What is a "cohomological type" automorphic representation?

What is an automorphic representation of CM type ?

• 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

• Answer:

  1.-- in the $Gl_2$-case, $\pi$ is of CM type if it is the automorphic induction of a Grossencharacter of a CM extension K of $F$. In terms of the Galois representation of $Gal(\bar F/F)$ attached to $\pi$, that means that $\pi$ is not the induced representation from a character of a subgroup $Gal(\bar F/K)$ of index $2$, where $K$ is a CM extension of $F$. In the general case, the notion of CM stratifies into many different notions. Read things about the Mumford-Tate groups for more about this. 2.-- because already in the $F=\mathbb{Q}$-case, the Sato-Tate conjecture excludes the CM case.