Dimensions of Quantized Tilting Modules

We will follow the notations of [7]. Let (Y,X, . . . ) be a simply connected root datum of finite type. Let p be a prime number bigger than the Coxeter number h. Let ζ be a primitive p-th root of unity in C. Let U be the quantum group with divided powers associated to these data. Let T be the category of tilting modules over U , see e.g. [1]. Recall that any tilting module is a sum of indecomposable ones, and indecomposable tilting modules are classified by their highest weights, see loc.cit. Let X+ be the set of dominant weights, and for any λ ∈ X+ let T (λ) denote the indecomposable tilting module with highest weight λ. The tensor product of tilting modules is again a tilting module. Let us introduce the following preorder relation ≤T on X+: λ ≤T μ iff T (λ) is a direct summand of T (μ)⊗(some tilting module). We say that λ ∼T μ if λ ≤T μ and μ ≤T λ. Obviously, ∼T is an equivalence relation on X+. The equivalence classes are called weight cells. The set of weight cells has a natural order induced by ≤T . It was shown in [8] that the partially ordered set of weight cells is isomorphic to the partially ordered set of two-sided cells in the affine Weyl group W associated with (Y,X, . . . ) (W is a semidirect product of the finite Weyl group Wf with the dilated coroot lattice pY ). Let G and g be the simply connected algebraic group and the Lie algebra (both over C) associated to (Y,X, . . . ), and let N be the nilpotent cone in g, i.e. the variety of ad−nilpotent elements. It is well known that N is a union of finitely many G−orbits called nilpotent orbits. Using the theory of support varieties one defines Humphreys map H : { the set of weight cells} → { the set of closed G−invariant subsets of N}, see [9]. The Conjecture due to J.Humphreys says that the image of this map consists of irreducible varieties, i.e. the closures of nilpotent orbits; moreover J.Humphreys conjectured that this map coincides with Lusztig’s bijection between the set of two-sided cells in the affine Weyl group and the set of nilpotent orbits, see [2] and [9]. In particular, Humphreys map should preserve Lusztig’s a−function; this function is equal to half of codimension in N of the nilpotent orbit and is defined purely combinatorially on the set of two-sided cells, see [4]. The aim of this note is to show that the Humphreys map does not decrease the a−function: for a weight cell A corresponding to a two-sided cell A in W we have the inequality codimNH(A) ≥ a(A). This inequality follows easily from the definition of H, Theorem 4.1 in [9] and the Main Theorem below: