in $E$.

If half the sum of the positive roots of $G$ is in the lattice $X^*(T)$, then these notions coincide. However there is no reason for them to coincide in general and we shall shortly see examples for ${\operatorname{GL}}_2({\mathbb{Q}}_p)$ where they do not.

Note also that it is not immediately clear that these notions are independent of the choice of $K$: perhaps there is some $\pi$ with a $K$-fixed vector and a $K'$-fixed vector for two non-conjugate hyperspecial maximal compacts (for example, the trivial 1-dimensional representation of ${\operatorname{SL}}_2({\mathbb{Q}}_p)$ has this property), and which is defined over $E$ (or has Satake parameter defined over $E$) for one choice but not for the other. The reader should bear in mind that for the time being these notions depend on the choice of $K$, although we will soon see (in Corollary \[Carith\_well\_defined\] and Lemma \[Larith\_well\_defined\]) that they are in fact independent of this choice.

We now discuss some other natural notions of being “defined over $E$” for $E$ a subfield of ${\mathbb{C}}$, and relate them to the notions above. So let $E$ be a subfield of ${\mathbb{C}}$ and let $\pi$ be a smooth irreducible admissible complex representation of $G(k)$ with a $K$-fixed 