Principal series and generalized principal series Whittaker functions with peripheral \(K\)-types on the real symplectic group of rank 2
The paper under review contains a variety of results pertaining to Whittaker functions belonging to the standard representations of the group \(\mathrm{Sp}(2,\mathbb{R})\). These results generalize those of \textit{T. Ishii} [J. Funct. Anal. 225, No. 1, 1--32 (2005; Zbl 1078.11031)] which performed similar computations for principal series representations having minimal \(K\)-types; those of \textit{T. Miyazaki} and \textit{T. Oda} [Tohoku Math. J., II. Ser. 50, No. 2, 243--260 (1998; Zbl 1003.11020)] with computations for representations induced from the Jacobi maximal parabolic subgroup; and \textit{T. Oda} [Tohoku Math. J., II. Ser. 46, No. 2, 261--279 (1994; Zbl 0812.11033)] with computations for the discrete series representations. The paper is divided into three chapters each containing a different main result. The first main result of the paper under review computes the explicit formulas of principal series Whittaker functions on \(\mathrm{Sp}(2,\mathbb{R})\) with arbitrary one-dimensional \(K\)-types. In fact, the author deduces these results from those of Ishii by applying various Maass shift operators to the functions whose explicit formulas were computed by Ishii. These shift operators move the \(K\)-type from the Whittaker functions with minimal scalar \(K\)-types to ones having arbitrary scalar \(K\)-types. Additionally, the effect that these shift operators have on the explicit formula for the Whittaker functionals from representations with minimal scalar \(K\)-type are tractable enough for the author to use to derive the explicit formulas for the Whittaker functions from representations with with arbitrary scalar \(K\)-types. The second main result of the paper under review pertains to vector-valued Whittaker functions for representations \(\Pi\) which are induced from a certain maximal parabolic subgroup and have peripheral \(K\)-types. The maximal parabolic subgroup in question is the Siegel parabolic subgroup (denoted \(P_S\)). This parabolic subgroup has a Levi factor isomorphic to \(\mathrm{GL}(2,\mathbb{R})\) and an abelian unipotent radical. In this case, the peripheral \(K\)-types of the \(P_S\)-principal series representation are those \(K\)-types whose dimension are smallest. In particular, the second main result is the explicit computation for the holonomic system for the radial part of the Whittaker functions described the previous few sentences. The author also conjectures that the dimension of the solution space of this holonomic system is equal to the dimension of a space algebraic Whittaker functions for the representation \(\Pi\). This space of algebraic Whittaker functions has dimension \(8\), the order of the Weyl group for \(\mathrm{Sp}(2,\mathbb{R})\). One of the key steps in this chapter involves analyzing the action of the Casimir element of the center of the universal enveloping algebra to the function in the space of \(\Pi\) that realizes a particular \(K\)-type of interest. The third main result of the paper under review gives explicit formulas for the Whittaker functions coming from a vary particular \(P_S\)-principal series representation \(\Pi\) having a very particular \(K\)=type. In this case, the author derives eight power series solutions to the holonomic system derived in Chapter II. In order to describe this result, let \(P_S=N_SA_SM_S\) be the Langlands decomposition of \(P_S\) where \(N_S\) is the unipotent radical, \(A_S\) is the connected component of the center of the Levi factor and \(M_S\simeq\mathrm{SL}^{\pm}(2,\mathbb{R})\), the set of \(2\times 2\) matrices with determinant in the set \(\{\pm 1\}\). For these results, \(\Pi\) is induced from the representation \(1_N\otimes e^{\nu_S+\rho}\otimes D_2\) on \(P_S\) where \(D_2\) is a particular discrete series representation of \(M_S\). The author finds explicit formulas for the Whittaker functions associated to a particular \(K\)-type that is described explicitly in Chapter II. The main tool in this section has to do with various embeddings of the representation \(\Pi\) into various principal series representations for \(\mathrm{Sp}(2,\mathbb{R})\). This paper contains a great deal of detail for all of the computations involved and is well-cited for the computations that are omitted. Having explicit results for Whittaker functions is a useful tool in several areas of automorphic forms appearing such diverse settings as the local computation of integrals representing \(L\)-functions to various Hodge structures of Shimura varieties.
scientific article; zbMATH DE number 5836518
scientific article