Comparison of the Sally-Shalika character formulas with the endoscopic character identities for ${\mathrm{S}\mathrm{L}}_2$

We consider the depth-zero supercuspidal $L$-packets of ${\mathrm{S}\mathrm{L}}_2(F)$ where $F$ is a non-archimedean local field of characteristic zero. We compare the explicit endoscopic character identities for ${\mathrm{S}\mathrm{L}}_2(F)$ with the classical character formulas of Sally-Shalika. Our main result concerns the supercuspidal $L$-packet of size $4$. For this $L$-packet, we show how the norm $1$ groups $H_1,H_2,H_3$ in the three quadratic extensions of $F$ play a crucial role in the endoscopic character identities for ${\mathrm{S}\mathrm{L}}_2(F)$.