International Congress of Mathematicians

Representations of Yangians 651irreducible representations W\ and W ß of the groups OJV+M and OM respectively.The vector space Horn 0M ( W ß , W\ ) (2.5)comes with a natural action of the group ON • This action of ON may be reducible.The vector space (2.5) is non-zero, if and only if X k ^ Pk and X' k — p' k ^ N foreach k = 1,2,... ; see [18]. Thus for a given N, the vector spaces (1.5) and (2.5)are zero or non-zero simultaneously. Further, for a given N, the dimension of (2.5)does not exceed that of (1.5). Our results provide an embedding of (2.5) into (1.5),compatible with the action of the orthogonal group ON in these two vector spaces.Denote by BJV(AT) the subalgebra of OM -invariants in the universal envelopingalgebra Y(SON+M)- Then Bjv(Af) contains the subalgebra U(sojv) C Y(SON+M),and is contained in the centralizer of the subalgebra U(SOM) C Y(SON+M)- Thealgebra Bjv(Af) naturally acts in the vector space (2.5). The Bjv(Af)-module (2.5)is either irreducible, or splits into a direct sum of two irreducible Bjv(Af)-modules.In the latter case, (2.5) is irreducible under the joint action of the algebra Bjv(M)and the subgroup ON C OJV+M-For every non-negative integer M, Olshanski [17] defined a homomorphismY(gljvj