I'm looking to define an appropriate measure for an integration involving matrices from $GL^+(2,\mathbb{R})$. Specifically, I have the following integral:
$$\int_M \left( \int_{GL^+(2,\mathbb{R})} \det \begin{bmatrix} a(x,q) & b(x,q) \\ c(x,q) & d(x,q) \end{bmatrix} \right) Dq \, dx $$
Where $M$ is some manifold and $q$ parametrizes the elements of $GL^+(2,\mathbb{R})$.
I'm uncertain about how to define the measure $Dq$ for the integration over $GL^+(2,\mathbb{R})$. Given that $q$ parametrizes the matrices and the integrand involves matrix entries, what would be an appropriate definition for $Dq$?
Is there a standard or recommended approach to define such a measure over the space of $GL^+(2,\mathbb{R})$ matrices?
Any insights or references are greatly appreciated.
The natural volume element comes from taking the wedge product of the left-invariant $1$-forms on the Lie group. These $1$-forms are the entries of the Maurer-Cartan matrix $1$-form $g^{-1}dg$. If you write the general element of $GL(2,\Bbb R)$ as $$g=\begin{bmatrix} x & y \\ z & w\end{bmatrix},$$ then \begin{align*} g^{-1}dg &= \frac1{xw-yz}\begin{bmatrix} w & -y \\ -z & x\end{bmatrix}\begin{bmatrix} dx & dy \\ dz & dw\end{bmatrix} \\ &= \frac1{xw-yz}\begin{bmatrix} w\,dx-y\,dz & w\,dy-y\,dw \\ -z\,dx+x\,dz & -z\,dy+x\,dw\end{bmatrix}. \end{align*} The left-invariant volume form then is (up to sign) \begin{align*} dV &= \frac1{(xw-yz)^4}\left|\begin{matrix} w &0&-y&0 \\ 0&w&0&-y \\ -z&0&x&0\\0&-z&0&x\end{matrix}\right|dx\wedge dy\wedge dz\wedge dw \\ &= \frac1{(xw-yz)^2}dx\wedge dy\wedge dz\wedge dw \\ &= \det(g)^{-2} dx\wedge dy\wedge dz\wedge dw. \end{align*} I don't know that you'll need this, but you can in fact check that this group is unimodular; that is, that volume form is right-invariant as well.