Invariant measure on affine charts of complex Grassmannian

Consider the complex Grassmannian $U(n)/U(k)\times U(n-k)$ with it’s $U(n)$-invariant measure. The affine chart corresponding to $i_1, \ldots, i_k$ is given by $n\times k$ matrices for which the submatrix given by columns corresponding to $i_1, \ldots, i_k$ is the $k\times k$ identity matrix $E_k$.

Consider for example the chart given by $1, \ldots, k$ which consists of matrices $(I_k | X),$ where $X$ is arbitrary $k\times (n-k)$ complex matrix. The span of it’s rows defines an open subset of $Gr(n, k)$ and the assignment $X \mapsto \mathrm{rowspan}\, (I_k|X)$ is bijective.

What is the formula for the restriction of the invariant measure in terms of entries of $X$?

Answer

Attribution
Source : Link , Question Author : Vít Tuček , Answer Author : Robert Bryant