International Congress of Mathematicians

650 Alaxim NazarovThen one can define an involutive automorphism TJN of the algebra X(gl N ,a)the assignment_n N : Sij(x) ^ Sij(-x - f ).byHowever, TJN does not determine an automorphism of Y(gl N ,a), because TJN doesnot preserve the ideal ofX(gl N ,a) generated by the coefficients of all series (2.3).For any formal power series f(x) £ C[[x -1 ]] with the leading term 1, theassignment Sy(x) H> f(x) • Sy(x) (2.4)defines an automorphism of the algebra X(gl N ,a). The defining relations of thealgebra X(gl N ,a) imply that the assignmentßN : Sy (x) H> öij • 1 + E ij -E Jidefines a homomorphism of associative algebras /3JV : X(gl N ,a) —¥ U(sojv). Bydefinition,the homomorphism /3JV is surjective. Moreover, /3JV factors through TTN •Note that the homomorphism Y(gl N ,a) —¥ U(sojv) corresponding to /3JV , cannot beobtained from O.N '• Y(gl N ) —¥ U(gl JV ) by restricting to the subalgebra Y(gl N ,a),because the image ofY(gl N ,a) relative to O.N is not contained in the subalgebraU(SOJV) C U(gl JV ); see [11]. An embedding U(sojv) —t Y(gl N ,a) can be defined byE t j-Ej t^T^-T^,cf. (1.12). The homomorphism Y(gl N ,a)identical on the subalgebra U(sojv) C—¥ U(sojv) corresponding to /3JV, is thenY(gl N ,a).2.2. For any partition v with v[ ^ N, the irreducible polynomial gl^-module V v canalso be regarded as a representation of the complex general linear Lie group GLN-Consider the subgroup ON C GLN preserving the standard symmetric bilinear form( , ) on C^. The subalgebra SON C gljv corresponds to this subgroup. Note thatthe complex Lie group ON has two connected components. In [20] the irreduciblefinite-dimensional representations of the group ON are labeled by the partitions v ofn = 0,1,2, ... such that v[ + v 2 ' ^ N. Denote by W v the irreducible representationof ON corresponding to v. As sojv-module, W v is irreducible unless 2v[ = N, inwhich case W v is a direct sum of two irreducible sojv-modules.Choose any embedding of the irreducible representation V v of the group GLNinto the space (C N )® n . Take any two distinct numbers k,l £ {1, ...,n). Byapplyingthe bilinear form ( , ) to a tensor w £ (C N )® n in the fcth and Ith tensorfactors, we obtain a certain tensor w £ (C N )® (" _2^. The tensor w is called traceless,if w = 0 for all distinct k and I. Denote by (C N )f n the subspace in (C N )® nconsisting of all traceless tensors, this subspace is ON -invariant. Then W v can beembedded into (C N )® n as the intersection V v n (C N )®. n , see [20].Let the indices i and j range over {1, ..., N + M}. Choose the embeddingof the Lie algebras gl^r and gl M into gijv+M as in Subsection 1.2. It determinesembeddings of groups GLN X GL M —^ GLN+M and ON X OM —^ OJV+M- Take anytwo partitions À and p such that A{ + X' 2 ^ N + M and p[ + p' 2 ^ M. Consider the