For the special linear group G we describe a system of positive roots associated to conjugacy classes in its Weyl group. Using this we can explicitly describe the algebra of regular functions on certain transverse slices to conjugacy classes in G as a polynomial algebra of invariants. These may be viewed as an algebraic group analogue of certain parabolic invariants that generate the W-algebra in type A described by J. Brundan and A. Kleshchev.