Endoscopic character relations are identities between characters of representations of different groups — a reductive group G and its endoscopic groups H. These relations are central to the Langlands program because they connect the automorphic spectrum of G to the simpler spectra of the endoscopic groups.
For toral supercuspidal representations — the most regular, best-behaved representations, constructed from characters of maximal tori — the twisted endoscopic character relation has an explicit form. The character of the L-packet on G equals a signed sum of characters on the endoscopic groups, with signs determined by the root system and the toral data.
The proof for classical groups uses the Kaletha-Schwein framework, which constructs supercuspidal L-packets with explicit internal parameterization. The twisted endoscopy adds a complication: the character relation must be compatible with an outer automorphism of G that permutes the L-packets. The compatibility is achieved by tracking how the automorphism acts on the toral data.
The verification is computational at its core: both sides of the character relation are expressed in terms of Gauss sums and root-system combinatorics, and the identity is verified by matching the combinatorial expressions term by term.
An identity between characters of different groups, verified through number-theoretic sums. The Langlands program's prediction — that automorphic spectra of different groups are related through endoscopy — confirmed for one more class of representations.