Invariant measure on affine charts of complex Grassmannian

Consider the complex Grassmannian U(n)/U(k)×U(nk) with it’s U(n)-invariant measure. The affine chart corresponding to i1,,ik is given by n×k matrices for which the submatrix given by columns corresponding to i1,,ik is the k×k identity matrix Ek.

Consider for example the chart given by 1,,k which consists of matrices (Ik|X), where X is arbitrary k×(nk) complex matrix. The span of it’s rows defines an open subset of Gr(n,k) and the assignment Xrowspan(Ik|X) is bijective.

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

Answer

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.)

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

Leave a Comment