In physics, I come across this kind of integration (in the nonlinear sigma model): \begin{equation} S[g] = \frac{1}{\lambda} \int d^dr\ \text{tr}[\triangledown g\triangledown g^{-1}] \end{equation} where $\lambda$ is a nonzero parameter and g, defined on a Euclidean space $\mathbb{R}^d$, takes value in a matrix representation of some compact Lie group $G$. Physicists sometimes need to split the matrix-valued field $g(\mathbf{r})$ into a ''fast'' mode component and a ''slow'' mode component, so they write $g(\mathbf{r})=g_s(\mathbf{r})g_f(\mathbf{r})$ where $g_s(\mathbf{r})$ and $g_f(\mathbf{r})$ have frequency components in the ranges $[-\Omega, \Omega]$ and $(-\infty, \Omega)\cap (\Omega,\infty)$, respectively. May I know how a Fourier transform of a matrix-represented compact lie group is defined?
Edit: $g: \mathbb{R}^d \rightarrow G$!! For more mathematical information, you can check the nonlinear sigma model. The ultimate goal is whether we have ''$g(\mathbf{r})=g_s(\mathbf{r})g_f(\mathbf{r})$ where $g_s(\mathbf{r})$ and $g_f(\mathbf{r})$ have frequency components in the ranges $[-\Omega, \Omega]$ and $(-\infty, \Omega)\cap (\Omega,\infty)$, respectively.''
