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#

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.