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.

Abstract. We prepare for a comparison of global trace formulas of general linear groups and their metaplectic coverings. In particular, we generalize the local metaplectic correspondence of Flicker and Kazhdan and describe the terms expected to appear in the invariant trace formulas of the above covering groups. The conjectural trace formulas are then placed into a form suitable for comparison.

To each Levi subgroup of a general linear group there corresponds a set of general linear groups of smaller order. One may therefore construct an irreducible representation of such a Levi subgroup by taking the tensor product of irreducible representations of the smaller general linear groups. We generalize this construction to the context of metaplectic coverings over a p-adic field.

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The standard theory of endoscopy for real groups has two parallel formulations. The original formulation of Langlands and Shelstad relies on methods in harmonic analysis. The subsequent formulation of Adams, Barbasch and Vogan relies on sheaf-theoretic methods. The original formulation was extended by Kottwitz and Shelstad to twisted endoscopy. We extend the sheaf-theoretic formulation to the context of twisted endoscopy and provide applications for computing Arthur packets.