In equation 23 of the following paper https://arxiv.org/pdf/1803.10743.pdf, the Haar integration over the quotient space $G/H$ is defined as follows
$$\int_{G / H} f(x) d x=\int_{G} f(g H) d g$$, where $H$ is a subgroup of $G$. Can anyone explain what does the notation $f(gH)$ where $g \in G$ stand for ? And can anybody justify the above formula ?
General elements of $G/H$ are usually written as left cosets. That is, $G/H=\{ gH : g \in G\}$. Now, you are integrating a measurable function $f: G/H \to \Bbb{C}$ over $G/H$, so it makes sense to talk about $f(gH)$. In fact when you put $f(x)$ you are assuming that $x\in G/H$, so $x$ looks like $gH$ for some $g\in G$. Notice that the map $g\mapsto f(gH)$ is now a measurable function from $G$ to $\Bbb{C}$, and you already have a Haar measure on $G$. That way, the formula you write is just telling you a way to integrate over $G/H$ using the Haar measure on $G$.