Proposition 5

Assume that the group H is compact and denote by $dh$ the Haar measure on H of total mass 1. Let $s$ be a section of E of class $C^r$. Denote by $s^{\sharp}$ the vector integral $\int_H^hs dh$. Then $s^{\sharp}$ is a section of E of class $C^r$, invariant under H; for $x\in X$, we have $s^{\sharp}(x) =\int_Hhs(h^{-1}x)dh\in E_x$. The endomorphism $s \rightarrow s^{\sharp}$ of $\mathscr{S}^r(X; E)$ is a projection onto the subspace of H-invariant sections.

Consider the map $h \rightarrow^hs$ from H to $\mathscr{S}^r(X; E)$; it is continuous by Lemma 4. Since the space $\mathscr{S}^r(X; E)$ is separated and complete, the integral $s^{\sharp}=$ $\int_H^hs dh$ belongs to $\mathscr{S}^r(X; E) ($Integration, Chap. III, §3, no. 3, Cor. 2). The linear map $s \rightarrow s(x)$ from $\mathscr{S}^r(X; E)$ to $E_x$ being continuous, we have $s^{\sharp}(x) =$ $\int_H^hs(x)dh$ for all $x\in X$. It is clear that $s^{\sharp}$ is invariant under H; if $s$ is an H-invariant section, we have $s^{\sharp}=s$, hence the last assertion.