High-energy or relativistic electron beam acceleration along and across a magnetic field, and the generation of an electric field transverse to the magnetic field, both induced by the Compton scattering of almost perpendicularly propagating extraordinary waves, are investigated theoretically based on kinetic wave equations and transport equations. Compton scattering occurs via nonlinear Landau damping of two extraordinary waves interacting nonlinearly with the electron beam, satisfying the resonance condition of $\omega _{{\bf k}} -\omega _{{\bf k}'} - (k_{ \bot } - k'_{\bot})\nu_{\rm d} - (k_{\|} - k'_{\|})\nu_{\rm b}\,{=}\,m\omega _{\rm ce}$$(m\,{=}\,0, \pm 1)$, where $\nu_{\rm b}$ and $\nu_{\rm d}$ are the parallel and perpendicular velocities of the electron beam, respectively. The transport equations can be derived from the single-particle theory and also from Vlasov–Maxwell equations. The transport equations show that two extraordinary waves accelerate the electron beam in the ${\bf k}''$ direction (${\bf k}''\,{=}\,{\bf k} \,{-}\,{\bf k}'$). Simultaneously, an intense cross-field electric field ${\bf E}_{0}\,{=}\,{\bf B}_{0}\,{\times}\,{ \bf v}_{\rm d}/c$ is generated via the dynamo effect owing to the perpendicular drift of the electron beam to satisfy the generalized Ohm's law, which means that this cross-field electron drift is identical to the ${\bf E}\,{\times}\, {\bf B}$ drift. The single-particle theory is very useful for an easy and straightforward understanding of the physical mechanism of the electron beam acceleration and the generation of cross-field electric field, although the rigorously exact transport equations are derived from Vlasov–Maxwell equations.