The volume of the complex Stiefel manifold (where $n>p$) $$\mathcal{S}(n,p)=\{\boldsymbol{Q}\in\mathbb{C}^{n\times p}|\boldsymbol{Q}^{\rm H}\boldsymbol{Q}=\boldsymbol{I}\} $$ is given by $$ {\rm Vol}(\mathcal{S}(n,p))=\prod_{k=n-p+1}^n\frac{2\pi^k}{(k-1)!}, $$ which is a standard result.
Now, I am trying to compute the volume of the following manifold $$ \mathcal{S}_{\boldsymbol{\Sigma}}=\{\boldsymbol{Q}\in\mathbb{C}^{n\times p}|\boldsymbol{Q}^{\rm H}\boldsymbol{Q}=\boldsymbol{\Sigma}\}, $$ where $\boldsymbol{\Sigma}\in\mathbb{C}^{p\times p}$ is a diagonal matrix with real positive diagonal entries. I am aware that it is a rescaled version of the Stiefel manifold, which should correspond to a certain metric transformation. But I had a really hard time trying to write that specific transformation down... Any hint will be greatly appreciated!
Many thanks.