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.

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.

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.

We derive Fredholm determinant representation for isomonodromic tau functions of Fuchsian systems with *n* regular singular points on the Riemann sphere and generic monodromy in GL (*N*,ℂ). The corresponding operator acts in the direct sum of *N* (*n* − 3) copies of *L*2 *(S*1). Its kernel has a block integrable form and is expressed in terms of fundamental solutions of *n* − 2 elementary 3-point Fuchsian systems whose monodromy is determined by monodromy of the relevant *n*-point system via a decomposition of the punctured sphere into pairs of pants. For *N* = 2 these building blocks have hypergeometric representations, the kernel becomes completely explicit and has Cauchy type. In this case Fredholm determinant expansion yields multivariate series representation for the tau function of the Garnier system, obtained earlier via its identification with Fourier transform of Liouville conformal block (or a dual Nekrasov–Okounkov partition function). Further specialization to *n* = 4 gives a series representation of the general solution to Painlevé VI equation.