Let $G$ be a topological group, and let $\mu_l, \mu_r$ be any left (right) Haar measure on $G$.
I have seen the modular character $\Delta: G \rightarrow (0,\infty)$ defined as the unique function such that $\mu_l(Ex) = \Delta(x) \mu_l(E)$ for all Borel sets $E$, equivalently such that $\mu_r(x^{-1}E) = \Delta(x)\mu_r(E)$.
I have also seen it defined as the unique function $\delta: G \rightarrow (0,\infty)$ such that $\mu_l(Ex^{-1}) = \delta(x)\mu_l(E)$, equivalently $\mu_r(xE) = \delta(x) \mu_r(E)$.
Examples: Induced Representations of Locally Compact Groups by Kaniuth and Taylor use the first definition. Introduction to Harmonic Analysis on Reductive $p$-adic Groups by Allen Silberger uses the second definition.
The Haar Integral by Nachbin distinguishes between them, calling the first one the "right-hand modulus" and the second one the "left-hand modulus."
Clearly $\Delta(x) = \delta(x)^{-1}$. So, which function do we want to call the "conventional" modular character? $\Delta$, or $\delta$? I keep reading papers and notes in representation theory which make use of the modular character and sometimes it's impossible to tell which one they mean. It's driving me nuts.
Is one of these definitions much more popular than the other? Is one definition more common when we are using the right Haar measure? If possible, I would like to be able to read a paper and immediately know what they mean by modular character.