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If we consider the group of upper triangular matrices $B=\bigl(\begin{smallmatrix} a&b\\ 0&a^{-1} \end{smallmatrix} \bigr)$ where $a$ and $b$ are either real or complex and $a\neq1$, then the left Haar measure is given by $a^{-2}\,da \,db$.

While I understand that this measure is invariant with respect to left translation, I am a little confused as to why the factor of $a^{-2}$ is necessary. Any clarifications would be appreciated, Thank you!

  • 3
    If $a \neq 1$ what's the identity of this group?2012-06-29
  • 1
    $a\ne0$ I presume.2012-06-29

1 Answers 1