Pinning and Pinned Automorphisms

A pinning of a split connected reductive $G$ is a triple $(B,T,\{X_\alpha\}_{\alpha\in\Delta})$ where $T\subset B$ is a maximal torus in a Borel, $\Delta$ are the simple roots, and each $X_\alpha$ is a nonzero root vector for $\alpha$.

A pinned automorphism is an automorphism of $G$ (equivalently of $\mathfrak g$) preserving $B$, $T$, and each $X_\alpha$.

In the letter, $\Omega$ is a group of such “pinning-preserving” automorphisms (with lattice conditions), and its contragredient action gives an action on the dual group .

Example: For many types, pinned automorphisms correspond to Dynkin diagram automorphisms.