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

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