Suppose G is a real reductive algebraic group, θ is an automorphism of G, and ω is a quasicharacter of the group of real points G(R). Under some additional assumptions, the theory of twisted endoscopy associates to this triple real reductive groups H. The Local Langlands Correspondence partitions the admissible representations of H(R) and G(R) into L-packets. We prove twisted character identities between L-packets of H(R) and G(R) comprised of essential discrete series or limits of discrete series.

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Journal Memoirs of the American Mathematical Society
Mezo, P. (2013). Character identities in the twisted endoscopy of real reductive groups. Memoirs of the American Mathematical Society, 222(1042), 1–96. doi:10.1090/S0065-9266-2012-00661-7