# Whittaker model for $\mathrm{GL}(2, \mathbb{R})$

The complete list of irreducible admissible representations of $$\mathrm{GL}(2, \mathbb{R})$$ are known - principal series, discrete series, limit of discrete series, and finite dimensional representations (symmetric power of standard representation). In Bump's book, chapter 2.8, he introduces the Whittaker model and shows that if the Whittaker model exists, then unique. However, it seems that he doesn't talk about existence at all. (He talked about the existence of Whittaker models for infinite dimensional representations over non-archimedean local fields later, by using the theory of Jacquet functor).
I believe that most of the irreducible admissible representations have Whittaker models, but I have no idea how to construct it.

For example, assume that we have a principal series $$\mathcal{P}_{\mu}(\lambda, \epsilon)$$, where $$\mu, \lambda\in \mathbb{C}$$ are scalars corresponds to the elements $$Z = \begin{pmatrix}1&0\\0&1\end{pmatrix}, \Delta\in Z(U\mathfrak{gl}(2, \mathbb{R}))$$, and $$\epsilon\in \{0, 1\}$$ represents its parity. Assume that it is irreducible (i.e. $$\lambda \neq \frac{k}{2}(1-\frac{k}{2})$$ for all $$k\equiv \epsilon$$ (mod 2). How can we construct Whittaker model for this?

• In brief, by constructing an intertwining operator from principal series to the Whittaker space. The main issue is that the relevant integrals do not converge for parameter values giving unitary principal series, so meromorphic continuation of the intertwinings is necessary. But/and this is elementary in this tiny case. In contrast, in higher rank, it's a huge production. E.g., the bulk of A. Knapp's Princeton Press book "Repns... by example" is dedicated to proving meromorphic continuation (and related) in more complicated situations. Feb 15, 2019 at 22:43
• @paulgarrett Could you recommend any reference for $\mathrm{GL}(2, \mathbb{R})$ case that shows how to define such integral explicitly? Feb 17, 2019 at 2:13
• I would have thought that the intertwining from principal series to the Whittaker space, for $GL_2$, would have appeared in Bump's book... but maybe not. Still, it may have been labelled in a different way, so you might look again. Also, you might look in Goldfeld's CUP book, and in Goldfeld-Hundley's CUP books, and in chapter 12 of my relatively recent CUP book (also available, legally, on-line...) Feb 17, 2019 at 22:33

For any local field $$k$$, an intertwining operator from the induced model for a principal series to the Whittaker model (such as referred to by Paul Garrett in his comment above) is explicitly: $$\phi \mapsto W_\phi,$$ where $$W_\phi(g) = \int_{k} \phi \left( ( \begin{smallmatrix} & -1 \\ 1 & \end{smallmatrix}) ( \begin{smallmatrix} 1 & x\\ & 1 \end{smallmatrix}) g \right) \psi(x)\,dx,$$ where $$\psi$$ is the chosen additive character for your Whittaker model. In general, this only converges conditionally, but if you interpret the integral as the limit over an increasing sequence of compact subsets of $$k$$ defined by $$|x|\leq R$$, $$R \to \infty$$, then it does give an intertwiner.