Explicit Plancherel Measure for $\mathrm{PGL}_2(\mathbb{F})$

In this paper we compute an explicit Plancherel formula for $\mathrm{PGL}_2(\mathbb{F})$ where $\mathbb{F}$ is a non-archimedean local field by a method developed by Busnell, Henniart and Kutzko. Let $G$ be connected reductive group over a non-archimedean local field $\mathbb{F}$. We show that we can obtain types and covers (as defined by Bushnell and Kutzko in “Smooth representations of reductive $p$-adic groups: structure theory via types” Pure Appl. Math, 2009) for $G/Z$ coming from types and covers of $G$ in a very explicit way. We then compute those types and covers for $\mathrm{GL}_2(\mathbb{F})$ which give rise to all types and covers for $\mathrm{PGL}_2(\mathbb{F})$ that are in the principal series. The Bernstein components $\mathfrak{\overline{s}}$ of $\mathrm{PGL}_2(\mathbb{F})$ that correspond to the principal series are of the form ${[\overline{\mathbb{T}}, \varphi ]}_{\overline{G}}$ where $\overline{\mathbb{T}}$ is the diagonal matrices in $\mathrm{GL}_2(\mathbb{F})$ modulo the center and $\varphi$ is a smooth character of $\overline{\mathbb{T}}$ . Let $\overline{J}, {\overline{\lambda}}_{\varphi}$ be an $\mathfrak{\overline{s}}$-type. Then the Hecke algebra $\mathcal{H}(\overline{G}, {\overline{\lambda}}_{\varphi})$ is a Hilbert algebra and has a measure associated to it called Plancherel measure of $\mathcal{H}(\overline{G}, {\overline{\lambda}}_{\varphi})$. We show that computing the Plancherel measure for $\mathrm{PGL}_2(\mathbb{F})$ essentially reduces to computing the Plancherel measure for $\mathcal{H}(\overline{G}, {\overline{\lambda}}_{\varphi})$. We get that the Hilbert algebras $\mathcal{H}(\overline{G}, {\overline{\lambda}}_{\varphi})$ come in two flavors; they are either $\mathbb{C}[\mathbb{Z}]$ or they are a free algebra in two generators $s_1, s_2$ subject to the relations $s^2_1 = 1$ and $s^2_2 = (q^{-{1/2}} - q^{-{1/2}})s_2 + 1$. We denote this latter algebra by $\mathcal{H}(q, 1)$. The Plancherel measure for $\mathbb{C}[\mathbb{Z}]$ as well as the Plancherel measure for $\mathcal{H}(q, 1)$ are known.

Keywords

Plancherel measure, $\mathrm{PGL}_2$, types and covers

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Published 27 September 2023