  The classical coinvariant ring $R_n$ is defined as the quotient of a
polynomial ring in $n$ variables by the positive-degree $S_n$-invariants. It
has a known basis that respects the decomposition of $R_n$ into irreducible
$S_n$-modules, consisting of the higher specht polynomials due to Ariki,
Terasoma, and Yamada.
  We provide an extension of the higher Specht basis to the generalized
coinvariant rings $R_{n,k}$. We also give a conjectured higher Specht basis for
the Garsia-Procesi modules $R_\mu$, and provide a proof of the conjecture in
the case of two-row partition shapes $\mu$. We then combine these results to
give a higher Specht basis for an infinite subfamily of the modules
$R_{n,k,\mu}$ recently defined by Griffin, which are a common generalization of
$R_{n,k}$ and $R_{\mu}$.
