Assume that GC is a simply connected complex semi-simple Lie group with Lie algebra [unk]. Let G ⊂ GC be a real form, H ⊂ G be a maximally split Cartan subgroup with Lie algebra [unk]0, [unk] = [unk]0 [unk] C, and P ⊆ [unk]* be the weight lattice. If X is an irreducible Harish-Chandra module with infinitesimal character λ ∈ [unk]*, one can associate to X a family {θ(μ): μ ∈ λ + P} of Z-linear combinations of distribution characters of G, so that θ(λ) = X. θ(μ) is irreducible when μ lies in Cλ, a certain positive “Weyl” chamber containing λ. In this case let Ann θ(μ) be its annihilator in U([unk]) and set p(μ) = Goldie rank of U([unk])/ Ann θ(μ). Let d = Gelfand-Kirillov dimension of X. For most x ∈ [unk]0 if exp tx is regular for small t > 0 then (i) c(μ) = limt→0+td θ(μ) (exp tx) exists for all μ ∈ λ + P; (ii) c(μ) extends to a homogeneous Weyl group harmonic polynomial on [unk]* of degree ½(dim G - dim H) - d; (iii) up to a constant, c = the polynomial extending p to [unk]*. c is said to be the character polynomial of Ann X.
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机译：假设GC是一个具有李代数[unk]的简单连接的复杂半简单Lie群。令G⊂GC为实数形式，H⊂G为李代数[unk] 0，[unk] = [unk] 0 [unk] C和P⊆[unk] * < / sup>是权重格。如果X是一个不可约的Harish-Chandra模块，具有极小字符λ∈[unk] * ，则可以将X线性组合的一个族{θ（μ）：μ∈λ+ P}与X关联。 θ（λ）=X。当μ位于Cλ（包含λ的某个正“ Weyl”腔）中时，θ（μ）不可约。在这种情况下，让Annθ（μ）成为U（[unk]）的an灭者，并设置p（μ）= U（[unk]）/ Annθ（μ）的Goldie秩。令d = X 的Gelfand-Kirillov维。对于大多数 x ∈[unk] 0，如果exp tx 对于小 t i ） c （μ）= lim t →0 + t d θ（μ ）（exp tx ）对于所有μ∈λ+ P 存在; （ ii ） c （μ）扩展到度数为½（dim G的[unk] * 的齐次Weyl群谐波多项式-昏暗的 H ）- d ; （ iii ）直到一个常数， c =将 p 扩展到[unk] * 的多项式。据说 c 是Ann X 的字符多项式。
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