Gelfand--Graev representation as a Hecke algebra module of simple types of a finite central cover of $\mathrm{G}\mathrm{L}(r)$

For an $n$-fold Kazhdan–Patterson cover or Savin’s cover of a general linear group over a non-archimedean local field of residual characteristic $p$ with $\mathrm{g}\mathrm{c}\mathrm{d}(n,p)=1$, we realize the Gelfand–Graev representation as a Hecke algebra module of a simple type and study its explicit expression. As a main corollary, we calculate the Whittaker dimension of every discrete series representation of such a cover. Using Zelevinsky’s classification, this theoretically gives the Whittaker dimension of every irreducible representation.