Bourbaki

Exercise 15, § 7

Tag 014R, permanent. Cite it as [Bourbaki, Tag 014R].

Lie Groups and Lie Algebras, VIII, 242

Let λ∈h∗\lambda \in \mathfrak{h}^* and let vv (resp. v′v') be a primitive element of Z(λ)Z(\lambda ) (resp. of E(λ)E(\lambda )) of weight λ\lambda. Let I (resp. I′I') be the annihilator of vv (resp. v′v') in U(g)U(\mathfrak{g}). a)I=U(g)n++∑h∈hU(g)(h−λ(h))a) I = U(\mathfrak{g})\mathfrak{n}_++\sum_{h\in\mathfrak{h}}U(\mathfrak{g})(h-\lambda (h)).

b)I′b) I' is the largest left ideal of U(g)U(\mathfrak{g}) distinct from U(g)U(\mathfrak{g}) and containing I.

c) If λ∈P++\lambda \in P_{++}, then

I′=I+∑α∈BU(g)X−αλ(Hα)+1=I+∑α∈BU(n−)X−αλ(Hα)+1 I'= I +\sum_{\alpha\in B}U(\mathfrak{g})X_{-\alpha}^{\lambda(H_{\alpha})+1}= I +\sum_{\alpha\in B}U(\mathfrak{n}_-)X_{-\alpha}^{\lambda(H_{\alpha})+1}

(Use the preceding exercise.)

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Transcribed from the printed volume. Not translated.