In 2001, Kato, Murase and Sugano published a paper describing certain multiplicity one results for special orthogonal groups over local fields of odd residue degree. In that paper, the authors define Whittaker-Shintani functions, complex-valued functions on the group which possess desirable symmetry properties on the right and left by various subgroups and prove that the space of all such functions is at most one-dimensional for each choice of parameters, and generically exactly one.
In this project, we carry an analogous study for quasi-split unitary groups, so that instead of a single local field we have a quadratic extension of local fields of odd residue degree. Gelfand-Graev functions are complex-valued functions on a unitary group characterized by the following collection of symmetry conditions.
1.Invariance under the maximal compact of the full group on the right and the maximal compact of a smaller unitary group on the left.
2.Transformation on the left by a particular unipotent subgroup acts by a specified character.
3. Transformation on the right by the Hecke algebra of the full group and the left by the Hecke algebra of a smaller unitary group on the left acts by specified Hecke characters.
We show that, for each choice of characters, the space of these functions is one-dimensional. Furthermore we give an explicit formula for typical such functions, which turns out to be rational with respect to the Hecke characters chosen.