  We study the orbifold Hirzebruch-Riemann-Roch (HRR) theorem for quotient
Deligne-Mumford stacks, explore its relation with the representation theory of
finite groups, and derive a new orbifold HRR formula via an orbifold Mukai
pairing. As a first application, we use this formula to compute the dimensions
of $ G $-equivariant moduli spaces of stable sheaves on a $ K3 $ surface $ X $
under the action of a finite subgroup $ G $ of its symplectic automorphism
group. We then apply the orbifold HRR formula to reproduce the number of fixed
points on $ X $ when $ G $ is cyclic without using the Lefschetz fixed point
formula. We prove that under some mild conditions, equivariant moduli spaces of
stable sheaves on $ X $ are irreducible symplectic manifolds deformation
equivalent to Hilbert schemes of points on $ X $ via a connection between
Gieseker and Bridgeland moduli spaces, as well as the derived McKay
correspondence.
