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## On the irreducible components of moduli schemes for affine spherical varieties

Transformation Groups. 2018. Vol. 23. No. 2. P. 299–327.

Roman Avdeev, Cupit-Foutou S.

We give a combinatorial description of all affine spherical varieties with prescribed weight monoid Γ. As an application, we obtain a characterization of the irreducible components of Alexeev and Brion’s moduli scheme M_Γ for such varieties. Moreover, we find several sufficient conditions for M_Γ to be irreducible and exhibit several examples where M_Γ is reducible. Finally, we provide examples of non-reduced M_Γ.

Roman Avdeev, Cupit-Foutou S., Advances in Mathematics 2018 Vol. 328 P. 1299–1352

Given a connected reductive algebraic group G and a finitely generated monoid Γ of dominant weights of G, in 2005 Alexeev and Brion constructed a moduli scheme M_Γ for multiplicity-free affine G-varieties with weight monoid Γ. This scheme is equipped with an action of an `adjoint torus' T_ad and has a distinguished T_ad-fixed point X_0. ...

Added: February 25, 2018

Roman Avdeev, Transformation Groups 2021 Vol. 26 No. 2 P. 403–431

Given a connected reductive complex algebraic group G and a spherical subgroup H⊂G, the extended weight monoid Λˆ+G(G/H) encodes the G-module structures on spaces of regular sections of all G-linearized line bundles on G/H. Assuming that G is semisimple and simply connected and H is specified by a regular embedding in a parabolic subgroup P⊂G, ...

Added: September 9, 2021

Roman Avdeev, Selecta Mathematica, New Series 2015 Vol. 21 No. 3 P. 931–993

A subgroup H of an algebraic group G is said to be strongly solvable if H is contained in a Borel subgroup of G. This paper is devoted to establishing relationships between the following three combinatorial classifications of strongly solvable spherical subgroups in reductive complex algebraic groups: Luna’s general classification of arbitrary spherical subgroups restricted ...

Added: July 8, 2015

Р.С. Авдеев, Петухов А. В., Математический сборник 2014 Т. 205 № 9 С. 3–48

For every finite-dimensional vector space V and every V-flag variety X we list all connected reductive subgroups in GL(V) acting spherically on X. ...

Added: October 22, 2014

Roman Avdeev, Petukhov A., Transformation Groups 2021 Vol. 26 No. 3 P. 719–774

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 ...

Added: September 2, 2020

Roman Avdeev, Communications in Contemporary Mathematics 2024 Vol. 26 No. 6 Article 2350029

In this paper, we obtain several structure results for a class of spherical subgroups of connected reductive complex algebraic groups that extends the class of strongly solvable spherical subgroups. Based on these results, we construct certain one-parameter degenerations of the Lie algebras corresponding to such subgroups. As an application, we exhibit explicit algorithms for computing the set of spherical ...

Added: December 27, 2023

Roman Avdeev, Petukhov A., Algebras and Representation Theory 2020 Vol. 23 No. 3 P. 541–581

Let G be a connected semisimple algebraic group and let H⊂G be a connected reductive subgroup. Given a flag variety X of G, a result of Vinberg and Kimelfeld asserts that H acts spherically on X if and only if for every irreducible representation R of G realized in the space of sections of a ...

Added: February 11, 2019

Colliot-Thélène J., Kunyavskiĭ B., Vladimir L. Popov et al., Compositio Mathematica 2011 Vol. 147 No. 2 P. 428–466

Let k be a field of characteristic zero, let G be a connected reductive algebraic group
over k and let g be its Lie algebra. Let k(G), respectively, k(g), be the field of k rational functions on G, respectively, g. The conjugation action of G on itself induces
the adjoint action of G on g. We investigate the ...

Added: March 17, 2013

Vladimir L. Popov, / Cornell University. Series math "arxiv.org". 2021. No. 2102.08032.

Several results on presenting an affine algebraic group variety as a product of algebraic varieties are obtained. ...

Added: February 17, 2021

Р.С. Авдеев, Горфинкель Н. Е., Функциональный анализ и его приложения 2012 Т. 46 № 3 С. 1–15

For all spherical homogeneous spaces G/H, where G is a simply connected semisimple algebraic group and H a connected solvable subgroup of G, we compute the spectra of representations of G on spaces of regular sections of homogeneous line bundles over G/H. ...

Added: February 25, 2014

Roman Avdeev, Indagationes Mathematicae 2012 Vol. 23 No. 1-2 P. 10–18

In 1994, based on Roberts’ counterexample to Hilbert’s fourteenth problem, A’Campo-Neuen constructed an example of a linear action of a 12-dimensional commutative unipotent group H_0 on a 19-dimensional vector space V such that the algebra of invariants k[V]^{H_0} is not finitely generated. We consider a certain extension H of H_0 by a one-dimensional torus and ...

Added: February 25, 2014

Buryak A., Rossi P., Bulletin of the London Mathematical Society 2021 Vol. 53 No. 3 P. 843–854

In this paper we compute the intersection number of two double ramification (DR) cycles (with different ramification profiles) and the top Chern class of the Hodge bundle on the moduli space of stable curves of any genus. These quadratic DR integrals are the main ingredients for the computation of the DR hierarchy associated to the ...

Added: February 1, 2021

V. L. Popov, Transformation Groups 2011 Vol. 16 No. 3 P. 827–856

Let G be a connected semisimple algebraic group over an algebraically closed field k. In 1965 Steinberg proved that if G is simply connected, then in G there exists a
closed irreducible cross-section of the set of closures of regular conjugacy classes. We prove
that in arbitrary G such a cross-section exists if and only if the ...

Added: March 16, 2013

Р.С. Авдеев, Известия РАН. Серия математическая 2010 Т. 74 № 6 С. 3–26

The extended weight semigroup of a homogeneous space G/H of a connected semisimple algebraic group G characterizes the spectra of the representations of G on spaces of regular sections of homogeneous line bundles over G/H, including the space of regular functions on G/H. We compute the extended weight semigroups for all strictly irreducible affine spherical ...

Added: February 25, 2014

Р.С. Авдеев, Математические заметки 2013 Т. 94 № 1 С. 22–35

For an arbitrary connected solvable spherical subgroup H of a connected semisimple algebraic group G, we compute the group N_G(H), the normalizer of H in G. Thereby we complete a classification of all (not necessarily connected) solvable spherical subgroups in semisimple algebraic groups. ...

Added: February 25, 2014

Р.С. Авдеев, Труды Московского математического общества 2010 Т. 71 С. 235–269

A spherical homogeneous space G/H of a connected semisimple algebraic group G is called excellent if it is quasi-affine and its weight semigroup is generated by disjoint linear combinations of the fundamental weights of the group G. All the excellent affine spherical homogeneous spaces are classified up to isomorphism. ...

Added: February 25, 2014

В. Л. Попов, Доклады Российской академии наук. Математика, информатика, процессы управления (ранее - Доклады Академии Наук. Математика) 2017 Т. 475 № 1 С. 14–16

Даны классификации неприводимых представлений простых алгебраических групп модальностей 0, 1 и 2. ...

Added: May 3, 2017

Р.С. Авдеев, Труды Московского математического общества 2011 Т. 72 № 1 С. 5–62

We develop a structure theory of connected solvable spherical subgroups in semisimple algebraic groups. Based on this theory, we obtain an explicit classification of all such subgroups up to conjugacy. ...

Added: February 25, 2014

Р.С. Авдеев, Математический сборник 2012 Т. 203 № 11 С. 3–22

For an affine spherical homogeneous space G/H of a connected semisimple algebraic group G, we consider the factorization morphism by the action on G/H of a maximal unipotent subgroup of G. We prove that this morphism is equidimensional if and only if the weight semigroup of G/H satisfies a simple condition. ...

Added: February 25, 2014

V. L. Popov, Doklady Mathematics 2017 Vol. 96 No. 1 P. 312–314

For connected simple algebraic groups defined over an algebraically closed field of characteristic
zero, the classifications of irreducible algebraic representations of modalities 0, 1, and 2 are obtained. ...

Added: June 30, 2017

Васильев Д. А., Siberian Mathematical Journal 2023 Vol. 64 No. 3 P. 525–541

We construct an infinite series of irreducible components of the moduli space of stable rank 3 sheaves on P3 with the zero first Chern class and establish the rationality of the components of this series. We also prove the rationality of the irreducible components of the moduli space of stable rank 2 sheaves on P3 belonging to an infinite subseries of the series ...

Added: May 29, 2023

Buryak A., Gubarevich D., Mathematical Physics Analysis and Geometry 2023 Vol. 26 No. 3 Article 23

One of many manifestations of a deep relation between the topology of the moduli spaces of algebraic curves and the theory of integrable systems is a recent construction of Arsie, Lorenzoni, Rossi, and the first author associating an integrable system of evolutionary PDEs to an F-cohomological field theory (F-CohFT), which is a collection of cohomology ...

Added: November 20, 2023

Vladimir L. Popov, / Cornell University. Series math "arxiv.org". 2017. No. 1707.06914 [math.AG].

We classify all connected affine algebraic groups G such that there are only finitely many G-orbits in every algebraic G-variety containing a dense open G-orbit. We also prove that G enjoys this property if and only if every irreducible algebraic G-variety X is modality-regular, i.e., the modality of X (in the sense of V. Arnol’d) ...

Added: July 24, 2017

Ivan V. Arzhantsev, Yulia I. Zaitseva, Kirill V. Shakhmatov, Proceedings of the Steklov Institute of Mathematics 2022 Vol. 318 No. 1 P. 13–25

Let X be an algebraic variety such that the group Aut(X) acts on X transitively. We define the transitivity degree of X as the maximum number m such that the action of Aut(X) on X is m-transitive. If the action of Aut(X) is m-transitive for all m, the transitivity degree is infinite. We compute the transitivity degree for all quasi-affine toric varieties and for many homogeneous spaces of algebraic groups. We also discuss a conjecture and ...

Added: November 5, 2022