Conjugates of strongly equivariant maps (Q1343686)

Let \(G_1\) and \(G_2\) be hermitian algebraic groups with associated symmetric domains \(X_1\) and \(X_2\), respectively. A holomorphic embedding \(\tau : X_1 \to X_2\) is called weakly equivariant if there exists a morphism of algebraic groups \(\rho : G_1 \to G_2\) compatible with \(\tau\). It is called strongly equivariant if, in addition, the image of \(X_1\) is totally geodesic in \(X_2\). If the groups \(G_i\) are defined over \(\mathbb{Q}\), then we can define the conjugate \(\tau^\sigma : X^\sigma_1 \to X_2^\sigma\) of \(\tau\), for an automorphism \(\sigma\) of \(\mathbb{C}\). The main theorem of this paper is that if \(\tau\) is strongly equivariant, then so is \(\tau^\sigma\). This was conjectured by \textit{Min Ho Lee} who proved the analogous statement for weakly equivariant maps [Pac. J. Math. 149, No. 1, 127-144 (1991; Zbl 0782.32026)].