# Invariant measure on affine charts of complex Grassmannian

Consider the complex Grassmannian $$U(n)/U(k)×U(n−k)U(n)/U(k)\times U(n-k)$$ with it’s $$U(n)U(n)$$-invariant measure. The affine chart corresponding to $$i1,…,iki_1, \ldots, i_k$$ is given by $$n×kn\times k$$ matrices for which the submatrix given by columns corresponding to $$i1,…,iki_1, \ldots, i_k$$ is the $$k×kk\times k$$ identity matrix $$EkE_k$$.

Consider for example the chart given by $$1,…,k1, \ldots, k$$ which consists of matrices $$(Ik|X),(I_k | X),$$ where $$XX$$ is arbitrary $$k×(n−k)k\times (n-k)$$ complex matrix. The span of it’s rows defines an open subset of $$Gr(n,k)Gr(n, k)$$ and the assignment $$X↦rowspan(Ik|X)X \mapsto \mathrm{rowspan}\, (I_k|X)$$ is bijective.

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