Kloosterman sheaves and Bessel functions for generic principal series of finite groups

Let $G$ be a quasi-split reductive group over a finite field and let $\widehat{G}$ be the Langlands dual group. Assuming the derived subgroup of $G$ is almost simple, we use techniques from the geometric Langlands program to relate special values of Bessel functions for generic principal series representations of $G$ to the trace of Frobenius acting on Kloosterman sheaves of Heinloth-Ngô-Yun for $\widehat{G}$. We also give explicit examples in essentially all possible cases, including exceptional groups.