Cyclotomic double affine Hecke algebras
We show that the partially spherical cyclotomic rational Cherednik algebra (obtained from the full rational Cherednik algebra by averaging out the cyclotomic part of the underlying
reflection group) has four other descriptions: (1) as a subalgebra of the degenerate DAHA of type A
given by generators; (2) as an algebra given by generators and relations; (3) as an algebra of differentialreflection operators preserving some spaces of functions; (4) as equivariant Borel-Moore homology of
a certain variety. Also, we define a new q-deformation of this algebra, which we call cyclotomic DAHA.
Namely, we give a q-deformation of each of the above four descriptions of the partially spherical rational Cherednik algebra, replacing differential operators with difference operators, degenerate DAHA
with DAHA, and homology with K-theory, and show that they give the same algebra. In addition, we
show that spherical cyclotomic DAHA are quantizations of certain multiplicative quiver and bow varieties, which may be interpreted as K-theoretic Coulomb branches of a framed quiver gauge theory.
Finally, we apply cyclotomic DAHA to prove new flatness results for various kinds of spaces of q-deformed quasiinvariants.
We propose a conjectural construction of various slices for double affine Grass- mannians as Coulomb branches of 3-dimensional N = 4 supersymmetric affine quiver gauge theories. It generalizes the known construction for the usual affine Grassman- nians, and makes sense for arbitrary symmetric Kac-Mody algebras.
We define an integral form of shifted quantum affine algebras of type A and construct Poincaré–Birkhoff–Witt–Drinfeld bases for them. When the shift is trivial, our integral form coincides with the RTT integral form. We prove that these integral forms are closed with respect to the coproduct and shift homomorphisms. We prove that the homomorphism from our integral form to the corresponding quantized K -theoretic Coulomb branch of a quiver gauge theory is always surjective. In one particular case we identify this Coulomb branch with the extended quantum universal enveloping algebra of type A. Finally, we obtain the rational (homological) analogues of the above results [proved earlier in Kamnitzer et al. (Proc Am Math Soc 146(2):861–874, 2018a; On category O for affine Grassmannian slices and categorified tensor products. arXiv:1806.07519, 2018b) via different techniques].
We study canonical intertwining operators between induced modules of the trigonometric Cherednik algebra. We demonstrate that these operators correspond to the Zhelobenko operators for the affine Lie algebra . To establish the correspondence, we use the functor of Arakawa, Suzuki and Tsuchiya which maps certain modules to modules of the Cherednik algebra.
This is a companion paper of [Part II]. We study Coulomb branches of unframed and framed quiver gauge theories of type ADE. In the unframed case they are isomorphic to the moduli space of based ra- tional maps from P^1 to the flag variety. In the framed case they are slices in the affine Grassmannian and their generalization. In the appendix, written jointly with Joel Kamnitzer, Ryosuke Kodera, Ben Webster, and Alex Weekes, we identify the quantized Coulomb branch with the truncated shifted Yangian.
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.
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.