Super Duality and Homology of Unitarizable Modules of Lie Algebras

Abstract

The $\mathfrak{u}$-homology formulas for unitarizable modules at negative levels over classical Lie algebras of infinite rank of types $\mathfrak{g}l(n)$, $\mathfrak{s}p(2n)$ and $\mathfrak{s}o(2n)$ are obtained. As a consequence, we recover the Enright's formulas for three Hermitian symmetric pairs of classical types $(SU(p,q),SU(p)\times SU(q))$, $(Sp(2n),U(n))$ and $(S{O}^{\ast }(2n),U(n))$.