In the paper "On the definition of transfer factors" Langlands and Shelstad define a certain number of factors $\Delta_{I}$, $\Delta_{II}$,$\Delta_{III,1}$,$\Delta_{III,2}$, which are roots of unity.
Let consider the easy case where $G=SL_{2}$ over a field $F$ and as a endoscopic group I take $H$ a one dimensional anisotropic torus splitted by a quadratic extension $E/F$.
My question concerns the computation of the transfer factor $\Delta_{III,2}$ in this particular case.
First, Langlands talk about a pairing between $T(F)$ and $H^{1}(W,\hat{T})$, where $\hat{T}$ is the dual torus, how can we compute explicitely this pairing?
Second, if I choose a $\chi$-data fix admissible embeddings $j:\hat{H}\rightarrow\hat{G}$ and $\gamma_{H}\in H(F)$ $G$-regular semisimple, I can define a cocycle $a\in H^{1}(W,\hat{T})$ (LS, sect. 3.4 p44)
and $\Delta_{III,2}$ is the pairing of $a$ and $\gamma:=j(\gamma_{H})$.
Can we compute it explicitely?
