For tame Kazhdan-Patterson and Savin covers of GL_r, the Gelfand-Graev representation decomposes as explicit simple-type Hecke algebra modules, yielding the Whittaker dimension of discrete series as |X(λ)/S_k|.
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Gelfand--Graev representation as a Hecke algebra module of simple types of a finite central cover of $\mathrm{GL}(r)$
For tame Kazhdan-Patterson and Savin covers of GL_r, the Gelfand-Graev representation decomposes as explicit simple-type Hecke algebra modules, yielding the Whittaker dimension of discrete series as |X(λ)/S_k|.