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Stable vectors in Moy–Prasad filtrations
- Part of
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- Journal:
- Compositio Mathematica / Volume 153 / Issue 2 / February 2017
- Published online by Cambridge University Press:
- 09 February 2017, pp. 358-372
- Print publication:
- February 2017
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- Article
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Let
$k$ be a finite extension of 
$\mathbb{Q}_{p}$, let 
${\mathcal{G}}$ be an absolutely simple split reductive group over 
$k$, and let 
$K$ be a maximal unramified extension of 
$k$. To each point in the Bruhat–Tits building of 
${\mathcal{G}}_{K}$, Moy and Prasad have attached a filtration of 
${\mathcal{G}}(K)$ by bounded subgroups. In this paper we give necessary and sufficient conditions for the dual of the first Moy–Prasad filtration quotient to contain stable vectors for the action of the reductive quotient. Our work extends earlier results by Reeder and Yu, who gave a classification in the case when 
$p$ is sufficiently large. By passing to a finite unramified extension of 
$k$ if necessary, we obtain new supercuspidal representations of 
${\mathcal{G}}(k)$.
