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.
Kottwitz, ``Stable trace formula: cuspidal tempered terms", Duke Math
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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.