Tame supercuspidal representations of 𝐺𝐿_{𝑛} distinguished by orthogonal involutions

Hakim, Jeffrey

doi:10.1090/S1088-4165-2013-00426-0

Tame supercuspidal representations of $\mathrm {GL}_n$ distinguished by orthogonal involutions
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by Jeffrey Hakim

Represent. Theory 17 (2013), 120-175

DOI: https://doi.org/10.1090/S1088-4165-2013-00426-0

Published electronically: March 4, 2013

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Abstract:

For a $p$-adic field $F$ of characteristic zero, the embeddings of a tame supercuspidal representation $\pi$ of $G= \textrm {GL}_n (F)$ in the space of smooth functions on the set of symmetric matrices in $G$ are determined. It is shown that the space of such embeddings is nonzero precisely when $-1$ is in the kernel of $\pi$ and, in this case, this space has dimension four. In addition, the space of $H$-invariant linear forms on the space of $\pi$ is determined whenever $H$ is an orthogonal group in $n$ variables contained in $G$.

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