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# Asian Journal of Mathematics

## Volume 20 (2016)

### Number 5

### Generalized pseudo-coefficients of discrete series of $p$-adic groups

Pages: 969 – 988

DOI: http://dx.doi.org/10.4310/AJM.2016.v20.n5.a7

#### Author

#### Abstract

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.

#### Keywords

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.

Published 22 February 2017