  We introduce higher Specht polynomials - analogs of Specht polynomials in
higher degrees - in two sets of variables $x_1,\ldots,x_n$ and $y_1,\ldots,y_n$
under the diagonal action of the symmetric group $S_n$. This generalizes the
classical Specht polynomial construction in one set of variables, as well as
the higher Specht basis for the coinvariant ring $R_n$ due to Ariki, Terasoma,
and Yamada, which has the advantage of respecting the decomposition into
irreducibles.
  As our main application of the general theory, we provide a higher Specht
basis for the hook shape Garsia--Haiman modules. In the process, we obtain a
new formula for their doubly graded Frobenius series in terms of new
generalized cocharge statistics on tableaux.
