In a recent paper of BL-Gee-Geraghty: "Sato-Tate for Hilbert modular forms" (JAMS 2011), a theorem is proved for regular algebrai cuspidal automorphic representation of $GL_2(\mathbb A_F)$ with $F$ a totally real field, which is not of CM type. I could not find any definition or reference for "CM type" in that paper. But I expect it should correspond to CM elliptic curve in the classical modular case.

My question is :

What is the precise definition for "an automorphic representation of CM type", both in the $GL_2$ case here and for general reductive group over number fields.

I prefer a definition "purely" in terms of representation-theory, not of arithmetic-geometry.

Why is the CM case excluded in that paper ?

Any comments or references will be very welcome. Thanks