Quadratic base change for 𝑝-adic 𝑆𝐿(2) as a theta correspondence I: Occurrence

Manderscheid, David

doi:10.1090/S0002-9939-99-04972-2

Quadratic base change for $p$-adic $\mathrm {SL}(2)$ as a theta correspondence I: Occurrence
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Proc. Amer. Math. Soc. 127 (1999), 1281-1288 Request permission

Abstract:

The local theta correspondence is considered for reductive dual pairs $\left ( \mathrm {SL}_{2}\left ( F\right ) ,\mathrm {O}\left ( F\right ) \right )$ where $F$ is a $p$-adic field of characteristic zero and $\mathrm {O}$ is the orthogonal group attached to a quaternary quadratic form with coefficients in $F$ and of Witt rank one over $F$. It is shown that certain representations of $\mathrm {SL}_{2}\left ( F\right )$ occur in the correspondence.

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