Corollary 3

Let H be a connected closed subgroup of G containing T, $\mathfrak{h}$ its Lie algebra, and $dh$ the Haar measure on H of total mass 1. Let $\varphi$ be an integrable central function on G, with values in a Banach space or in $\overline{\mathbf{R}}$. Then the function $h \rightarrow \varphi (h$)det(Ad$_{\mathfrak{g}/\mathfrak{h}}(h)-1$) is integrable and central on H and we have

$\int_G\varphi (g)dg=\frac{w(H)}{w(G)}\int_H\varphi (h$)det(Ad$_{\mathfrak{g}/\mathfrak{h}}(h)-1$)$dh$. (7)

Indeed, the function $h \rightarrow \varphi (h$)det(Ad$_{\mathfrak{g}/\mathfrak{h}}(h)-1$) is a central function on H whose restriction to T is the function $t \rightarrow \varphi (t)\delta_G(t)\delta_H(t)^{-1}$. Thus, the corollary follows from Cor. 2 applied to G and to H.