  We investigate Landau-quantized excitonic absorption and luminescence of
monolayer WSe$_2$ under magnetic field. We observe gate-dependent quantum
oscillations in the bright exciton and trions (or exciton-polarons) as well as
the dark trions and their phonon replicas. Our results reveal spin- and
valley-polarized Landau levels (LLs) with filling factors $n = +0, +1$ in the
bottom conduction band and $n = -0$ to $-6$ in the top valence band, including
the Berry-curvature-induced $n = \pm0$ LLs of massive Dirac fermions. The LL
filling produces periodic plateaus in the exciton energy shift accompanied by
sharp oscillations in the exciton absorption width and magnitude. This peculiar
exciton behavior can be simulated by semi-empirical calculations. The
experimentally deduced g-factors of the conduction band (g ~ 2.5) and valence
band (g ~ 15) exceed those predicted in a single-particle model (g = 1.5, 5.5,
respectively). Such g-factor enhancement implies strong many-body interactions
in gated monolayer WSe$_2$. The complex interplay between Landau quantization,
excitonic effects, and many-body interactions makes monolayer WSe$_2$ a
promising platform to explore novel correlated quantum phenomena.
