Proposition 1

Tag 00SJ, permanent. Cite it as [Bourbaki, Tag 00SJ].

Lie Groups and Lie Algebras, chapter VII, § 1. Primary decomposition of linear representations

For all $\lambda \in P$ ,

$(V\otimes_kk')^{\lambda}(S) = V^{\lambda}(S)\otimes_kk'$ and $(V\otimes_kk')_{\lambda}(S) = V_{\lambda}(S)\otimes_kk'$ .

Let $(a_i)$ be a basis of the $k$ -vector space $k'$ . If $v\in V\otimes_kk',v$ can be expressed uniquely in the form $\sum v_i\otimes a_i$ where $(v_i)$ is a finitely-supported family of elements of V. For all $s\in S$ ,

(r'(s)-\lambda (s))^n(v) =\sum(r(s)-\lambda (s))^nv_i\otimes a_i

It follows that

$v\in (V\otimes_kk')^{\lambda}(S)\Leftarrow \Rightarrow v_i\in V^{\lambda}(S)$ for all $i$ ,

$v\in (V\otimes_kk')_{\lambda}(S)\Leftarrow \Rightarrow v_i\in V_{\lambda}(S)$ for all $i$ ,

which implies the proposition.

Bourbaki sets the proof directly after the statement, and the corpus does not separate them, so what is above is the statement and its proof together.

Cited by

Proposition 4, § 1
Proposition 6, § 1
Proposition 9, § 1
Corollary 2, § 1
Corollary 3, § 1

Cites

Nothing.