Let $G$ be a reductive group over a $p$-adic field. My understanding is that the Plancherel measure on $G$ is a measure on the unitary dual $\hat{G}$.


But at the same time, for example, in his famous 1990 Annals paper, Shahidi defines a Plancherel measure using induced representations and intertwining operators.


Are those two notions of Plancherel measure related? Or are they a mere coincidence of terminology?



A question on Plancherel measure for $p$-adic group
They are related but not the same. The Pleancherel measure is strictly speaking a measure on $\hat{G}$. If one were to parametrize $\hat{G}$, say by an integral on the real line, then one could get the Plancherel measure to be given by density functions. In other words, there exists some function $\mu$ such that the Plancherel measure of a set $A \subset \hat{G}$ is $$\int\_{A}\mu(x)dx,$$
where $dx$ is the Lebesgue measure. Now for connected reductive groups over a local field Harish-Chandra proved this density function comes from intertwining operators of induced *tempered* representations after tensoring by some unramified character. These unramified characters can be parametrized by the real numbers, and this way you will get a subset of $\hat{G}$. Shahidi goes further and defines "Plancherel measure" not only for tempered representations but for unitary ones, which are parametrized by various copies of the complex numbers. Shahidi's density functions are from $\mathbb{C}^n$ into $\mathbb{C}$, and not necessarily functions from the reals into the positive reals like Harish-Chandra's.


