In this paper we consider an open SL(2,ℝ) spin chain, mainly the simplest case of one particle. Eigenfunctions of the model can be constructed using the so-called reflection operator. We obtain several representations of this operator and show its relation to the hypergeometric function. Besides, we prove orthogonality and completeness of one-particle eigenfunctions and connect them to the index hypergeometric transform. Finally, we briefly state the formula for the eigenfunctions in many-particle case. Bibliography: 17 titles.

We introduce and study a new family of commuting Baxter operators in the Ruijsenaars hyperbolic system, different from that considered by us earlier. Using a degeneration of Rains integral identity we verify the commutativity between the two families of Baxter operators and explore this fact for the proof of the coupling constant symmetry of the wave function. We also establish a connection between new Baxter operators and Noumi-Sano difference operators.

In the previous paper we showed that the wave functions of the quantum Ruijsenaars hyperbolic system diagonalize Baxter Q-operators. Using this property and duality relation we prove orthogonality and completeness relations for the wave functions or, equivalently, unitarity of the corresponding integral transform.

In these notes we review the technique of Baxter Q-operators in the Ruijsenaars-Sutherland hyperbolic systems in the cases of one and two particles. Using these operators we show in particular that eigenfunctions of these systems admit two dual integral representations and prove their orthogonality and completeness.

We give a description of the Halln\"as--Ruijsenaars eigenfunctions of the 2-particle hyperbolic Ruijsenaars system as matrix coefficients for the order 4 element $S\in SL(2,\mathbb{Z})$ acting on the Hilbert space of $GL(2)$ quantum Teichm\"uller theory on the punctured torus. The $GL(2)$ Macdonald polynomials are then obtained as special values of the analytic continuation of these matrix coefficients. The main tool used in the proof is the cluster structure on the moduli space of framed $GL(2)$-local systems on the punctured torus, and an $SL(2,\mathbb{Z})$-equivariant embedding of the $GL(2)$ spherical DAHA into the quantized coordinate ring of the corresponding cluster Poisson variety.

We present a proof of hypergeometric identities which play a crucial role in the theory of Baxter operators in the Ruijsenaars model.

Using Zhelobenko-Stern formulas for the action of the generators of orthogonal Lie algebra in corresponding Gelfand-Tsetlin basis, we derive Mellin-Barnes presentations for the wave functions of $B_n$ Toda lattice. They are in accordance with Iorgov-Shadura formulas.

We introduce Baxter Q -operators for the quantum Ruijsenaars hyperbolic system. We prove that they represent a commuting family of integral operators and also commute with Macdonald difference operators, which are gauge equivalent to the Ruijsenaars Hamiltonians of the quantum system. The proof of commutativity of the Baxter operators uses a hypergeometric identity on rational functions that generalize Ruijsenaars kernel identities.

In the previous paper, we introduced a commuting family of Baxter Q -operators for the quantum Ruijsenaars hyperbolic system. In the present work, we show that the wave functions of the quantum system found by M. Hallnäs and S. Ruijsenaars also diagonalize Baxter operators. Using this property, we prove the conjectured duality relation for the wave function. As a corollary, we show that the wave function solves bispectral problems for pairs of dual Macdonald and Baxter operators. Besides, we prove the conjectured symmetry of the wave function with respect to spectral variables and obtain new integral representation for it.

We obtain certain Mellin-Barnes integrals that present wave functions for GL(n, R)hyperbolic Sutherland model with arbitrary positive coupling constant.

We obtain certain Mellin-Barnes integrals which present Whittaker wave functions related to classical real split forms of simple complex Lie groups.

We present a construction of an integrable model as a projective type limit of spin Calogero-Sutherland model withNfermionic particles, whereNtends to infinity. It is implemented in the multicomponent fermionic Fock space. Explicit formulas for limits of Dunkl operators and the Yangian generators are presented by means of fermionic fields.

This paper is a survey of the mathematical work of Grigori Iosifovich Olshanski, the author of fundamental research papers concerning representations of infinite-dimensional groups, determinantal point processes, and multidimensional special functions. We shall also briefly discuss new directions and new opportunities that have arisen in connection with his discoveries. Olshanski studied in the Faculty of Mechanics and Mathematics of Moscow State University in 1964–1969. In his diploma thesis (see [1]) he obtained Frobenius duality for spaces of type L(G/Γ), where G is a nilpotent Lie group and Γ is a lattice. In 1972 he finished his postgraduate studies in the Department of the Theory of Functions and Functional Analysis of the faculty, with A. A. Kirillov as his advisor, and in 1973 he defended his Ph.D. thesis, Representations of reductive groups over local non-Archimedean fields. He then investigated representations of the group of automorphisms of Bruhat–Tits trees in [2], a paper which definitely attracted attention; on trees he also refined and perfected some of the methods that he later used in his investigations of classical groups [4]. Perhaps the main work of Olshanski was development of the theory of representations of infinite-dimensional classical groups. His further studies were connected with this work in one way or another, and in many respects it also determined the originality of his views on classical groups, special functions, random processes, and combinatorics. The work was published in a series of papers over the years 1978–1991 (beginning in [3], with the final publications1 being [19] and [21], and moreover, this was the topic of his D.Sc. thesis [15], defended in 1990 at the Leningrad Branch of the Steklov Mathematical Institute).

We present a construction of an integrable model as a projective type limit of Calogero-Sutherland models of $N$ fermionic particles, when $N$ tends to infinity. Explicit formulas for limits of Dunkl operators and of commuting Hamiltonians by means of vertex operators are given.

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.

We define contravariant forms on diagonal reduction algebras, algebras of h-deformed differential operators and on standard modules over these algebras. We study properties of these forms and their specializations. We show that the specializations of the forms on the spaces of h-commuting variables present zero singular vectors iff they are in the kernel of the specialized form. As an application we compute norms of highest weight vectors in the tensor product of an irreducible finite dimensional representation of the Lie algebra gln with a symmetric or wedge tensor power of its fundamental representation.

Arakawa, Suzuki and Tsuchiya defined a correspondence between certain modules of the trigonometric Cherednik algebra C-N depending on a parameter kappa is an element of C, and certain modules of the affine Lie algebra (sl) over cap (m) of level kappa - m. We give a detailed proof of this correspondence by working with the affine Lie algebra (gl) over cap (m) alongside of (sl) over cap (m). We also relate this construction to a correspondence between certain modules of the degenerate affine Hecke algebra (SIC)(N) and all modules of sl(m) or gl(m). The latter correspondence was constructed earlier by Cherednik.

We describe the diagonal reduction algebra D(gl n ) of the Lie algebra gl n in the R-matrix formalism. As a byproduct we present two families of central elements and the braided bialgebra structure of D(gl n ).

We present a construction of a new integrable model as an infinite limit of Calogero models of N particles with spin. It is implemented in the multicomponent Fock space. Explicit formulas for Dunkl operators, the Yangian generators in the multicomponent Fock space are presented. The classical limit of the system is examined.