The multiplicity one theorem for the Whittaker model, being such a useful tool in the representation theory of linear groups, fails for covering groups.
Gelfand-Graev representation V of a linear or covering group G admits any irreducible generic smooth representation as a quotient.
We study the space of Iwahori-fixed vectors of V for the case G is a covering group, and present several applications of the description, among them
1. determination of dimensions of Whittaker spaces of constituents of principal series ( regular and unitary cases)
2. conceptual construction of Chinta-Gunnells action.
3. recursive relations for the spherical Whittaker function for coverings of SL(2).
This is a joint work with Fan Gao and Edmund Karasiewicz.