Principal series Whittaker functions on \(Sp(2,\mathbb{C})\)
Let \(G= \text{Sp}(2,\mathbb{C})\) be the complex symplectic group with Lie algebra \(\mathfrak g\). Let \(P= MAN\) be a Borel subgroup and let \(K\) be a maximal compact subgroup. Let \(\Pi= \text{Ind}^G_P(\sigma_d\otimes e^v\otimes 1_N)\) be the standard normalized induced principal series representation realized on a Hilbert subspace of \(L^2(K)\). Let \(\Pi^{\infty}\) denote the subspace of smooth vectors and let \(\Pi_K\) denote its Harish-Chandra module. We fix a non-degenerate unitary character \(\Psi: N\to \mathbb{C}^{\times}\). Let \(C= C^{\infty}(N\setminus G;\psi)\) be the space of continuous functions \(F\) on \(G\) such that \(F(ng)= \psi(n)F(g)\) for all \(n\in\mathbb{N}\) and \(g\in G\). The group \(G\) acts on \(C\) by right translation. Let \(I= \Hom_{({\mathfrak g},K)}(\Pi_K,C)\) and \(I^{\infty}= \Hom_G(\Pi, C)\). It is a result of Kostant that \(\dim_{\mathbb{C}} I=|W|\), where \(W\) is the Weyl group. In this case \(|W|= 8\). Let \(\text{Wh}(\Pi)=\{\Phi(f)\in C: \Phi\in I\), \(f\in\Pi_K\}\) and \(\text{Wh}(\Pi)^{\infty}= \{\Phi(f)\in C: \Phi\in I^{\infty}\), \(f\in\Pi^{\infty}\}\). The functions in \(\text{Wh}(H)\) are called the Whittaker functions of \(\Pi\). A result of Wallach states that the space of \(K\)-finite vectors in \(\text{Wh}(\Pi)^{\infty}\) is the Whittaker functions of moderate growth in \(\text{Wh}(\Pi)\). Let \(V_d\) denote its minimal \(K\)-type of \(\Pi\). In this paper the author picks a basis \(\{\Phi_w: w\in W\}\) of \(\text{Wh}(\Pi)\) and a basis \(\{v_{{\mathbf L}}\}\) of \(V_d\). By the Iwasawa decomposition \(G= NAK\), \(\Phi_w(v_{{\mathbf L}})\) as a function on \(G\) is essentially determined by its values on the split torus \(A\). The main result of this paper is Theorem 6.2 where the author computes the Fourier expansion of \(\Phi_w(v_{{\mathbf L}})\) with respect to the split torus \(A\). The Fourier coefficients are expressed using gamma functions and it is too complicated to state in this review. These formulas are proved by first expressing the Whittaker functions as solutions of a system of differential equations and then solve them. Combining with a result of Shalika, the space of Whittaker functions of moderate growth is one-dimensional and the second main result of the paper is to express one such vector as an explicit linear combination of the basis \(\{\Phi_w\}\). In the final section of this paper, the author uses his main results to compute the Novodvorsky's local zeta integrals at the Archimedean place.
scientific article; zbMATH DE number 5919155
scientific article