Let $G$ be a connected linear algebraic group over the field of complex number ${\Bbb C}$.
Let $G({\Bbb C})$ denote the complex Lie group of ${\Bbb C}$-points of $G$.
Let $\sigma$ be an *anti-holomorphic involution* of $G({\Bbb C})$, that is, a an automorphism of the real Lie group
$$\sigma\colon G({\Bbb C})\to G({\Bbb C})$$
such that $\sigma$ is anti-holomorphic and $\sigma^2={\rm id}$.

The anti-holomorphic involution $\sigma$ naturally acts on the ring of holomorphic function on $G({\Bbb C})$: $$({}^\sigma\!\! f)(g)=\overline{f(\sigma^{-1}(g))},$$ where the bar denotes complex conjugation (and, of course, $\sigma^{-1}=\sigma$).

We say that $\sigma$ as above is *anti-regular*, if, when acting on the ring of holomorphic functions on $G$,
$\sigma$ preserves the subring of regular functions (recall that $G$ is an algebraic group).

Question.Are all anti-holomorphic involutions anti-regular in the following cases: (1) $G$ is a connected linear algebraic group; (2) $G$ is a (connected) reductive algebraic group; (3) $G$ is a (connected) semisimple algebraic group?

**Remark.**
An anti-regular involution $\sigma$ of $G({\Bbb C})$ defines by Galois descent a real structure on $G$.
Indeed, we may put
$$ G_{\Bbb R}={\rm Spec}\,({\Bbb C}[G]^\sigma),$$
where ${\Bbb C}[G]^\sigma$ is the subring of fixed points of $\sigma$ in the ring of regular functions ${\Bbb C}[G]$ on $G$.

Conversely, an algebraic group $G_{\Bbb R}$ over ${\Bbb R}$ defines a complex algebraic group $G:=G_{\Bbb R}\times_{\Bbb R} {\Bbb C}$, and the complex conjugation on ${\Bbb C}$ induces by functoriality an anti-regular involution $\sigma$ on $G({\Bbb C})$.

Iwork with anti-regular involutions. My working definition of a real algebraic group is: a pair $(G, \sigma)$, where $G$ is a complex algebraic group and $\sigma$ is ananti-regularinvolution. $\endgroup$ – Mikhail Borovoi Sep 23 '19 at 19:07anti-holomorphicinvolution. I would like to be sure that this is the same, at least for semisimple groups. $\endgroup$ – Mikhail Borovoi Sep 23 '19 at 19:11