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For simplicity, let $G_n=GL(n,\mathbb{R})$, $N_n$ be the upper trianguler unipotent subgroup, $P_{n-i}$ be the standard parabolic subgroup associated to partition $n=(n-i,i)$, and finally let $K=O(n)$ be the maximal compact subgroup.

Consider the left coset space $N_n\backslash G_n$ with a right invariant Haar measure, denoted as $dg$ on it. Decompose $N_n\backslash G_n=N_n\backslash P_{n-1} \bullet P_{n-1}\backslash G_n$, correspondingly there is a decomposition of $dg$ in terms of right Haar measures $dp$ on $ N_n\backslash P_{n-1}$ and $dh$ on $P_{n-1}\backslash G_n$, if we write $g=ph$ with $p\in P_{n-1}$ and $h\in P_{n-1}\backslash G_n$

$$\int_{N_n\backslash G_n}f(g)dg=\int_{P_{n-1}\backslash G_n}\int_{N_n\backslash P_{n-1}}f(ph)|det p|^{-1}dpdh$$

The first question I have is that why do we have the factor $|det p|^{-1}$ on the right side? Don't we just have a formula like $$\int_{N_n\backslash G_n}f(g)dg=\int_{P_{n-1}\backslash G_n}\int_{N_n\backslash P_{n-1}}f(ph)dpdh \ \ \ \ ?$$

And what's the formular if we use decomposition $G_n=P_{n-1}K$?

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1 Answer 1

You've performed a change of variables; the extra factor is analogous to the Jacobian determinant. The extra factor is $\Delta_1(h)/\Delta_2(ph)$ where $\Delta_1,\Delta_2$ are the modular functions for $N_n\backslash P_{n-1}$ and $P_{n-1}\backslash G_n$, respectively. See Theorem 8.32 in Lie groups beyond an introduction by Anthony Knapp.

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thanks for your answer,but I'm still a little confused. First what's the modular function of $N_n\backslash P_{n-1}$? Note that this quotient is isomorphic to $N_{n-1}\backslash G_{n-1}\times G_1$. Secondly, what's the modular function on $P_{n-1}\backslash G_n$? Note that this homogeneous space doesn't admit a $G$ invariant measure. The final question is that in your mentioned theorem 8.32, it is required the intersection of $S,T$ is compact, which is not satisfied in our case. –  liegroupstu Jun 24 '13 at 13:49

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