Asian Journal of Mathematics
Volume 20 (2016)
Generalized pseudo-coefficients of discrete series of $p$-adic groups
Pages: 969 – 988
Let $G$ be a connected reductive group over a $p$-adic field $F$ of characteristic $0$ and let $M$ be an $F$-Levi subgroup of $G$. Given a discrete series representation $\sigma$ of $M(F)$, we prove that there exists a locally constant and compactly supported function on M(F), which generalizes a pseudo-coefficient of $\sigma$. This function satisfies similar properties to the pseudo-coefficient, and its lifting to $G(F)$ is applied to the Plancherel formula.
pseudo-coefficient, discrete series, $p$-adic group, trace Paley–Wiener theorem, orbital integral, character function, Plancherel formula
2010 Mathematics Subject Classification
Primary 22E50. Secondary 22E35.