Whittaker category for the Lie algebra of polynomial vector fields

Abstract

For any positive integer n, let A n = C[t 1 , . . . , t n ],a of the category of (A n , W n )-Whittaker modules with finite dimensional Whittaker vector spaces is equivalent to the category of finite dimensional modules over L n , where L n is the Lie subalgebra of W n consisting of vector fields vanishing at the origin. As a corollary, we classify all simple non-singular Whittaker W nmodules with finite dimensional Whittaker vector spaces using gl nmodules. We also obtain an analogue of Skryabin's equivalence for the non-singular block Ω W a .