Let \(G\) be a simply connected complex semisimple Lie group of rank 1. Fix a Borel subgroup \(B\) of \(G\) and its maximal torus \(H\) and denote by \(\omega_ j\), \(j=1,\dots,l\), the fundamental weights and by \(W\) the Weyl group. Let \(P\) be a standard parabolic subgroup of \(G\) and \(S\subset \{1,\dots,l\}\) the subset corresponding to those simple roots whose root vectors generate the nilradical of \(P\). Denote \(\omega=\sum_{j\in S} \omega_ j\), \(\rho^ \omega: G\to\text{GL}(V^ \omega)\) the irreducible representation with highest weight \(\omega\), \(\Pi^ \omega\) the set of weights of \(\rho^ \omega\). The representation \(\rho^ \omega\) determines an embedding of the variety \(G/P\) into \(\mathbb{P}(V^ \omega)\); let \(\pi_ \mu\) \((\mu\in\Pi^ \omega)\) be the homogeneous coordinates in \(\mathbb{P}(V^ \omega)\) in the weight basis. A torus orbit is the closure \(X\) of an orbit of \(H\) in \(G/P\) containing a point \(x\) which satisfies \(\pi_{W(\omega)}(x)\neq 0\) for all \(w\in W/W_ S\), where \(W_ S\) is the Weyl group of \(P\).
The main results of the paper are as follows. One describes: the torus orbit \(X\) as toric variety by means of a rational polyhedral fan; the divisor defining the embedding \(X\to\mathbb{P}(V^ \omega)\); the generators of the ideal of the image of this embedding. One proves the formula \((D_ 1\cdots D_ l)=| W|\text{det }C\), where \(D_ j\) are the divisors on \(X\) corresponding to \(\omega_ j\) and \(C\) is the Cartan matrix.