On Splitting Invariants and Sign Conventions in Endoscopic Transfer

The transfer factors for standard endoscopy involve, among other things, the Langlands-Shelstad splitting invariant. This note introduces a twisted version of that splitting invariant. The twisted splitting invariant is then used to define a better twisted factor ${\mathrm{\Delta }}_I$. In addition we correct a sign error in the definition of twisted transfers. There are two ways to correct the sign error. One way yields twisted transfer factors ${\mathrm{\Delta }}^{\mathrm{\prime }}$ that are compatible with the classical Langlands correspondence. The other way yields twisted transfer factors ${\mathrm{\Delta }}_D$ that are compatible with a renormalized version of the Langlands correspondence.