Abstract. The deformed quantum Calogero-Moser-Sutherland problems related to the root systems of the contragredient Lie superalgebras are introduced. The construction is based on the notion of the generalized root systems suggested by V. Serganova. For the classical series a recurrent formula for the quantum integrals is found, which implies the integrability of these problems. The corresponding algebras of the quantum integrals are investigated, the explicit formulas for their Poincare series for generic values of the deformation parameter are presented.

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It is shown that the deformed Calogero-Moser-Sutherland (CMS) operators can be described as the restrictions on certain affine subvarieties (called generalised discriminants) of the usual CMS operators for infinite number of particles. The ideals of these varieties are shown to be generated by the Jack symmetric functions related to the Young diagrams with special geometry. A general structure of the ideals which are invariant under the action of the quantum CMS integrals is discussed in this context. The shifted super-Jack polynomials are introduced and combinatorial formulas for them and for super-Jack polynomials are given. r 2004 Elsevier Inc. All rights reserved.

Abstract. It is shown that the deformed Macdonald-Ruijsenaars operators can be described as the restrictions on certain affine subvarieties of the usual Macdonald-Ruijsenaars operator in infinite number of variables. The ideals of these varieties are shown to be generated by the Macdonald polynomials related to Young diagrams with special geometry. The super Macdonald polynomials and their shifted version are introduced, the combinatorial formulas for them are given.

The Grothendieck rings of finite dimensional representations of the basic classical Lie superalgebras are explicitly described in terms of the corresponding generalized root systems. We show that they can be interpreted as the subrings in the weight group rings invariant under the action of certain groupoids called super Weyl groupoids.

Chevalley's theorem states that for any simple finite dimensional Lie algebra g: (1) the restriction homomorphism of the algebra of polynomials S(g *) −→ S(h *) onto the Cartan subalgebra h induces an isomorphism S(g *) g ∼ = S(h *) W , where W is the Weyl group of g; (2) each g-invariant polynomial is a linear combination of the polynomials trρ(x) k , where ρ is a finite dimensional representation of g. None of these facts is necessarily true for simple Lie superalgebras. We reformulate Chevalley's theorem as formula (*) below to include Lie superalgebras. Let h be the split Cartan subalgebra of g; let R = R + ∪ R − be the set of nonzero roots of g, the union of positive and negative ones. Set R + = {α ∈ R + | −α ∈ R − }. For each root α ∈R + denote by g(α) the Lie superalgebra generated by h and the root superspaces gα and g −α. Let the image of S(g(α) *) g(α) under the restriction homomorphism S(g(α) *) −→ S(h *) be denoted by I α (h *) and the image of S(g *) g by I(h *). Then I(h *) = α∈R + I α (h *). (*) Chevalley's theorem for anti-invariant polynomials is also presented.

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We define the Dunkl and Dunkl-Heckman operators in infinite number of variables and use them to construct the quantum integrals of the Calogero-Moser-Sutherland problems at infinity. As a corollary we have a simple proof of integrability of the deformed quantum CMS systems related to classical Lie superalgebras. We show how this naturally leads to a quantum version of the Moser matrix, which in the deformed case was not known before.