Twists and conjugates of automorphic representations

Question

I am trying to understand the definition of RAECSDC (regular, algebraic, essentially conjugate self-dual, cuspidal), which appears in many papers on automorphy lifting. Let $\pi$ be an automorphic representation of $GL_n(\mathbb A_F)$, regarded as a $(\mathfrak g,K_\infty)\times GL_n(\mathbb A_F^\infty)$-module, where $F$ is a number field.

1. Let $\chi:F^\times\backslash \mathbb A_F^\times \to \mathbb C^\times$ be a Hecke character. How is $\pi\otimes (\chi\circ det)$ defined?
2. If $F$ is an imaginary CM field, then how is the conjugate $\pi^c$ defined?
3. If $\sigma\in Aut(\mathbb C)$, then how is $^\sigma\pi$ defined?

I am especially confused about the action of $\mathfrak g$ on them. Sorry if these are trivial questions, but I can't find the definitions anywhere.

Answer 1

Two of these three concepts are easy, the third is much deeper.

For (1): the twist is defined in "the obvious manner" as a representation; i.e. $\pi \otimes \chi$ has the same underlying space as $\pi$ but the action of $g \in G(\mathbb{A}_F^\infty)$ is multiplied by $\chi(g)$, and similarly for the $(\mathfrak{g}, K_\infty)$ structure. This defines a representation of $G(\mathbb{A}_F^\infty) \times (\mathfrak{g}, K_\infty)$; what is not quite obvious is that $\pi \times \chi$ is automorphic when $\pi$ is, but that isn't too hard to check.

For (2): if $F$ is a CM-field with totally real subfield $F^+$, then conjugation $c$ acts on $\mathbb{A}_F$, with the invariants being $\mathbb{A}_{F^+}$. So you get an action on automorphic forms by defining $\phi^c(g) = \phi(c g)$, and if $\phi$ generates an automorphic representation $\pi$, then $\phi^c$ also generates an automorphic representation which we can call $\pi^c$.

For (3): if $\pi$ is any automorphic representation of $\operatorname{GL}_n(\mathbb{A}_F)$, then you can extract a collection of numbers $a_v(\pi)$ indexed by the finite places $v$ of $F$, with $a_v(\pi)$ being the trace of a Hecke operator on the $GL_n(\mathbb{Z}_v)$-invariants of $\pi_v$ (understood as 0 if this space is zero). Now, if $\pi$ is RAECSDC, there is a unique ${}^\sigma \pi$ satisfying $a_v({}^\sigma\pi) = \sigma(a_v(\pi))$ for all finite $v$. However, the existence of ${}^\sigma \pi$ is a very hard theorem and it does not admit a simple representation-theoretic description. (Note that essential conjugate self-duality is not needed here, but "regular algebraic" is vital: if $\pi$ is some more general kind of automorphic representation, e.g. generated by a Maass form, then $\sigma(a_v(\pi))$ still makes sense as a collection of numbers, but it will not necessarily be the Hecke traces of any automorphic representation.)

Note that $a_v(\pi^c) = a_{c(v)}(\pi)$, which is usually a different thing from $c(a_v(\pi))$, showing the difference between (2) and (3).