Invariants of the Cox rings of double flag varieties of low complexity for exceptional groups

We find the algebras of unipotent invariants of the Cox rings for all double flag varieties of complexity $0$ and $1$ for the exceptional simple algebraic groups; namely, we obtain presentations of these algebras in terms of generators and relations. It is well known that such an algebra is free in the case of complexity $0$. In this paper, we show that, in the case of complexity $1$, the algebra in question is either a free algebra or a hypersurface. A similar result for classical groups was previously obtained by the author. Knowing the structure of this algebra enables one to decompose tensor products of some irreducible representations effectively into irreducible summands and to obtain some branching rules.
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