Hypergeometric systems of Gelfand-type equations and series in orthogonal polynomials

We describe multivariable hypergeometric series in orthogonal polynomials. These series are solutions of special systems of Gelfand-type equations. The difference andq-analogs are also given.     

Cohomology in nonunitary representations of semisimple Lie groups (the group U(2, 2))

A method for constructing special nonunitary representations of semisimple Lie groups by using representations of Iwasawa subgroups is suggested. As a typical example, the group U(2, 2) is studied.     

On the compactness of the set of invariant Einstein metrics

Let $M = G/H$ be a connected simply connected homogeneous manifold of a compact, not necessarily connected Lie group $G$ . We will assume that the isotropy $H$ -module $\mathfrak{g/h }$ has a simple spectrum, i.e. irreducible submodules are mutually non-equivalent. There exists a convex Newton polytope $N=N(G,H)$ , which was used for the estimation of the number of isolated complex solutions of the algebraic Einstein equation for invariant metrics on $G/H$ (up to scaling). Using the moment map, we identify the space $\mathcal{M }_1$ of invariant Riemannian metrics of volume 1 on $G/H$ with the interior of this polytope $N$ . We associate with a point ${x \in \partial N}$ of the boundary a homogeneous Riemannian space (in general, only local) and we extend the Einstein equation to $\partial N$ . As an application of the Alekseevsksky–Kimel’fel’d theorem, we prove that all solutions of the Einstein equation associated with points of the boundary are locally Euclidean. We describe explicitly the set $T\subset \partial N$ of solutions at the boundary together with its natural triangulation. Investigating the compactification ${\overline{\mathcal{M }}}_{1} = N$ of $\mathcal{M }_1$ , we get an algebraic proof of the deep result by Böhm, Wang and Ziller about the compactness of the set $\mathcal{E }_1 \subset \mathcal{M }_1$ of Einstein metrics. The original proof by Böhm, Wang and Ziller was based on a different approach and did not use the simplicity of the spectrum. In Appendix, we consider the non-symmetric flag manifolds $G/H$ with the second Betti number $b_2=1$ . We calculate the normalized volumes $2,6,20,82,344$ of the corresponding Newton polytopes and discuss the number of complex solutions of the algebraic Einstein equation and the finiteness problem.     

Integral models of representations of current groups

We suggest a new construction of nonlocal representations of the current group. Instead of the Fock space, which is usually used in this situation, we consider the direct integral of countable tensor products of representations over the trajectories of some stochastic process. The construction substantially uses the invariance of the so-called infinite-dimensional Lebesgue measure.     

GG Functions and their Relations to General Hypergeometric Functions

In this Letter, we present a new approach to the notion of hypergeometric functions.