# Two definitions of automorphic forms on Lie groups

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:

1. left $$\Gamma$$-invariant,
2. $$K$$-finite,
3. $$Z(\mathfrak{g})$$-finite,
4. 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 $$j:G/K\times G\to K_\mathbb{C}$$.

• Clarify right or left invariant and finite. For a modular form $f\in M_k(\Gamma)$ with $\Gamma$ a finite index subgroup of $SL_2(\Bbb{Z})$ then $F(\gamma)=(c_\gamma i + d_\gamma)^{-k} f(\gamma . i)$ is an automorphic form $\in C^\infty( \Gamma\backslash SL_2(\Bbb{R}))$ and if $f$ is a cusp form then $F\in L^2( \Gamma\backslash SL_2(\Bbb{R}))$. The right translates of $F$ then generate an Hilbert space $V$ and the right action of $SL_2(\Bbb{R})$ on $V$ is the automorphic representation. Dec 2 '20 at 22:17
• Does this answer your question? mathoverflow.net/q/124754/6518 Dec 2 '20 at 22:54
• $SL(2)$ is ill-formated and should be $\mathrm{SL}(2)$
– YCor
Dec 3 '20 at 9:47

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.
• Thank you. I think you mean that a given automorphic on the domain can NOT always extend to the one on a group. But can we get the one on a domain from the one defined on the group? I guess we can just divide it by a cocycle as $F(g)=j(g,i)f(i)$ for $G=SL_2(\mathbb{R})$ (here $i$ is just the coset $0\in G/K$). BTW: Your short notes help me quite a lot. Thank you so much!