On the action of the Koszul map on the enveloping algebra of the general linear Lie algebra

We describe a linear equivariant isomorphism K from the enveloping algebra UðglðnÞÞ to the algebra C½Mn, n ffi SymðglðnÞÞ of polynomials in the entries of a “generic” square matrix of order n. The isomorphism K maps any Capelli bitableau ½SjT in UðglðnÞÞ to the (determinantal) bitableau ðSjTÞ in C½Mn, n and any Capelli -bitableau ½SjT in UðglðnÞÞ to the (permanental) -bitableau ðSjTÞ in C½Mn, n: These results are far-reaching generalizations of the pioneering result of Koszul on the Capelli determinant in UðglðnÞÞ: We introduce column Capelli bitableaux and -bitableaux in Section 6; since they are mapped by the isomorphism K to monomials in C½Mn, n, this isomorphism can be regarded as a sharpened version of the PBW isomorphism for the enveloping algebra UðglðnÞÞ: Since the center fðnÞ of UðglðnÞÞ equals the subalgebra of invariants UðglðnÞÞAdglðnÞ , then K½fðnÞ ¼ C½Mn, nadglðnÞ :