The local Langlands conjecture for GSp(4)GSp(4) III: Stability and twisted endoscopy

In this paper, we complete the analysis of the local Langlands correspondence for GSp(4)GSp(4) and its inner form over a non-archimedean local field of characteristic 0, which was shown in two earlier papers of the second author with S. Takeda and W. Tantono. We show that the L-packets satisfy the local character identities postulated in the theory of (twisted) endoscopy.