On the homomorphisms between scalar generalized Verma modules

We study the homomorphisms between scalar generalized Verma modules. We conjecture that any homomorphism between is composition of elementary homomorphisms. The purpose of this article is to show the conjecture is affirmative for many parabolic subalgebras under the assumption that the infinitesimal characters are regular.Comment: 46 pages, A reference is adde

Derived functor modules as irreducible constituents of degenerate principal series of the maximal Gelfand-Kirillov dimension (joint work with Peter E. Trapa)\ud

Introduction\ud \ud In the representation theory of real reductive Lie groups, there are two fundamental construction of representations; namely the parabolic induction and the cohomological induction. In particular, roughly speaking, we call an induced representation (respectively a cohomologically induced representation) from one-dimensional representation a degenerate principal series representation (respectively, derived functor module). They have natural geometrical interpretations. This suggests there is some relation between , the situation is two constructions of representations. However, for a group like complicated and we cannot say much about such a relation. However, for a group quite like $Sp(p, q)$ , whose Cartan subgroups are always connected, the situation is rather simple. In this talk, we explain the relation of degenerate principal series and derived functor modules for $Sp(p, q)$ . (Similar results holds for $SO^{*}(2n).$ )\ud $Sp(n, \mathbb{R})$\u

Extremal Trigonometrical and Power Polynomials of Several Variables

We consider the set of the power non-negative polynomials of several variables and its subset that consists of polynomials which can be represented as a sum of squares. It is shown in the classic work by D.Hilbert that it is a proper subset. Both sets are convex. In our paper we have made an attempt to work out a general approach to the investigation of the extremal elements of these convex sets. We also consider the class of non-negative rational functions. The article is based on the following methods: 1.We investigate non-negative trigonometrical polynomials and then with the help of the Calderon transformation we proceed to the power polynomials. 2.The way of constructing support hyperplanes to the convex sets is given in the paper