QUASIFINITE HIGHEST WEIGHT MODULES OVER THE LIE-ALGEBRA OF MATRIX DIFFERENTIAL-OPERATORS ON THE CIRCLE

We give a complete description of the quasifinite highest weight modul es over the central extension of the Lie algebra of M X M-matrix diffe rential operators on the circle and obtain them in terms of representa tion theory of the Lie algebra (g) over cap l((infinity),R-m) of infin ite matrices with only finitely many nonzero diagonals over the algebr a R-m = C[t]/(t(m+1)). We also classify the unitary ones, and construc t them in terms of charged free fermions. This construction provides a large (and conjecturally complete) family of irreducible modules over the associated vertex algebra W-1+infinity,c(M), where c is a positiv e integer. (C) 1998 American Institute of Physics.