Harich-Chandra introduced the notion of $(\mathfrak{g}, K)$-module to study representations of a real Lie group $G$ via its Lie algebra $\mathfrak{g}$ and a maximal compact subgroup $K$. One also has classification of irreducible admissible $(\mathfrak{g}, K)$-modules for some cases, e.g. when $G = \mathrm{GL}_2(\mathbb{R})$. Now, if one have a holomorphic modular form / Maass form $f$, then the corresponding automorphic representation $\pi_f$ of $\mathrm{GL}_2(\mathbb{A})$ decomposes as a restricted tensor product of local representations $\pi_{f} = \otimes_v \pi_{f, v}$, and the archimedean part $\pi_{f, \infty}$ is a discrete series / principal series representation.
Now, for half-integral weight modular forms, one can also attach automorphic representations $\bar{\pi}$ of $\overline{\mathrm{GL}}_2(\mathbb{A})$, the 2-fold covering of $\mathrm{GL}_2(\mathbb{A})$. It also decomposes as product of local components, and one might expect that $\bar{\pi}_{\infty}$ becomes a discrete series representation of $\overline{\mathrm{GL}}_2(\mathbb{R})$. However, I can' find a reference that dealt with $(\mathfrak{g}, K)$-module of $\overline{\mathrm{GL}}_2(\mathbb{R})$. Instead, Gelbart's book directly considers representations of $\mathrm{SL}_2(\mathbb{R})^{\pm}$ and give some classifications that is quite similar to that of $(\mathfrak{gl}_2(\mathbb{R}), O(2))$-module. I think there should be a theory of $(\mathfrak{g}, K)$-module for covering groups, which would give almost same classification for $\overline{\mathrm{GL}}_2(\mathbb{R})$ as in Gelbart's book. Could you help me to find such a references? Thanks in advance.