For residual characteristic 2, each depth-zero supercuspidal representation of SL(2,F) restricts to a maximal compact subgroup as a direct sum of explicitly constructed representations I(1,u,ℓ), indexed by square classes, with counts that grow without bound when char(F)=2.
21, Cambridge University Press, Cambridge, 1991
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Branching rules for irreducible depth-zero supercuspidal representations of $\mathrm{SL}(2,F)$, when $F$ has residual characteristic $2$
For residual characteristic 2, each depth-zero supercuspidal representation of SL(2,F) restricts to a maximal compact subgroup as a direct sum of explicitly constructed representations I(1,u,ℓ), indexed by square classes, with counts that grow without bound when char(F)=2.