Proposition 14

Tag 09G8, permanent. Cite it as [Bourbaki, Tag 09G8].

Integration, chapter VII, § 2. Quotient of a space by a group; homogeneous spaces

— Suppose that the conditions of Prop. 13 are fulfilled and that, in addition, Y is normal.

a) The restriction of $\mu_G$ to $\Omega$ is, up to a constant factor, the image of $\mu_X \otimes \mu_Y$ under the mapping $(x, y) \mapsto xy$ of $X \times Y$ onto $\Omega$ .

b) For $x \in X$ and $y \in Y$ ,

\Delta_G(xy) = \Delta_X(x)\Delta_Y(y) \operatorname{mod}(i_x),

where $i_x$ denotes the automorphism $v \mapsto x^{-1}vx$ of Y.

We have $\Delta_G = \Delta_Y$ on Y (Prop. 10 b)), therefore a) follows from (23). Let $x_0 \in X,\ y_0 \in Y$ . Denote by $p$ the mapping $(x, y) \mapsto xy$ of $X \times Y$ onto $\Omega$ . Since

xy(x_0y_0)^{-1} = xx_0^{-1}(x_0yy_0^{-1}x_0^{-1}) = xx_0^{-1}i_{x_0^{-1}}(yy_0^{-1}),

we have

\begin{align*} \Delta_G(x_0y_0)p(\mu_X \otimes \mu_Y) &= \delta(x_0y_0)p(\mu_X \otimes \mu_Y) \\ &= p(\delta(x_0)\mu_X \otimes i_{x_0^{-1}}\delta(y_0)\mu_Y) \\ &= p(\Delta_X(x_0)\mu_X \otimes \Delta_Y(y_0)(\operatorname{mod}\ i_{x_0})\mu_Y) \\ &= \Delta_X(x_0)\Delta_Y(y_0)(\operatorname{mod}\ i_{x_0})p(\mu_X \otimes \mu_Y), \end{align*}

whence b).

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

Proposition 14

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