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Let $G$ be a connected, reductive group, split over a local field $F$. Let $B = TU$ be a Borel subgroup defined over $F$ with maximal torus $T$ and unipotent radical $U$. Let $P$ be a parabolic subgroup of $G$ containing $F$, and let $N = R_u(P)$. For $\alpha \in \Phi(T,U)$, let $U_{\alpha}$ be the root subgroup corresponding to $\alpha$.

Fix an $F$-isomorphism $x_{\alpha}: \mathbb{A}^1\rightarrow U_{\alpha}$ such that $t x_{\alpha}(a)t^{-1} = x_{\alpha}(\alpha(t)a)$ for all $t \in T, a \in \mathbb{A}^1$. Each $x_{\alpha}$ allows us to transfer a Haar measure on $F$ to one on $U_{\alpha}$. Since the product map (in any order)

$$\prod\limits_{\alpha \in \Phi(T,N)} U_{\alpha}(F) \rightarrow N(F)$$

is a homeomorphism, one can transfer the Radon product measure on $\prod\limits U_{\alpha}(F)$ to a Borel measure $\mu$ on $N(F)$. Then $\mu$ is a Haar measure on $N(F)$.

One can verify this directly for parabolic subgroups of split classical groups, but how can one argue that $\mu$ is a Haar measure for arbitrary split groups? More generally in the non-split case, is it possible to define a Haar measure on $N(F)$ using root subgroups for relative roots?

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  • $\begingroup$ Argue by downwards induction on $n$ that it is a Haar measure on the subgroup of $N$ generated by roots of height $n$. The base case, where $n$ is the height of the highest root, follows from the fact that the group in that case is a direct product. I think that the argument is exactly the same in the non-split case, but the root groups, and hence their Haar measures, can be more complicated. (Basically, you have to handle an $\mathrm{SU}_3$ relative root group.) $\endgroup$ – LSpice Feb 14 '17 at 3:34
  • $\begingroup$ The inductive step comes from the fact that the effect of the commutator of ‘lower’ roots is just translation in certain ‘higher’ root groups. $\endgroup$ – LSpice Feb 14 '17 at 3:40
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As in my comment, for each $h > 0$, let $N(F)_{\ge h}$ be the subgroup of $N(F)$ generated by root subgroups corresponding to roots of height at least $h$. If $h$ is at least the height of the highest root, then $N(F)_{\ge h}$ is a direct product of root groups, so the analogue of your measure on $N(F)_{\ge h}$ is obviously a Haar measure. In general, we have that $N(F)_{\ge h}/N(F)_{> h}$ is isomorphic to the direct product of all root groups coming from roots of height $h$, so that the result follows from the equality $$ \int_{N(F)_{\ge h}} f(n_h)\mathrm dn_{\ge h} = \int_{N(F)_{\ge h}/N(F)_{> h}} \int_{N(F)_{> h}} f(n_{= h}n_{> h})\mathrm dn_{> h}\mathrm dn_{= h} $$ (with hopefully obvious notation).

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