Whittaker newforms for archimedean representations of GL(2)
Let \(F\) be \(\mathbb{R}\) or \(\mathbb{C}\) and let \(\psi\) denote a nontrivial additive character of \(F\). Let \(G= \text{GL}(2, F)\). Let \(T\) denote the set diagonal matrices in \(G\). Let \(K_1= \text{SO}(2)\) if \(F= \mathbb{R}\) and \(K_1= \text{SU}(2)\) if \(F= \mathbb{C}\). Fix an infinite-dimensional irreducible representation \((\pi,V)\) of \(G\). Let \(W(\pi)\) denote its \(\psi\)-Whittaker model. For \(W\in W(\pi)\), the ``Melin transform'' is defined as \[ \Psi_W(s,g)= \int_{F^\times} W\left(\begin{pmatrix} x & 0\\ 0 & 1\end{pmatrix} g\right)|x|^{s-{1\over 2}} d^\times x. \] It converges absolutely if \(\text{Re}(s)\) is large. The \(L\)-function \(L(s,\pi)\) is defined in a way such that \(\Phi_W(s, g):= \Psi_W(s, g)/L(s,\pi)\) has meromorphic continuation to \(s\in\mathbb{C}\) for all \(W\in W(\pi)\) and satisfies a functional equation, and for some \(W\), \(\Phi_W(s, g)\) is an exponential function of \(s\) [see \textit{H. Jacquet} and \textit{R. P. Langlands}, Automorphic forms on \(\text{GL}(2)\), Lect. Notes Math. 114. Berlin-Heidelberg-New York: Springer-Verlag (1970; Zbl 0236.12010)]. A function \(\Psi_W\in W(\pi)\) is called a Whittaker newform if \(\Psi_W(s,I)= L(s,\pi)\), where \(I\) is the identity matrix. The main result of this paper is to determine a Whittaker newform \(W\) for every infinite-dimensional \(\pi\). We will describe \(W\) when \(\pi\) has trivial central character. Consider \(\pi|_{K_1}= \sum^\infty_{n=-\infty} V_n\), where \(V_n\) is the irreducible finite-dimensional representation of \(K_1\) with highest weight \(n\) with respect to \(T\cap K_1\). Let \(n_0\) denote the smallest nonnegative \(n\) such that \(V_n\) has a nonzero \(T\cap K_1\) fixed vector. Then \(W\) is one of these fixed vectors in \(V_{n_0}\). The author interprets the finding of Whittaker newforms as a problem of finding test vectors. The latter problem was studied in the non-Archimedean case by \textit{B. Gross} and \textit{D. Prasad} [Math. Ann. 291, No. 2, 343--355 (1991; Zbl 0768.22004)].
scientific article; zbMATH DE number 5285735