Hamiltonian reduction is used to project a trivially integrable system on the Heisenberg double of SU(n; n), to obtain a system of Ruijsenaars type on a suitable quotient space. This system possesses BCn symmetry and is shown to be equivalent to the standard three-parameter BCn hyperbolic Sutherland model in the cotangent bundle limit.

See http://link.springer.com/article/10.1007/s00220-014-2126-6#

We construct a filtration on an integrable highest weight module of an affine Lie algebra whose adjoint graded quotient is a direct sum of global Weyl modules. We show that the graded multiplicity of each global Weyl module there is given by the corresponding level-restricted Kostka polynomial. This leads to an interpretation of level-restricted Kostka polynomials as the graded dimension of the space of conformal coinvariants. In addition, as an application of the level one case of the main result, we realize global Weyl modules of current algebras of type ADEADE in terms of Schubert subvarieties of thick affine Grassmanian, as predicted by Boris Feigin.

In Gurevich and Saponov (J Geom Phys 138:124–143, 2019) the notion of braided Yangians of Reflection Equation type was introduced. Each of these algebras is associated with an involutive or Hecke symmetry R. In these algebras quantum analogs of certain symmetric polynomials (elementary symmetric ones, power sums) were defined. In the present paper we show that these quantum symmetric polynomials commute with each other and consequently generate a commutative Bethe subalgebra. As an application, we get some Gaudin-type models.

Let gg be a complex simple Lie algebra. We study the family of Bethe subalgebras in the Yangian Y(g)Y(g) parameterized by the corresponding adjoint Lie group *G*. We describe their classical limits as subalgebras in the algebra of polynomial functions on the formal Lie group G1[[t−1]]G1[[t−1]]. In particular we show that, for regular values of the parameter, these subalgebras are free polynomial algebras with the same Poincaré series as the Cartan subalgebra of the Yangian. Next, we extend the family of Bethe subalgebras to the De Concini–Procesi wonderful compactification G¯¯¯¯⊃GG¯⊃G and describe the subalgebras corresponding to generic points of any stratum in G¯¯¯¯G¯ as Bethe subalgebras in the Yangian of the corresponding Levi subalgebra in gg. In particular, we describe explicitly all Bethe subalgebras corresponding to the closure of the maximal torus in the wonderful compactification.

In 2012, Gamayun, Iorgov, and Lisovyy conjectured an explicit expression for the Painlevé VI τ function in terms of the Liouville conformal blocks with central charge c = 1. We prove that the proposed expression satisfies Painlevé VI τ function bilinear equations (and therefore prove the conjecture). The proof reduces to the proof of bilinear relations on conformal blocks. These relations were studied using the embedding of a direct sum of two Virasoro algebras into a sum of Majorana fermion and Super Virasoro algebra. In the framework of the AGT correspondence, the bilinear equations on the conformal blocks can be interpreted in terms of instanton counting on the minimal resolution of $${\mathbb{C}^2/\mathbb{Z}_2}$$C2/Z2 (similarly to Nakajima–Yoshioka blow-up equations). © 2015, Springer-Verlag Berlin Heidelberg.

Determinantal point processes are characterized by a special structural property of the correlation functions: they are given by minors of a correlation kernel. However, unlike the correlation functions themselves, this kernel is not defined intrinsically, and the same determinantal process can be generated by many different kernels. The non-uniqueness of a correlation kernel causes difficulties in studying determinantal processes. We propose a formalism which allows to find a distinguished correlation kernel under certain additional assumptions. The idea is to exploit a connection between determinantal processes and quasifree states on CAR, the algebra of canonical anticommutation relations. We prove that the formalism applies to discrete N-point orthogonal polynomial ensembles and to some of their large-N limits including the discrete sine process and the determinantal processes with the discrete Hermite, Laguerre, and Jacobi kernels investigated by Borodin and Olshanski (CommunMath Phys 353:853–903, 2017). As an application we resolve the equivalence/disjointness dichotomy for some of those processes.

We study determinantal point processes on D induced by the reproducing kernels of generalized Bergman spaces. In the first case, we show that all reduced Palm measures *of the same order* are equivalent. The Radon–Nikodym derivatives are computed explicitly using regularized multiplicative functionals. We also show that these determinantal point processes are rigid in the sense of Ghosh and Peres, hence reduced Palm measures *of different orders* are singular. In the second case, we show that all reduced Palm measures, *of all orders*, are equivalent. The Radon–Nikodym derivatives are computed using regularized multiplicative functionals associated with certain Blaschke products. The quasi-invariance of these determinantal point processes under the group of diffeomorphisms with compact supports follows as a corollary.

It this paper we present a new construction of a hamiltonian hierarchy associated to a cohomological field theory. We conjecture that in the semisimple case our hierarchy is related to the Dubrovin-Zhang hierarchy by a Miura transformation and check it in several examples.