vector, for our fixed choice of $K$. Let $V$ be the underlying vector space for $\pi$.

\[definedoverE\] The following are equivalent:

\(i) The representation $\pi$ is defined over $E$.

\(ii) There is an $E$-subspace $V_0$ of $V$ which is $G(k)$-stable and such that $V_0\otimes_E{\mathbb{C}}=V$.

\(iii) For any (possibly discontinuous) field automorphism $\sigma$ of ${\mathbb{C}}$ which fixes $E$ pointwise, we have $\pi\cong\pi^{\sigma}=\pi\otimes_{{\mathbb{C}},\sigma}{\mathbb{C}}$ as ${\mathbb{C}}$-representations.

That (ii) implies (iii) is clear—it’s an abstract representation-theoretic fact. Conversely if (iii) holds, then (ii) follows from Lemma I.1 of [@Waldspurger:MR783510] (note: his $E$ is not our $E$), because $V^K$ is 1-dimensional. This latter lemma of Waldspurger also shows that if (ii) holds then $V_0^K$ is 1-dimensional over $E$, and hence (ii) implies (i). To show that (i) implies (ii) we look at the explicit construction giving $\pi$ from the algebra homomorphism $H_{{\mathbb{Q}}}(G(k),K)\to{\mathbb{C}}$ given in [@cartier:corvallis]. Given a homomorphism $H_{{\mathbb{Q}}}(G(k),K)\to{\mathbb{C}}$ with image landing in $E$, the resulting spherical function $\Gamma:G(k)\to{\mathbb{C}}$ defined in equation (30) of [@cartier:corvallis] is also $E$-valued. Now if we define $V_0$ to be the $E$-valued functions on $G(k)$ of the form $f(g)=\sum_{i=1}^nc_i\Gamma(gg_i)$ for $c_i\in E$ and $g_i\in G(k)$, then $G(k)$ acts on $V_0$ by right translations, $V_0\otimes_E{\mathbb{C}}$ is the $V_\Gamma$ of §4.4 of 