Let \(G\) be a connected almost simple algebraic group over an algebraically closed field, with a chosen pinning. Let \(\sigma\) be a Dynkin automorphism of \(G\). To this datum one can attach a smaller almost simple group \(G_\sigma\). The paper under review focuses on the space of tensor invariants \(V^G_{\underline\lambda}=(V_{\lambda_1}\otimes\cdots\otimes V_{\lambda_n})^G\), where \(\underline\lambda=\lambda_1,\ldots,\lambda_n\) is a sequence of dominant weights of \(G_\sigma\), and \(V_{\lambda_i}\) is the corresponding irreducible representation of \(G\). Similarly, one can define \(W_{\underline\lambda}^{G_\sigma}=(W_{\lambda_1}\otimes\cdots\otimes W_{\lambda_n})^{G_\sigma}\) by taking the irreducible representations \(W_{\lambda_i}\) of \(G_\sigma\).
The first main result is a twining character formula for tensor invariants. First, observe that \(\sigma\) induces an action on each \(V_{\lambda_i}\), as well as on \(V_{\underline\lambda}^G\). Theorem 1.1 asserts that \[ \mathrm{trace}\left(\sigma\mid V^G_{\underline\lambda}\right)=\dim W_{\underline\lambda}^{G_\sigma}. \] To state the second main result, recall that a connected reductive group \(G\) is said to have saturation factor \(k\) if for any sequence \(\underline\lambda\) of dominant weights \(\lambda_1,\ldots,\lambda_n\) whose sum lies in the root lattice and such that \(V_{N\underline\lambda}^G\neq 0\) for some \(N\in\mathbb Z_{\geq 1}\), then \(V_{k\underline\lambda}^G\neq 0\). It is conjectured that simply-laced \(G\) has saturation factor \(1\). The Theorem 1.2 asserts: if \(G\) has saturation factor \(k\), then \(G_\sigma\) has saturation factor \(c_\sigma k\) where \[ c_\sigma:=\begin{cases} 2,&\mathrm{type}(G)\neq\mathrm{A}_{2n},\quad\mathrm{order}(\sigma)=2,\\ 3,&\mathrm{order}(\sigma)=2, \\ 4,&\mathrm{type}(G)=\mathrm{A}_{2n},\quad\mathrm{order}(\sigma)=2.\end{cases} \] This allows us to study the saturation factor for a group \(H\) by folding some bigger group \(G\) using appropriate \(\sigma\). Using this, it is established that the non-simply laced group \(\mathrm{Spin}_{2n+1}\) has saturation factor \(2\).
The methods of this paper are innovative. The first step is to realize the spaces \(V^G_{\underline\lambda}\) via the geometric Satake correspondence, namely as the top Borel-Moore homology of the cyclic convolution variety \(\mathrm{Gr}_{G^\vee,\underline\lambda}\). The irreducible components of this variety furnish the Satake basis for \(V^G_{\underline\lambda}\). The second step is to use a bijection between the Satake basis and the \(G\)-laminations, the latter being the tropical points of a certain configuration space of decorated flags. Here a decorated flag means a point in the basic affine space \(G/U\). Finally, these correspondences for \(G\) are shown to be compatible with \(\sigma\)-actions. We are reduced to study the \(\sigma\)-action on \(G\)-laminations, which is of combinatorial nature.