Given a finite subgroup of SL 3 C, we determine how an arbitrary finite dimensional irreducible representation of SL 3 C decomposes under the action of . To the subgroup we associate a generalized McKay matrix C . Then, generalizing a method used by B. Kostant for SL 2 C, we decompose C as a sum of products of reflections associated to mutually orthogonal roots: this is a sort of algebraic McKay correspondence in dimension 3.

Within the framework of McKay correspondence we determine, for every finite subgroup Γ of SL4C, how the finitedimensional irreducible representations of SL4C decompose under the action of Γ. Let h be a Cartan subalgebra of sl4C and let 1, 2, 3 be the corresponding fundamental weights. For (p, q, r) ∈ N 3 , the restriction πp,q,r|Γ of the irreducible representation πp,q,r of highest weight p 1 + q 2 + r 3 of SL4C decomposes as πp,q,r|Γ = l i=0 mi(p, q, r)γi, where {γ0,. .. , γ l } is the set of equivalence classes of irreducible finite-dimensional complex representations of Γ. We determine the multiplicities mi(p, q, r) and prove that the series PΓ(t, u, w)i = ∞ p=0 ∞ q=0 ∞ r=0 mi(p, q, r)t p u q w r are rational functions. This generalizes the results of Kostant for SL2C and the results of our preceding works for SL3C. У рамках вiдповiдностi Маккея для кожної скiнченної пiдгрупи Γ групи SL4C визначено, яким чином скiнченновимiрне незвiдне зображення SL4C розкладається пiд дiєю Γ. Нехай h-картанова пiдалгебра sl4C, а 1, 2, 3-вiдповiднi фундаментальнi ваги. Для (p, q, r) ∈ N 3 звуження πp,q,r|Γ незвiдного зображення πp,q,r найбiльшої ваги p 1 + q 2 + r 3 в SL4C розкладається у виглядi πp,q,r|Γ = l i=0 mi(p, q, r)γi, де {γ0,. .. , γ l }-множина класiв еквiвалентностi незвiдних скiнченновимiрних комплексних зображень Γ. Визначено кратностi mi(p, q, r) та доведено, що ряди PΓ(t, u, w)i = ∞ p=0 ∞ q=0 ∞ r=0 mi(p, q, r)t p u q w r є рацiональними функцiями. Це є узагальненням результатiв Костанта для SL2C, а також результатiв наших попереднiх робiт для SL3C.

Let g be a finite-dimensional semi-simple Lie algebra, h a Cartan subalgebra of g, and W its Weyl group. The group W acts diagonally on V := h ⊕ h * , as well as on C[V ]. The purpose of this article is to study the Poisson homology of the algebra of invariants C[V ] W endowed with the standard symplectic bracket.To begin with, we give general results about the Poisson homology space in degree 0, denoted by HP 0 (C[V ] W ), in the case where g is of type B n − C n or D n , results which support Alev's conjecture.Then we are focusing the interest on the particular cases of ranks 2 and 3, by computing the Poisson homology space in degree 0 in the cases where g is of type B 2 (so 5 ), D 2 (so 4 ), then B 3 (so 7 ), and D 3 = A 3 (so 6 sl 4 ). In order to do this, we make use of a functional equation introduced by Y. Berest, P. Etingof and V. Ginzburg. We recover, by a different method, the result established by J. Alev and L. Foissy, according to which the dimension of HP 0 (C[V ] W ) equals 2 for B 2 . Then we calculate the dimension of this space and we show that it is equal to 1 for D 2 . We also calculate it for the rank 3 cases, we show that it is equal to 3 for B 3 − C 3 and 1 for D 3 = A 3 .