### CAPELLI IDENTITIES FOR CLASSICAL LIE ALGEBRAS

Let *g* be either the symplectic Lie algebra *sp*_{N}
or the orthogonal Lie algebra *so*_{N}
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 (1^{n} ) 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 **C**^{M} and **C**^{N} ,
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 *gl*_{N} .
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`