Characteristic Classes and Primitive Ideals

In this chapter, G is again a semisimple group with Lie algebra g, and we use the notations introduced in 1.5. As in chapter 3 and 4, we denote by Λ the lattice in t* of integral weights (3.1) of our maximal torus T ⊂ G, by Ω ⊂ Λ the “dominant integral weights” (3.16) with respect to the ordering fixed by our choice of a Borel subgroup B ⊃ T, and by p ∈ Ω half the sum of weights in b. We furthermore denote by U(g) the enveloping algebra of g, that is the ring of differential operators on G invariant under right translations. Our purpose is to study g—modules, that is to say U(g)—modules. In particular, we are interested in the annihilators in U(g) of simple g—modules, called primitive ideals. We denote by L(λ) the simple g-module of highest weight λ (which is defined as the unique simple quotient of the universal (or Verma-) module M(λ) = U(g) ⨂ _{U (b)}k_λ, where k_λ is a one—dimensional b—module of weight λ). Then the center of U(g) is a polynomial ring in dim (T) variables (Harish—Chandra, Chevalley); this center acts by a character on L(λ) which is denoted x_λ; we note that by Harish—Chandra’s theorem, x _λ =x _µ if and only if µ = w.λ, for some Weyl group element w ∈ W, where the “shifted Weyl group action” w.λ:= w(λ+ρ)-ρ is used. Finally, it will be convenient to identify g* with g, and t* with t by means of the Killing form. We apply analogous notations to the group G×G, so for instance (λ,/µ) ∈ t* × t* defines a central character x_(λ,µ) of U(g × g) etc..