We analyze conditions under which a projection from the vector-valued Jack orMacdonald polynomials to scalar has useful properties, specially commuting with actions of symmetric group or Hecke algebra, respectively, and Cherednik operators for these are eigenfunctions. In framework representation theory we study relation between singular nonsymmetric Macdonald highest weight polynomials. Moreov...

We define regular Kac-Moody superalgebras and classify them using integrable modules. We give conditions for irreducible highest weight modules of regular Kac-Moody superalgebras to be integrable. This paper is a major part of the proof for the classification of finite-growth contragredient Lie superalgebras. The results of this paper are a crucial part of the proof for the classification of co...

Since Kashiwara introduced the theory of crystal base ([2]) in 1990, one of the most fundamental problems has been to describe the crystal base associated with the given integrable highest weight module as explicitly as possible. In order to answer this, many kinds of new combinatorial objects have been invented, e.g., in [9] some analogues of Young tableaux were introduced in order to describe...

The purpose of this note is to define and construct highest weight modules for Felder’s elliptic quantum groups. This is done by using exchange matrices for intertwining operators between modules over quantum affine algebras. A similar problem for the elliptic quantum group corresponding to Belavin’s R-matrix was posed in [7]. This problem, as well as its analogue for Felder’s R-matrix was solv...

Let g be a Lie algebra over a field k of characteristic zero, and a fix positive integer N. The Lie algebra ĝ = g ⊗k k[t]/t N+1 k[t] is called a truncated current Lie algebra. In this paper a highest-weight theory for ĝ is developed when the underlying Lie algebra g possesses a triangular decomposition. The principal result is the reducibility criterion for the Verma modules of ĝ for a wide cla...