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## Сферические действия на многообразиях флагов

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.

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 question whether or not the field extensions k(G)/k(G)^G and k(g)/k(g)^G are purely transcendental. We show that the answer is the same for k(G)/k(G)^G and k(g)/k(g)^G, and reduce the problem to the case where G is simple. For simple groups we show that the answer is positive if G is split of type A_n or C_n, and negative for groups of other types, except possibly G_2. A key ingredient in the proof of the negative result is a recent formula for the unramified Brauer group of a homogeneous space with connected stabilizers. As a byproduct of our investigation we give an affirmative answer to a question of Grothendieck about the existence of a rational section of the categorical quotient morphism for the conjugating action of G on itself.

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 universal covering isogeny Ĝ → G is bijective; this answers Grothendieck's question cited in the epigraph. In particular, for char k = 0, the converse to Steinberg's theorem holds. The existence of a cross-section in G implies, at least for char k = 0, that the algebra k[G]G of class functions on G is generated by rk G elements. We describe, for arbitrary G, a minimal generating set of k[G]G and that of the representation ring of G and answer two Grothendieck's questions on constructing generating sets of k[G]G. We prove the existence of a rational (i.e., local) section of the quotient morphism for arbitrary G and the existence of a rational cross-section in G (for char k = 0, this has been proved earlier); this answers the other question cited in the epigraph. We also prove that the existence of a rational section is equivalent to the existence of a rational W-equivariant map T- - - >G/T where T is a maximal torus of G and W the Weyl group.

Lie theory, inaugurated through the fundamental work of Sophus Lie during the late nineteenth century, has proved central in many areas of mathematics and theoretical physics. Sophus Lie’s formulation was originally in the language of analysis and geometry; however, by now, a vast algebraic counterpart of the theory has been developed. As in algebraic geometry, the deepest and most far-reaching results in Lie theory nearly always come about when geometric and algebraic techniques are combined. A core part of Lie theory is the structure and representation theory of complex semisimple Lie algebras and Lie groups, which is an exemplary harmonious field in modern mathematics. It has deep ties to physics, and many areas of mathematics, such as combinatorics, category theory, and others. This field has inspired many generalizations, among them the representation theories of affine Lie algebras, vertex operator algebras, locally finite Lie algebras, Lie superalgebras, etc. This volume originates from a pair of sister conferences titled “Algebraic Modes of Representations” held in Israel in July 2017. The first conference took place at the Weizmann Institute of Science, Rehovot, July 16–18, and the second conference took place at the University of Haifa, July 19–23. Both conferences were dedicated to the 75th birthday of Anthony Joseph, who has been one of the leading figures in Lie Theory from the 1970s until today. The conferences were supported by the United States–Israel Binational Science Foundation and the Chorafas Institute for Scientific Exchange (Weizmann part) and by the Israel Science Foundation (Haifa part). Joseph has had a fundamental influence on both classical representation theory and quantized representation theory. A detailed description of his work in both areas has been given in the articles by W. McGovern and D. Farkash–G. Letzter in the volume “Studies in Lie theory,” Progress in Mathematics, vol. 243, Birkhauser. Concerning Joseph’s contribution to classical representation theory, it is impossible not to mention his classification of primitive ideals of the universal enveloping algebra of sl(n). The essential new ingredient here is the introduction of a partition of the Weyl group into left cells, corresponding to the Robinson map from the symmetric group to the standard Young tableaux. Joseph further extended this result to other simple Lie algebras using similar techniques, and this has since then become a powerful tool in Lie theory. As for quantized representation theory, Joseph’s monograph “Quantum Groups and Their Primitive Ideals,” Ergebnisse der Mathematik und Ihrer Grenzgebiete, 3rd series, vol. 29, has had a fundamental influence over the field since its appearance in 1995. The present volume contains 14 original papers covering a broad spectrum of current aspects of Lie theory. The areas discussed include primitive ideals, invariant theory, geometry of Lie group actions, crystals, quantum affine algebras, Yangians, categorification, and vertex algebras. The authors of this volume are happy to dedicate their works to Anthony Joseph.

We first establish several general properties of modality of algebraic group actions. In particular,we introduce the notion of a modality-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, exploring a finer geometric structure of linear actions, we generalize to the case of any cyclically graded semisimple Lie algebra the notion of a packet (or a Jordan/decomposition class) and establish the properties of packets.

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 prove that H is epimorphic in SL(V). In particular, the homogeneous space SL(V)/H provides a new example of a homogeneous space with epimorphic stabilizer that admits no projective embeddings with small boundary.

The following topics about subgroups of the Cremona groups are discussed: (1) maximal tori; (2) conjugacy and classification of diagonalizable subgroups of codimensions 0 and 1; (3) conjugacy of finite abelian subgroups; (4) algebraicity of normalizers of diagonalizable subgroups; (5) torsion primes.

The aim of these notes is to give an introduction into Schubert calculus on Grassmannians and flag varieties. We discuss various aspects of Schubert calculus, such as applications to enumerative geometry, structure of the cohomology rings of Grassmannians and flag varieties, Schur and Schubert polynomials. We conclude with a survey of results of V. Kiritchenko, V. Timorin and the author on a new approach to Schubert calculus on full flag varieties via combinatorics of Gelfand-Zetlin polytopes.

A model for organizing cargo transportation between two node stations connected by a railway line which contains a certain number of intermediate stations is considered. The movement of cargo is in one direction. Such a situation may occur, for example, if one of the node stations is located in a region which produce raw material for manufacturing industry located in another region, and there is another node station. The organization of freight traﬃc is performed by means of a number of technologies. These technologies determine the rules for taking on cargo at the initial node station, the rules of interaction between neighboring stations, as well as the rule of distribution of cargo to the ﬁnal node stations. The process of cargo transportation is followed by the set rule of control. For such a model, one must determine possible modes of cargo transportation and describe their properties. This model is described by a ﬁnite-dimensional system of diﬀerential equations with nonlocal linear restrictions. The class of the solution satisfying nonlocal linear restrictions is extremely narrow. It results in the need for the “correct” extension of solutions of a system of diﬀerential equations to a class of quasi-solutions having the distinctive feature of gaps in a countable number of points. It was possible numerically using the Runge–Kutta method of the fourth order to build these quasi-solutions and determine their rate of growth. Let us note that in the technical plan the main complexity consisted in obtaining quasi-solutions satisfying the nonlocal linear restrictions. Furthermore, we investigated the dependence of quasi-solutions and, in particular, sizes of gaps (jumps) of solutions on a number of parameters of the model characterizing a rule of control, technologies for transportation of cargo and intensity of giving of cargo on a node station.