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I just started reading "On the functional equations satisfied by Eisentstein series" by Langlands http://publications.ias.edu/sites/default/files/Eisenstein-ps.pdf . I wasn't sure of some notation/convention which may be standard for some people. It's really not clear to me since there seems to be no explanation, so I would greatly appreciate if someone could possibly clarify them for me.

On page 3 of the document he sets up the assumptions and my questions are from this page.

1) In the very first sentence "Let $G$ be a Lie group with Lie algebra $\mathfrak{g}$." It's really not clear to me which field $\mathfrak{g}$ is over.... Which field is $\mathfrak{g}$ is over?

2) In the first sentence of the second paragraph, he defines the parabolic subgroup of $G$: define a parabolic subgroup $P$ of $G$ to be the normalizer in $G$ of a subalgebra $\mathfrak{p}$ of $\mathfrak{g}$ such that the complexification $\mathfrak{p}_c = \mathfrak{p} \otimes_{R} C$ of $\mathfrak{p}$ contains a Cartan subalgebra $\mathfrak{j}_c$ of $\mathfrak{g}_c$ together with the root vectors belonging to the roots of $\mathfrak{j}_c$ whcih are positive with respect to some order on $\mathfrak{j}_c$.

I don't understand what $\mathfrak{g}_c$ is... What is $\mathfrak{g}_c$ here?

Any comments and explanation are appreciated. Thank you.

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    $\begingroup$ I haven't read that, but would expect that $\mathfrak{g}_{\mathbb{C}} = \mathfrak{g} \otimes_{\mathbb{R}} \mathbb{C}$ is the complexification of $\mathfrak{g}$ (see also the notation for $\mathfrak{p}_c$) --- and that $\mathfrak{g}$ would therefore probably be defined over the reals. $\endgroup$
    – Jules Lamers
    Commented Mar 19, 2019 at 23:01

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