Document Zbl 1200.11034

Whittaker functions on real semisimple Lie groups of rank two. (English) Zbl 1200.11034

Can. J. Math. 62, No. 3, 563-581 (2010).

Let \(G\) be one of the three split Lie groups of rank 2: \(\text{SL}_3(\mathbb R)\), \(\text{Sp}_2(\mathbb R)\) and \(G_2(\mathbb R)\). Let \(P_0= MAN\) be a of \(G\). For a weight \(v\) of \(A\), let \(\pi_v= \text{Ind}^G_{MAN}1_M\otimes a^v\otimes 1_N\) denote the spherical normalized induced principal series representation. We fix a non-degenerate \(\eta: N\to\mathbb C^\times\). Let \(C^\infty_\eta(N\setminus G\)) denote the space of smooth functions \(f: G\to\mathbb C\) such that \(f(ng)= \eta(n)f(g)\) for all \(n\in G\). We define the space of class one as \[ \text{Wh}(G,v,\eta)= \{\Phi(v_0): \Phi\in\text{Hom}_G(\pi_v, C^\infty_\eta(N\setminus G)\}, \] where \(v_0\) is a spherical vector of \(\pi_v\). By the \(G= NAK\), \(w=\Phi(v_0)\) as a function on \(G\) is uniquely determined by its restriction to \(A\). Using the two simple co-roots, we identify \(A\simeq(\mathbb R^+)^2\). Hence we may regard \(w\) as a function on \((\mathbb R^+)^2\).
We refer to the paper for the definition of a regular character \(v\). For a regular \(v\), M. Hashizume defines a basis of \(\text{Wh}(G,v,\eta)\) in [Hiroshima Math. J. 12, 259–293 (1982; Zbl 0524.43005)] where each is expressed as a power series on \(A\simeq(\mathbb R^+)^2\). The first main result of this paper is to write each basis vector of \(\text{Wh}(G_2(\mathbb R),v,\eta)\) (resp. \(\text{Wh}(\text{Sp}_2(\mathbb R),v,\eta)\)) as an infinite linear combination (whose coefficients are functions on \((\mathbb R^+)^2\)) of those of \(\text{Wh}(\text{SL}_3(\mathbb R),v',\eta)\). Here \(v'\) ranges over a lattice of weights of \(A\) and we remind that a vector in \(\text{Wh}(G,v,\eta)\) is a function on \((\mathbb R^+)^2\). In this way, the author is able to express the class one Whittaker functionals in the existing literatures of \(G_2(\mathbb R)\) and \(\text{Sp}_4(\mathbb R)\) in terms of those of \(\text{SL}_3(\mathbb R)\). Such relations will be useful in the study of liftings of automorphic representations. As an application, the author uses such circle of idea to derive an explicit formula for an archimedean zeta integral on \(\text{GSp}_2\) considered by D. Bump, S. Friedberg and D. Ginzburg [Math. Ann. 313, No. 4, 731–761 (1999; Zbl 1055.11516)].

Reviewer: Hung Yean Loke (Singapore)

Cited in 3 Documents