For every reductive group over a p-adic field, a character has depth zero if and only if it is trivial on the pro-unipotent radical of every parahoric subgroup, and this agrees with the torus-based definition.
Langlands, ``Representations of abelian algebraic groups", Olga Taussky-Todd: in memoriam, Pacific J
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On depth-zero characters of p-adic groups
For every reductive group over a p-adic field, a character has depth zero if and only if it is trivial on the pro-unipotent radical of every parahoric subgroup, and this agrees with the torus-based definition.