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Put $B_p := \left\{ \begin{pmatrix} a & b \\ 0 & c \end{pmatrix} \in GL_2(Q_p) : a, b, c \in Q_p \right\}$ the subgroup of upper triangular matrices in $GL_2(Q_p)$, $Q_p$ denoting the $p$-adic rationals. I have already figured out that the modularity function is

$\Delta \begin{pmatrix} a & b \\ 0 & c \end{pmatrix} = (|a|/|c|)^\lambda$ i.e. if $\mu$ is the Haar measure on $B_p$ and $M$ is a measurable set then for any $x \in B_p$, $\mu(Mx)=\Delta(x)\mu(M)$

Does anybody know how to figure out that $\lambda=1$?

Cheers,

Fabian Werner

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    So, let us set $B_N := \{ x \in B_p: |det(x)| \geq p^{-N}\}$, and let $\nu$ be the restriction of the measure on $GL_2$ then we see that $B_p = \cup_{N \in \mathbb{N}} B_N$ and consequently $\nu(B_N) = \int_{GL_2(Q_p)} 1_{B_N} dx/|det(x)| \leq const \int_{GL_2(Q_p)} 1_{B_N} dx = 0$ by the last comment. $|\cdot|$ denotes the p-adic norm.2012-12-18

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