By computing reducibility points of parabolically induced representations, we construct, to within at most two unramified quadratic characters, the Langlands parameter of an arbitrary depth zero irreducible cuspidal representation π of a classical group (which may be not-quasisplit) over a non-archimedean local field of odd residual characteristic. From this, we can explicitly describe all the irreducible cuspidal representations in the union of one, two, or four L-packets, containing π. These results generalize the work of DeBacker-Reeder (in the case of classical groups) from regular to arbitrary tame Langlands parameters.

We show that for any tame regular discrete series parameter of GSp4 or its inner form GU2(D), the L-packet attached by the local Langlands conjecture [GT], [GTan] agrees with the L-packet of depth zero supercuspidal representations constructed by DeBacker and Reeder [DR].

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