CAPELLI IDENTITIES FOR CLASSICAL LIE ALGEBRAS


Dr Maxim Nazarov, University of York


Let g be either the symplectic Lie algebra spN or the orthogonal Lie algebra soN over the complex field C. For every value of a complex parameter u we introduced a family of distinguished central elements in the universal enveloping algebra U(g). The elements of this family are parametrised by partitions of n=0,1,2, ... with not more than N parts. The degree in U(g) of the central element corresponding to a partition of n does not exceed n for any u . For generic u the linear span of this family is exactly the subalgebra of all elements in U(g), which are invariant under the adjoint action of the corresponding classical Lie group G. For every partition of n and any value of the parameter u we computed the image of the corresponding central element in the space of symmetric polynomials of degree n in [N/2] variables under the Chevalley correspondence. For the simplest partitions (1n ) and (n) with certain special values of the parameter u , our central elements admit ``good'' representation by the differential operators with polynomial coefficients on the tensor product of the vector spaces CM and CN , for any positive integer M . These particular central elements generate the subalgebra of G-invariants in U(g ). We computed the images of these central elements under the Harish-Chandra homomorphism. Our formulas for the corresponding differential operators are the g-counterparts of the Capelli identities (1890) for the general linear Lie algebra glN . To obtain these new identities we used the representation theory of the twisted Yangian corresponding to g . Our results are available on the e-print archive Mathematics, see


EPSRC Grant and Duration: 5575 over 36 months from 1.1.1997, reference GR/L36970


For further information please contact: Dr Maxim Nazarov, Department of Mathematics, University of York, York YO10 5DD, England; maxim.nazarov@york.ac.uk