Definition 3

Tag 0A9U, permanent. Cite it as [Bourbaki, Tag 0A9U].

Lie Groups and Lie Algebras, chapter III, § 3. Passage from a Lie group to its Lie algebra

Let $t \in \mathscr{T}^{(s)}(G)$ with $s \leq r$ . Let $f : X \to F$ be a function of class C^r with values in a Hausdorff polynormed space (for example $F = \mathbf{K}$ ). The convolution of t and f, denoted by $t * f$ , is the function on X with values in F defined by

(t * f)(x) = \langle t^\vee * \varepsilon_x, f \rangle.

Then

\begin{align*} (t * f)(x) &= \langle \varrho(x)_*(t^\vee), f \rangle \quad \text{(no. 3, Proposition 14 (ii))} \\ &= \langle t^\vee, f \circ \varrho(x) \rangle \quad \text{(Diff. \& Anal. Man., R, 13.2.3)} \\ &= \langle t, (f \circ \varrho(x))^\vee \rangle \quad \text{(Diff. \& Anal. Man., R, 13.2.3)}. \end{align*}

Note that Definition 3 can also be written in the more symmetric form

\langle \varepsilon_x, t * f \rangle = \langle t^\vee * \varepsilon_x, f \rangle.

The function

(g, x) \mapsto f(gx) = (f \circ \varrho(x))(g)

G \times X

is of class C^r. By Differentiable and Analytic Manifolds, R, 13.4.4, the function

x \mapsto \langle t^\vee, f \circ \varrho(x) \rangle

is therefore of class

C^{r-s}

s < \infty

. In other words, if

s < \infty

t * f

is of class

C^{r-s}

Clearly $t * f$ depends linearly on $t$ and $f$ .

Formula (4) implies in particular, for $g \in G$ ,

(\varepsilon_g * f)(x) = f(g^{-1}x)

that is

\varepsilon_g * f = \gamma(g)f.

Suppose that $K = \mathbf{R}$ or $\mathbf{C}$ , that $G$ and $X$ are finite-dimensional and that $X$ has a positive measure invariant under $G$ . The definition of $\varepsilon_g * f$ agrees with that of Integration, Chapter VIII, § 4, no. 1 (cf. formula (2), loc. cit.).

Bourbaki puts the proof directly after the statement, so what is above is the statement and its proof together.

Cited by

Nothing.

Cites

Integration, Chapter VIII, § 4, no. 1 not yet on this site: INT VIII
no. 3, Proposition 14