### YANGIANS AND AFFINE HECKE ALGEBRAS

The aims of the project were threefold. The primary objective
was to develop the representation theory of the Yangian of the
general linear Lie algebra *gl*_{N} , and of the
analogues of this Yangian for the other classical Lie algebras
and superalgebras. In this direction, several important results
were obtained. In particular, we classified those irreducible
finite-dimensional representations of the Yangian of *gl*_{N}
where a certain distinguished maximal commutative subalgebra
acts semi-simply. The second objective was to study the theory
of the affine Hecke algebras, which are dual to these Yangians
in the sense of Drinfeld. Here, in particular,
we developed the fusion procedure introduced
earlier by Cherednik. The last objective was to apply these theories
to the classical representation theory. Here we made a number of
remarkable discoveries. In particular, we found the analogues
of the classical Capelli identity for *gl*_{N} (1890)
for the symplectic and orthogonal Lie algebras.
Some of our printed articles are also available on the e-print archive
*Mathematics*, see

**EPSRC Grant and Duration:**
Advanced Research Fellowship over 60 months from 1.10.1995,
reference B/95/AF/1978

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