Euler characteristics of the generalized Kloosterman sheaves for symplectic and orthogonal groups

We study the monodromy of certain $\mathrm{\ell }$-adic local systems attached to the generalized Kloosterman sheaves constructed by Yun and calculate their Euler characteristics under standard representations in the cases of symplectic and split/quasi-split orthogonal groups. This provides evidence for the conjectural description of their Swan conductors at $\mathrm{\infty }$ which is predicted by Reeder-Yu on the Langlands parameters attached to the epipelagic representations.