I'm looking for a reference concerning a calculation found in Furstenberg's 1963 paper "Non-commuting random products".

Lemma 8.8 of this paper states that if one takes a $d\times d$ invertible real matrix $A$, then the Jacobian of its action on the space $(d-1)$-dimensional subspaces at a subspace $S$ of $\mathbb{R}^d$ is given by

$J_A(S) = |det(A)|^{d-1}/|det_S(A)|^d$,

where $det_S(A)$ is the determinant of $A$ restricted to the subspace $S$ (and one uses the inner-product from the ambient space to define the $(d-1)$-dimensional volume on $S$ and its image under $A$).

For example if $d = 2$ and $S$ is a 1-dimensional subspace of $\mathbb{R}^2$ genereted by a unit vector $v$ then

$J_A(S) = |det(A)|/\|Av\|^2$.

The proof is left to the reader.

While, I am able to reproduce the result, I'm looking for a reference where these types of calculations are treated systematically.

Does anyone know of such a thing?

Thanks in advance!

  • $\begingroup$ It might help to clarify a bit what you mean by "these types of calculations". In what level of generality do you want to work? For $G/P$ a flag manifold of a reductive group $G$, the action of $P$ on the tangent space at a point is equal to its action on the Lie algebra quotient $\mathfrak{g}/\mathfrak{p}$. The Jacobian may be computed as the determinant of this action. Is that the kind of direction you're thinking of going? $\endgroup$ – Will Sawin Dec 28 '18 at 0:55
  • $\begingroup$ @WillSawin Sure. As long as one can in the end get explicit formulas such as the one I give above, I'm not afraid to dig into to a reference at the level of abstraction of your comment. However, as far as I can tell your answer doesn't seem to depend on the particular flag. If this is so it can't be right. $\endgroup$ – Pablo Lessa Dec 28 '18 at 2:22
  • $\begingroup$ The answer I was giving is for the standard flag. For a general flag you can conjugate the matrix by any matrix that translates the flag into standard form. $\endgroup$ – Will Sawin Dec 28 '18 at 13:05
  • $\begingroup$ @WillSawin The image of the flag doesn't matter either? Or do you need to put isometries (surely a general matrix will change the Jacobian, right?) on either side so that the standard flag is fixed? After this, how does one express the determinant of the action on the quotient of lie algebras in terms of geometric characteristics of the transformation at the flag in question? Can you give me a reference to help work these things out? Thanks for the pointers. $\endgroup$ – Pablo Lessa Dec 28 '18 at 16:11
  • 1
    $\begingroup$ A simple formula for the derivatives of a linearly-induced map on projective space: Proposition 3.4 in <arxiv.org/abs/1507.02967> A similar formula should exist for the Grassmannian G(d-1,d). Is this enough for you, or do you also want to consider "intermediate" Grassmannians? Here is some information about their geometry: <doi.org/10.1073/pnas.57.3.589> $\endgroup$ – Jairo Bochi Dec 28 '18 at 20:10

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