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I am reading Bump's book on Automorphic forms and Representations. I don't have a clear understanding of Haar measures and so, I am finding it difficult to do some of the exercises. Can somebody help me in proving that $\prod_{1\leq i < j\leq n}dx_{ij}$ is a Haar measure on $N(F)$, the group of upper triangular unipotent matrices in $GL_n(F)$ (where $F$ is a local field) and that $N(F)$ is unimodular?

Shiva
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  • Shiva, if you don't have a clear understanding of Haar measures, obtain an intuition for the same, and then try doing such questions. For example, it is like one trying to do multivariable calculus without any idea of what integration and/or differentiation actually means. I will try to write an answer that hopefully provides an intuition for Haar measures, but I will in no way attempt to prove what you want to prove. –  Aug 27 '14 at 20:19
  • Won't this follow by induction on $n$ using Bump's Exercise 1.2. from page five? I would do this by using the same technique as in this answer. – Jyrki Lahtonen Aug 27 '14 at 20:21
  • @JyrkiLahtonen I didn't find the exercise that you mentioned. But I was able to solve the problem by induction on $n$. Thank you! – Shiva Aug 28 '14 at 06:54
  • Sorry, Shiva. Wrong book :-) I somehow assumed Bump's book on Lie Groups. My bad. Glad to hear that you solved your problem! – Jyrki Lahtonen Aug 28 '14 at 10:06

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See my comment above.

The Haar measure assigns a notion of volume to a subset of a locally compact topological group. Let $\mu$ be a (left) Haar measure. Then there's an integral $\int_G f(x)d\mu(x)$, where $G$ is a locally compact topological group. Compare this to the definition of the integral of a real-valued function $f$ on a smooth manifold $M$: $\int_Mf\mathrm{d}vol_M$. Do you see the analogy? (Comparing the integral suggests a pretty good way of thinking about Haar measures: $\mu$ can be interpreted as some sort of volume measure on a locally compact topological group.)

I did not see this question.

Community
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