# Classification of $(\mathfrak{g},K)$-module of $\mathrm{GL}(2, \mathbb{R})$

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?

• 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. – Peter Humphries Feb 15 at 16:21
• @PeterHumphries Thank you! – Seewoo Lee Feb 15 at 18:56