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The character formulas of Sally and Shalika are an early triumph in $p$-adicharmonic analysis, but, to date, the calculations underlying the formulas havenot been available. In this paper, which should be viewed as a precursor of theforthcoming volume by the authors and Alan Roche, we leverage modern technology(for example, the Moy-Prasad theory) to compute explicit character tables. Aninteresting highlight is the computation of the 'exceptional' supercuspidalcharacters, i.e., those depth-zero representations not arising byinflation-induction from a Deligne-Lusztig representation of finite$\operatorname{SL}_2$; this provides a concrete application for the recent workof DeBacker and Kazhdan.

Supercuspidal characters of $\operatorname{SL}_2$ over a $p$-adic field