The index of representations associated with stabilisers

Let $Q$ be an algebraic group and $V$ a $Q$-module. The index of $V$ is the minimal codimension of the $Q$-orbits in the dual space $V^*$. There is a general inequality, due to Vinberg, relating the index of $V$ and the index of $Q_v$-module $V/q.v$ for any $v\in V$. In this article, we study conditions that guarantee us the equality. It was recently proved by Charbonnel (Bull. Soc. Math. France. v.132, 2004) that such an equality holds for the adjoint representation of a semisimple group. Another proof for the classical series was given by the second author, see math.RT/0407065. One of our goals, which is almost achieved, is to understand what is going on in the case of isotropy representations of symmetric spaces.Comment: 22 pages, some classification results added in Sections 6 and

On maximal commutative subalgebras of Poisson algebras associated with involutions of semisimple Lie algebras

For any involution $\sigma$ of a semisimple Lie algebra $\mathfrak g$, one constructs a non-reductive Lie algebra $\mathfrak k$, which is called a $\mathbb Z_2$-contraction of $\mathfrak g$. In this paper, we attack the problem of describing maximal commutative subalgebras of the Poisson algebra $S(\mathfrak k)$. This is closely related to the study of the coadjoint representation of $\mathfrak k$ and the set, $\mathfrak k^*_{reg}$, of the regular elements of $\mathfrak k^*$. By our previous results, in the context of $\mathbb Z_2$-contractions, the argument shift method provides maximal commutative subalgebras of $S(\mathfrak k)$ whenever $codim(\mathfrak k^* \setminus \mathfrak k^*_{reg})\ge 3$. Our main result here is that $codim(\mathfrak k^* \setminus \mathfrak k^*_{reg})\ge 3$ if and only if the Satake diagram of $\sigma$ has no trivial nodes. (A node is trivial, if it is white, has no arrows attached, and all adjacent nodes are also white.) The list of suitable involutions is provided. We also describe certain maximal commutative subalgebras of $S(\mathfrak k)$ if the (-1)-eigenspace of $\sigma$ in $\mathfrak g$ contains regular elements.Comment: 18 pages, to appear in Bull. Sci. Mat

Prospects for the development of a modern system of higher education in Russia

В данной статье рассматривается вопрос эффективности подготовки будущих специалистов в рамках современного российского образовательного пространства высшей школы, а так же результаты реформ, связанных с присоединением России к Болонскому процессу.This article examines the effectiveness of future specialists training in the context of the contemporary Russian educational space of higher education and the results of reforms related to Russia’s accession to the Bologna Process