  We study the Schur index, the Zhu's $C_2$ algebra, and the Macdonald index of
a four dimensional $\mathcal{N}=2$ Argyres-Douglas (AD) theories from the
structure of the associated two dimensional $W$-algebra. The Schur index is
derived from the vacuum character of the corresponding $W$-algebra and can be
rewritten in a very simple form, which can be easily used to verify properties
like level-rank dualities, collapsing levels, and S-duality conjectures. The
Zhu's $C_2$ algebra can be regarded as a ring associated with the Schur sector,
and a surprising connection between certain Zhu's $C_2$ algebra and the Jacobi
algebra of a hypersurface singularity is discovered. Finally, the Macdonald
index is computed from the Kazhdan filtration of the $W$-algebra.
