This is a classification of $(\mathfrak{g},K)$-module of $\mathrm{GL}(2, \mathbb{R})$ in Bump:


Here we assume that two complex numbers $s_{1}, s_{2}$ are given, and $\lambda = s(1-s), \mu = s_{1} + s_{2}$ where $s = \frac{1}{2}(s_{1} - s_{2} + 1)$. Also, $\epsilon\in \{0, 1\}$ is a parity of the module and $\chi_{i}$ is a character of $\mathbb{R}^{\times}$ defined as $\chi_{i}(y) = \mathrm{sgn}(y)^{\epsilon_{i}}|y|^{s_{i}}$, where $\epsilon_{1} + \epsilon_{2} \equiv \epsilon$ mod 2 and $\epsilon_{i}\in \{0, 1\}$. (There are two choices for given $\epsilon$.) $\pi(\chi_{1}, \chi_{2})$ is a principal series defined as an induced representation $\mathrm{Ind}_{B}^{\mathrm{GL}(2, \mathbb{R})} \chi$ where $$ \chi\left(\begin{pmatrix}y_{1} &z \\& y_{2}\end{pmatrix}\right) = \chi_{1}(y_{1})\chi_{2}(y_{2}), $$ and $\mathcal{D}_{\mu}(k)$ is a infinite dimensional irreducible admissible submodule of the induced representation where $s = k/2$ for some integer $k\geq 1$ with $k\equiv \epsilon$ (mod 2). Note that the Casimir element $\Delta$ and $Z = \begin{pmatrix} 1&0\\0&1\end{pmatrix}$ in $\mathfrak{g}$ acts as scalars $\lambda$ and $\mu$ respectively.

Now I have no idea why we have to assume $\mu$ is a real number for the discrete series. For $\mathrm{GL}(2, \mathbb{R})^{+}$ case, as I know, there was no such restriction on $\mu$. Am I wrong?

  • $\begingroup$ I think you're right. Obviously there is an additional condition on $\mu$ if you wish for it to be unitary, and if it is the local component of an automorphic representation of $\mathrm{GL}_2(\mathbb{A_Q})$, then there are further restrictions. $\endgroup$ – Peter Humphries Feb 15 at 16:21
  • $\begingroup$ @PeterHumphries Thank you! $\endgroup$ – Seewoo Lee Feb 15 at 18:56

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