Stability conditions for slodowy slices and real variations of stability

Anno R; Bezrukavnikov Roman; Mirkovic I.

Article

Stability conditions for slodowy slices and real variations of stability

Moscow Mathematical Journal. 2015. Vol. 1. No. 15. P. 187-203.

Anno R., Bezrukavnikov R., Mirkovic I.

Important example applications of this idea is the theory of character sheaves Lusztig, which provides a geometric formula for characters of representations of finite groups Shemale, i.e. groups of rational points of a reductive algebraic group over a finite field, for example the group GL(n,F_q) of invertible matrices with coefficients in a finite field. Highlighting the main provisions of this theory, I turn to the description of a new approach to the classification of character sheaves, is based on joint work with V. Stricom and M. Finkelberg. Then I will discuss the project of the application of these ideas to a group of loops (joint with Y. Varshavsky and D. Kazhdan), which aims to describe algebraic-geometric phenomena behind the theory of endoscopy. This is the last important Chapter of harmonic analysis on p-adic groups associated with the duality of Langlands; the correspondence between features and beams have been used in this area B.-C. NGO devoted to the proof of the so-called fundamental Lemma (for which in 2004 he was awarded the fields)."

Priority areas: mathematics

Language: English

Keywords: slodowy slices

Publication based on the results of: Теория представлений и математическая физика(2015)

Coulomb Branch of a Multiloop Quiver Gauge Theory

Goncharov E. A., Finkelberg M. V. Functional Analysis and Its Applications. 2019. Vol. 53. P. 241-249.

We compute the Coulomb branch of a multiloop quiver gauge theory for the quiver with a single vertex, r loops, one-dimensional framing, and dim V = 2. We identify it with a Slodowy slice in the nilpotent cone of the symplectic Lie algebra of rank r. Hence it possesses a symplectic resolution with 2r fixed points with respect to a Hamiltonian torus action. We also identify its flavor deformation with a base change of the full Slodowy slice.

Added: Jul 12, 2020

Absolutely convergent Fourier series. An improvement of the Beurling-Helson theorem

Vladimir Lebedev. arxiv.org. math. Cornell University, 2011. No. 1112.4892v1.

We obtain a partial solution of the problem on the growth of the norms of exponential functions with a continuous phase in the Wiener algebra. The problem was posed by J.-P. Kahane at the International Congress of Mathematicians in Stockholm in 1962. He conjectured that (for a nonlinear phase) one can not achieve the growth slower than the logarithm of the frequency. Though the conjecture is still not confirmed, the author obtained first nontrivial results.

Added: Apr 12, 2012

Новые информационные технологии. Тезисы докладов XVIII международной студенческой конференции-школы-семинара

М.: МИЭМ, 2010.

Added: Apr 12, 2012

Обоснование адиабатического предела для гиперболических уравнений Гинзбурга-Ландау

Пальвелев Р., Сергеев А. Г. Труды Математического института им. В.А. Стеклова РАН. 2012. Т. 277. С. 199-214.

Added: Feb 23, 2013

Метод параметрикса для диффузий и цепей Маркова

Конаков В. Д. STI. WP BRP. Издательство попечительского совета механико-математического факультета МГУ, 2012. № 2012.

Added: Dec 5, 2012

Hypercommutative operad as a homotopy quotient of BV

Khoroshkin A., Markaryan N. S., Shadrin S. arxiv.org. math. Cornell University, 2012. No. 1206.3749.

We give an explicit formula for a quasi-isomorphism between the operads Hycomm (the homology of the moduli space of stable genus 0 curves) and BV/Δ (the homotopy quotient of Batalin-Vilkovisky operad by the BV-operator). In other words we derive an equivalence of Hycomm-algebras and BV-algebras enhanced with a homotopy that trivializes the BV-operator. These formulas are given in terms of the Givental graphs, and are proved in two different ways. One proof uses the Givental group action, and the other proof goes through a chain of explicit formulas on resolutions of Hycomm and BV. The second approach gives, in particular, a homological explanation of the Givental group action on Hycomm-algebras.

Added: Aug 29, 2012

Is the function field of a reductive Lie algebra purely transcendental over the field of invariants for the adjoint action?

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 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.

Added: Mar 17, 2013

Cross-sections, quotients, and representation rings of semisimple algebraic groups

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 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.

Added: Mar 16, 2013

Введение в математическую статистику

Ивченко Г. И., Медведев Ю. И. М.: ЛКИ, 2010.

Added: Apr 12, 2012

Новые информационные технологии. Тезисы докладов XIX международной студенческой конференции-школы-семинара

М.: МИЭМ, 2011.

Added: Apr 12, 2012