We obtain a partial solution of the problem on the growth of the norms of exponential functions with a continuous phase in the Wiener algebra. The problem was posed by J.-P. Kahane at the International Congress of Mathematicians in Stockholm in 1962. He conjectured that (for a nonlinear phase) one can not achieve the growth slower than the logarithm of the frequency. Though the conjecture is still not confirmed, the author obtained first nontrivial results.

Let Γ be an arithmetic group of affine automorphisms of the n-dimensional future tube *T*. It is proved that the quotient space *T*/Γ is smooth at infinity if and only if the group Γ is generated by reflections and the fundamental polyhedral cone (“Weyl chamber”) of the group *d*Γ in the future cone is a simplicial cone (which is possible only for *n* ≤ 10). As a consequence of this result, a smoothness criterion for the Satake–Baily–Borel compactification of an arithmetic quotient of a symmetric domain of type IV is obtained.

To an arbitrary involutive stereotype algebra *A* the *continuous envelope*operation assigns its nearest, in some sense, involutive stereotype algebra Env*C**A* so that homomorphisms to various C*-algebras separate the elements of Env*C* A but do not distinguish between the properties of A and those of Env*C**A*.

If A is an involutive stereotype subalgebra in the algebra *C*(*M*) of continuous functions on a paracompact locally compact topological space *M*, then, for *C*(*M*) to be a continuous envelope of *A*, i.e., Env*C**A* = *C*(*M*), it is necessary but*not sufficient* that *A* be dense in *C*(*M*). In this note we announce a necessary and sufficient condition for this: the involutive spectrum of *A* must coincide with *M* up to a weakening of the topology such that the system of compact subsets in *M* and the topology on each compact subset remains the same.

It is well--known that certain properties of continuous functions on the circle T, related to the Fourier expansion, can be improved by a change of variable, i.e., by a homeomorphism of the circle onto itself. One of the results in this area is the Jurkat--Waterman theorem on conjugate functions, which improves the classical Bohr--P\'al theorem. In the present work we propose a short and technically very simple proof of the Jurkat--Waterman theorem. Our approach yields a stronger result.

Asymptotic properties of products of random matrices ξ k = X k …X 1 as k → ∞ are analyzed. All product terms X i are independent and identically distributed on a finite set of nonnegative matrices A = {A 1, …, A m }. We prove that if A is irreducible, then all nonzero entries of the matrix ξ k almost surely have the same asymptotic growth exponent as k→∞, which is equal to the largest Lyapunov exponent λ(A). This generalizes previously known results on products of nonnegative random matrices. In particular, this removes all additional “nonsparsity” assumptions on matrices imposed in the literature.We also extend this result to reducible families. As a corollary, we prove that Cohen’s conjecture (on the asymptotics of the spectral radius of products of random matrices) is true in case of nonnegative matrices.

This work is motivated by the observation that the character of an irreducible gl*n*-module (a Schur polynomial), being the sum of exponentials of integer points in a Gelfand–Tsetlin polytope, can be expressed by using Brion’s theorem. The main result is that, in the case of a regular highest weight, the contributions of all nonsimplicial vertices vanish, while the number of simplicial vertices is *n*! and the contributions of these vertices are precisely the summands in Weyl’s character formula.

We give an alternative proof of the main result of [1]; the proof relies on Brion’s theorem about convex polyhedra. The result itself can be viewed as a formula for the character of the Feigin-Stoyanovsky subspace of an integrable irreducible representation of the affine Lie algebra widehatsln(C). Our approach is to assign integer points of a certain polytope to vectors comprising a monomial basis of the subspace and then compute the character by using (a variation of) Brion’s theorem.

We compute the Coulomb branch of a multiloop quiver gauge theory for the quiver with a single vertex, *r* loops, one-dimensional framing, and dim *V* = 2. We identify it with a Slodowy slice in the nilpotent cone of the symplectic Lie algebra of rank *r*. Hence it possesses a symplectic resolution with 2*r* fixed points with respect to a Hamiltonian torus action. We also identify its flavor deformation with a base change of the full Slodowy slice.

The degenerate Lie group is a semidirect product of the Borel subgroup with the normal abelian unipotent subgroup. We introduce a class of the highest weight representations of the degenerate group of type A, generalizing the PBW-graded representations of the classical group. Following the classical construction of the flag varieties, we consider the closures of the orbits of the abelian unipotent subgroup in the projectivizations of the representations. We show that the degenerate flag varieties $\Fl^a_n$ and their desingularizations $R_n$ can be obtained via this construction. We prove that the coordinate ring of $R_n$ is isomorphic to the direct sum of duals of the highest weight representations of the degenerate group. In the end, we state several conjectures on the structure of the highest weight representations.

The Thoma cone is a certain infinite-dimensional space that arises in the representation theory of the infinite symmetric group. The present note is a continuation of a paper by A. M. Borodin and the author (Electr. J. Probab. 18 (2013), no. 75), where a 2-parameter family of continuous-time Markov processes on the Thoma cone was constructed. The purpose of the note is to show that these processes are diffusions.

The paper contains an explicit construction of Strebel differentials on one-parameter families of hyperelliptic curves of even genus. Descriptions of the corresponding separatrices are presented.

The boundary of the Gelfand–Tsetlin graph is an infinite-dimensional locally compact space whose points parameterize the extreme characters of the infinite-dimensional group *U*(∞). The problem of harmonic analysis on the group *U*(∞) leads to a continuous family of probability measures on the boundary—the so-called zw-measures. Recently Vadim Gorin and the author have begun to study a *q*-analogue of the zw-measures. It turned out that constructing them requires introducing a novel combinatorial object, the extended Gelfand–Tsetlin graph. In the present paper it is proved that the Markov kernels connected with the extended Gelfand–Tsetlin graph and its *q*-boundary possess the Feller property. This property is needed for constructing a Markov dynamics on the *q*-boundary. A connection with the B-splines and their *q*-analogues is also discussed.

Let $G$ be a connected reductive algebraic group over $\mathbb{C}$. Let $\Lambda^{+}_{G}$ be the monoid of dominant weights of $G$. We construct the integrable crystals $\mathbf{B}^{G}(\lambda),\ \lambda\in\Lambda^{+}_{G}$, using the geometry of generalized transversal slices in the affine Grassmannian of the Langlands dual group. We construct the tensor product maps $\mathbf{p}_{\lambda_{1},\lambda_{2}}\colon \mathbf{B}^{G}(\lambda_{1}) \otimes \mathbf{B}^{G}(\lambda_{2}) \rightarrow \mathbf{B}^{G}(\lambda_{1}+\lambda_{2})\cup\{0\}$ in terms of multiplication of generalized transversal slices. Let $L \subset G$ be a Levi subgroup of $G$. We describe the restriction to Levi $\operatorname{Res}^G_L\colon\operatorname{Rep}(G)\rightarrow\operatorname{Rep}(L)$ in terms of the hyperbolic localization functors for the generalized transversal slices.

We give a complete solution to the inverse problem for finite Jacobi operators with matrix-valued coefficients.