Adjacency for special representations of a Weyl group

0.1. Let G be a connected reductive group over C. Let UG be the set of unipotent conjugacy classes of G. Let C ∈ UG and let C ∈ UG be maximal with the property that C ⊂ C̄ − C (C̄ is the closure of C). A remarkable result of Kraft and Procesi [KP81], [KP82] is that when G is a classical group, then either dim(C) = dim(C) + 2 or the singularity of C at a point of C is the same as the singularity at 1 of a minimal unipotent class in a smaller reductive group. This result has been recently extended to exceptional groups by Fu, Juteau, Levy and Sommers [FJLS]. In the late 1980’s, inspired by [KP81], [KP82], I showed (unpublished) that the results of loc.cit. have a (weak) analogue in the case where UG is replaced by U G (the set of special unipotent classes of G). The analogues in this case of the pairs C,C above can be viewed as edges of a graph with set of vertices U G . One feature that was not present in loc.cit. (except in type A) is that U G has an order reversing involution which preserves the graph structure and that if two edges of the graph are interchanged by this involution then at least one of them is associated to a pair C,C with dim(C) = dim(C)± 2. (see Theorem 0.3). This property allows us to construct the graph above purely in terms of the involution above and truncated induction of Weyl group representations, see Theorem 5.4. (Since U G is naturally in bijection with the set of special representations of the Weyl groupW , we formulate our results in terms ofW . This has the advantage that our results make sense for any finite Coxeter group.)