essential difference from the finite-dimensional case is that already for ℓ = 2, the condition that G acts on X1 × X2 with finitely many orbits is a rather restrictive condition on the pair P1, P2. We describe this condition explicitly. Using the description we tackle the most interesting case where ℓ = 3, and present the answer in the form of a table. For ℓ ≥ 4 there always are infinitely many G-orbits on X1 × × Xℓ. PubDate: 2022-09-01

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Abstract: Abstract We prove a motivic analogue of the Weyl character formula, computing the Euler characteristic of a line bundle on a generalized ag manifold G/B multiplied either by a motivic Chern class of a Schubert cell, or a Segre analogue of it. The result, given in terms of Demazure–Lusztig (D–L) operators, identifies an Euler characteristic above to a formula of Brubaker, Bump and Licata for the Iwahori–Whittaker functions of the principal series representation of the p-adic Langlands dual group. As a corollary, we recover the classical Casselman–Shalika formula for the spherical Whittaker function. The proofs are based on localization in equivariant K-theory, and require a geometric interpretation of how the Hecke inverse of a D–L operator acts on the class of a point. We prove that the Hecke inverse operators give Grothendieck–Serre dual classes of the motivic classes, a result which might be of independent interest. In an Appendix jointly authored with Dave Anderson, we show that if the line bundle is trivial, we recover a generalization of a classical formula by Kostant, Macdonald, Shapiro and Steinberg for the Poincaré polynomial of G/B; the generalization we consider is due to Akyıldız and Carrell and replaces G/B by any smooth Schubert variety. PubDate: 2022-09-01

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Abstract: Abstract The rank n symplectic oscillator Lie algebra ð”¤n is the semidirect product of the symplectic Lie algebra ð”°ð”2n and the Heisenberg algebra Hn. In this paper, we first study weight modules with finite-dimensional weight spaces over ð”¤n. When the central charge \( \dot{z} \) ≠ 0, it is shown that there is an equivalence between the full subcategory ð’ªð”¤n \( \left[\dot{z}\right] \) of the BGG category ð’ªð”¤n for ð”¤n and the BGG category ð’ªð”°ð”2n for ð”°ð”2n. Then using the technique of localization and the structure of generalized highest weight modules, we give the classification of simple weight modules over ð”¤n with finite-dimensional weight spaces. As a byproduct we also determine all simple ð”¤n-modules (not necessarily weight modules) that have a simple Hn-submodule. PubDate: 2022-09-01

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Abstract: Abstract We prove that under certain assumptions a supermanifold of flags is rigid, that is, its complex structure does not admit any non-trivial small deformation. Moreover under the same assumptions we show that a supermanifold of flags is a unique non-split supermanifold with given retract. PubDate: 2022-09-01

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Abstract: Abstract We investigate compact projective generators in the category of equivariant -modules on a smooth affine variety. For a reductive group G acting on a smooth affine variety X, there is a natural countable set of compact projective generators indexed by finite dimensional representations of G. We show that only finitely many of these objects are required to generate; thus the category has a single compact projective generator. The proof goes via an analogous statement about compact generators in the equivariant derived category, which holds in much greater generality and may be of independent interest. PubDate: 2022-09-01

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Abstract: Abstract 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 homotopy equivalent to the exotic Springer fiber. This homotopy equivalent space admits an action of the type C Weyl group inducing Kato’s original exotic Springer representation on cohomology. Our results are described in terms of the diagrammatics of the one-boundary Temperley–Lieb algebra (also known as the blob algebra). This provides a first step in generalizing the geometric versions of Khovanov’s arc algebra to the exotic setting. PubDate: 2022-09-01

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Abstract: Abstract We give a formula for the crystal structure on the integer points of the string polytopes and the *-crystal structure on the integer points of the string cones of type A for arbitrary reduced words. As a byproduct, we obtain defining inequalities for Nakashima–Zelevinsky string polytopes. Furthermore, we give an explicit description of the Kashiwara *-involution on string data for a special choice of reduced word. PubDate: 2022-09-01

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Abstract: Abstract Deligne’s category \( \underset{\_}{\mathrm{Rep}}\left({S}_t\right) \) is a tensor category depending on a parameter t “interpolating” the categories of representations of the symmetric groups Sn. We construct a family of categories Cλ (depending on a vector of variables λ = (λ1, λ2, … , λl), that may be specialised to values in the ground ring) which are module categories over \( \underset{\_}{\mathrm{Rep}}\left({S}_t\right). \) The categories Cλ are defined over any ring and are constructed by interpolating permutation representations. Further, they admit specialisation functors to Sn-mod which are tensor-compatible with the functors \( \underset{\_}{\mathrm{Rep}}\left({S}_t\right)\to {S}_n-\operatorname{mod}. \) We show that Cλ can be presented using the Kostant integral form of Lusztig’s universal enveloping algebra \( \dot{U}\left({\mathfrak{gl}}_{\infty}\right), \) and exhibit a categorification of some stability properties of Kronecker coefficients. PubDate: 2022-09-01

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Abstract: Abstract We provide a combinatorial description of the monoidal category generated by the fundamental representation of the small quantum group of \( \mathfrak{sl} \) 2 at a root of unity q of odd order. Our approach is diagrammatic, and it relies on an extension of the Temperley–Lieb category specialized at δ = −q − q−1. PubDate: 2022-09-01

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