Consider the group $GL_n(\mathbb{Q}_p)$ of $n \times n$ invertible matrices over the $p$-adic field $\mathbb{Q}_p$.
My goal is to prove that the subgroup $P_0$ of upper triangular matrices is not unimodular. This is often left as an exercise in scribed lecture notes.
Here is my so-far-unsuccessful attempt: fix a left-Haar measure $\mu$, and normalize it so that $\mu(A)=1$ for $A=P_0 \cap GL_n(\mathcal{O})$ (the subgroup of upper triangular matrices with integral entries). Now the modular function $$\Delta: P_0 \to \mathbb{R}_{>0}$$ is simply $\Delta(x)=\mu(Ax)=\mu(x^{-1}Ax)$. Note that $x^{-1}Ax$ is a conjugate subgroup.
To start with, let us take $n=2$ and $x$ to be a diagonal matrix in $P_0$. In this case, $\mu(Ax)$ is the (left) index of a congruence subgroup of $P_0$: that is, $P_0$ quotiented by a subgroup of matrices whose off-diagonal entry is a fixed power of $p$ (which is determined by the $x$).
But I am unable to proceed with this argument or use it to actually compute the modular character, even for the simple case of $n=2$ and $x$ being diagonal.
So how do we explicitly compute the modular character of the upper triangular subgroup?