Principal series Whittaker functions on \(Sp(2,\mathbb{C})\)
type: Item
instance of: scholarly article
title: Principal series Whittaker functions on \(Sp(2,\mathbb{C})\)
zbMATH Open document ID: 1235.22017
author: Tadashi Miyazaki
published in: Journal of Functional Analysis
publication date: 2011-07-08T00:00:00Z
review text: 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…
Mathematics Subject Classification ID: 22E45
Mathematics Subject Classification ID: 33C99
zbMATH DE Number: 5919155
zbMATH Keywords: Whittaker functions
zbMATH Keywords: automorphic forms
reviewed by: Hung Yean Loke
MaRDI profile type: Publication
full work available at URL: j.jfa.2011.04.013
OpenAlex ID: W1598178616
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DOI: 10.1016/J.JFA.2011.04.013
description: scientific article; zbMATH DE number 5919155