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## On modality of representations

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.

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

Vladimir L. Popov, Modality of representations / Cornell University. Series math "arxiv.org". 2017. No. arXiv:1707.07720v1 [math.RT] 24 Jul 2017.

We first establish several general properties of modality of al gebraic group actions. In particular, we introduce the notion of a modali ty-regular action and prove that every visible action is modality-regular.
Then, using these results, we classify irreducible linear representations of
connected simple algebraic groups of every fixed modality < 3. Next, ex ploring a finer geometric structure of ...

Added: July 26, 2017

Vladimir L. Popov, Algebraic groups whose orbit closures contain only finitely many orbits / 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

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

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

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

Roman Avdeev, Degenerations of spherical subalgebras and spherical roots / Cornell University. Series math "arxiv.org". 2019. No. 1905.01169.

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 an explicit algorithm for computing the set ...

Added: June 1, 2019

Vladimir L. Popov, Number of components of the nullcone / Cornell University. Series math "arxiv.org". 2015. No. 1503.08303.

For every pair (G, V ) where G is a connected simple
linear algebraic group and V is a simple algebraic G-module with
a free algebra of invariants, the number of irreducible components
of the nullcone of unstable vectors in V is found. ...

Added: March 31, 2015

Vladimir L. Popov, The Jordan property for Lie groups and automorphism groups of complex spaces / Cornell University. Series math "arxiv.org". 2018. No. 1804.00323v1.

We prove that the family of all connected n-dimensional real Lie groups is uniformly Jordan for every n. This implies that all algebraic groups (not necessarily affine) over fields of cha\-racte\-ristic zero and some transformation groups of complex spaces and Riemannian manifods are Jordan. ...

Added: April 3, 2018

V. L. Popov, Mathematical notes 2018 Vol. 103 No. 5 P. 811-819

We prove that the family of all connected n-dimensional real Lie groups is uniformly Jordan for every n. This
implies that all algebraic (not necessarily affine) groups over fields of characteristic zero and some
transformation groups of complex spaces and Riemannian manifolds are Jordan. ...

Added: April 13, 2018

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

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

Vladimir L. Popov, Embeddings of automorphism groups of free groups into automorphism groups of affine algebraic varieties / Cornell University. Series math "arxiv.org". 2022. No. 2207.13072.

A new infinite series of rational affine algebraic varieties is constructed whose automorphism group contains the auto\-morphism group Aut(F_n) of the free group F_n of rank n. The automorphism groups of such varieties are nonlinear and contain the braid group B_n on n strands for n > 2, and are nonamenable for n > 1. ...

Added: July 27, 2022

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

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

Arzhantsev I., Celik D., Hausen J., Journal of Algebra 2013 Vol. 387 P. 87-98

Given an action of an affine algebraic group with only trivial characters on a factorial variety, we ask for categorical quotients. We characterize existence in the category of algebraic varieties. Moreover, allowing constructible sets as quotients, we obtain a more general existence result, which, for example, settles the case of afinitely generated algebra of invariants. ...

Added: November 13, 2013

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

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, 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

Kham T., Социальные и гуманитарные науки: теория и практика 2019 № 1(3) С. 167-183

The article examines the problems of defining the term computer simulations of scientific experiments. The first part analyzes the original method for classifying variations of terms proposed by Duran as the most successful for demonstrating significant existing contradictions among philosophers regarding the place and role of computer simulations in the philosophy of science. In the ...

Added: December 11, 2019

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

Vladimir L. Popov, Bass' triangulability problem / Cornell University. Series math "arxiv.org". 2015. No. 1504.03867.

Exploring Bass' Triangulability Problem on unipotent algebraic subgroups of the affine Cremona groups, we prove a triangulability criterion, the existence of nontriangulable connected solvable affine algebraic subgroups of the Cremona groups, and stable triangulability of such subgroups; in particular, in the stable range we answer Bass' Triangulability Problem is the affirmative. To this end we ...

Added: April 16, 2015

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

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