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.
Germ expansion for SL(2) in arbitrary characteristics
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abstract
Let $F$ be a local field of characteristic $p$ and $G$ be a connected reductive group over $F$. Recall that Shalika's germ expansion of orbital integrals of regular semi-simple elements near the identity, when it exists, is a sum indexed by the set of unipotent conjugacy classes in $G(F)$. Observe that if $G=SL(2)$ this set is always compact; it is finite if $p\ne2$ while it is uncountable if $p= 2$. As a consequence, Shalika's germ expansion for elliptic elements does not make sense if $p=2$. On the other hand the endoscopic expansion of elliptic orbital integrals always exists and yields a germ expansion equivalent if $p\ne2$ (up to a Fourier transform) to Shalika's germ expansion but is new if $p=2$. A conjecture for arbitrary groups is stated.
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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.