TITLE: Two definitions of automorphic forms on Lie groups
QUESTION [6 upvotes]: My question is the about the equivalence of two definitions of automorphic forms on a semisimple Lie group.
The most common definition of automorphic forms on a semisimple Lie group $G$ with respect to a discrete subgroup $\Gamma$ is given by a smooth function $f:G\to \mathbb{C}$ with the following properties:

left $\Gamma$-invariant,
$K$-finite,
$Z(\mathfrak{g})$-finite,
slowly increasing.

Is this equivalent to the ones defined for $\operatorname{SL}(2,\mathbb{R})$ and $\operatorname{SL}(2,\mathbb{Z})$ where we have an automorphy factor $J(g,z)=cz+d$?
I also find a general definition with automorphy factors as follows. [Borel, Armand. "Introduction to automorphic forms." Proc. Symp. Pure Math. Vol. 9. No. 199210. 1966.]
I think given a classical automorphic form on $\operatorname {SL}(2,\mathbb{R})$, one could get a $\Gamma$-invariant form by multiplying an automorphy factor $j$. See page 283 of [Bump, Automorphic forms and representations].
So I guess the two definitions differs by such a automorphic factor $j:G/K\times G\to K_\mathbb{C}$.

REPLY [3 votes]: There are two notions: automorphic (or modular) form on a domain $G/K$, which would potentially have an "automorphy factor/cocycle" $\mu:\Gamma\times G/K\to \mathbb C^\times$ (or to a larger $GL_n$ for vector-valued automorphic forms), and automorphic forms on_a_group $G$, which have no cocycle, but instead are left $\Gamma$-invariant, and right $K$-equivariant.
When the cocycle extends from $\Gamma\times G/K$ to $G\times G/K$, the procedure in Borel's Boulder article gives an automorphic form on the group attached to an automorphic form on the domain.
A meaningful instance of a cocycle not extending from the discrete subgroup $\Gamma_o(4)$ to the natural ambient Lie group $SL_2(\mathbb R)$ is the example of half-integral-weight automorphic forms "on the upper half-plane", as in G. Shimura's papers around 1973.