Generalized Whittaker functions on \(\mathrm{SU}(2,2)\) with respect to the Siegel parabolic subgroup (Q2784246)

The author studies generalized Whittaker functions for an irreducible admissible representation \(\tau\) of the special quasi-split unitary group in four variables, over \(\mathbb{R}\), \(G= \text{SU}(2,2)\). These Whittaker functions are obtained through intertwining operators of Gelfand-Graev type, coming off the Siegel radical. The Siegel radical \(N_S\), in this case, is naturally isomorphic to the space \(X\) of two by two Hermitian matrices. Thus, a character of \(N_S\), may be identified with a Hermitian matrix \(\xi\). The Levi subgroup of the Siegel parabolic subgroup is isomorphic to \(\text{GL}(2,\mathbb{C})\), and it acts on \(X\) by \(g(x)= gx\overline g^t\) (\(g\in \text{GL}(2,\mathbb{C})\), \(x\in X\)). Assume that \(\xi\) is invertible. Then the connected component of the stabilizer of \(\xi\) in \(\text{GL}(2,\mathbb{C})\) is the special unitary group \(\text{SU}(\xi)\) in two variables, corresponding to \(\xi\). Let \(\chi\) be an irreducible unitary representation of \(\text{SU}(\xi)\). Let \(R_\xi= \text{SU}(\xi)N_S\). Then \(\chi\otimes\xi\) is a representation of \(R_\xi\). The corresponding Gelfand-Graev operators in this case are the \(({\mathcal G},K)\) intertwining maps from \(\pi\) to the representation differentiably induced from \(R_\xi\) and \(\chi\otimes\xi\). The main results of this work give the dimension of the space of these maps, when \(\pi\) is in the discrete series. This dimension is, in general, at most two. If \(\xi\) is positive definite, and \(\pi\) is in the holomorphic, or anti-holomorphic discrete series, then this dimension is at most one, and if \(\pi\) is generic, i.e. it has an ordinary Whittaker model, then the dimension is either zero, or two, but once one adds the requirement that the generalized Whittaker functions obtained above are of moderate growth, the dimension drops from two to one. The remaining discrete series, which are non-holomorphic and non-generic do not admit Gelfand-Graev models with respect to definite \(\xi\), but they may have ones with respect to indefinite \(\xi\), and then the dimension is at most one. The author writes precise formulae of generalized Whittaker functions, given the Harish-Chandra parameter and the minimal \(K\)-type. In general, the author obtains in each case a system of partial differential equations satisfied by such a generalized Whittaker function. He then solves the system and finds a unique solution (up to scalars). The large body of this work consists of long detailed case by case computations done in order to write these systems of partial differential equations and their solutions.