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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 $X$?

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I don't have the book handy, but I seem to remember that this formula is written out explicitly in S.-s Chern's Complex manifolds without potential theory (Second Edition), in the chapter "The Grassmann manifold".

Added after finding a copy of the book: Yes, as I expected, the formula the OP seeks is developed on pages 78–81 of the above source. (One may need to read a bit before this to make sure that one understands Chern's notation.)

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  • $\begingroup$ Thank you very much! $\endgroup$
    – Vít Tuček
    Commented Sep 2, 2021 at 17:41

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