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In a Dynkin diagram there is a special involution often constructed as follows: Let $C$ be the Weyl chamber corresponding to our diagram. There is a unique element of the Weyl group $w$ for which $wC = -C$. Then $-w$ preserves $C$ and so permutes the simple roots giving us a diagram automorphism (indeed an involution). For most diagrams this is just the identity except $D_n$, ($n$ odd) where it swaps the two end nodes and $A_n$, $E_6$ where it is the natural symmetry of the diagram.

This involution is quite handy as applying it to decorated Dynkin (or Satake) diagrams capture the notion of duality. Firstly, if we use use the diagram to indicate a representation by placing integers over the nodes, applying the involution gives us the dual representation. Secondly, if we represent parabolic subalgebras (or their conjugacy classes) by crossing nodes on the diagram, the involution gives us the dual class containing the complementary/opposite parabolic subalgebras.

I have seen this involution referred to in many places but cannot recall ever seeing a name for it. Does anyone know of one?

Callum
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As I had said in a comment, this special involution is also quite helpful in the classification of forms of simple Lie algebras over non-algebraically closed fields, because any Satake diagram a.k.a. Tits index has to be stable under it. I opt for calling it the opposition involution, following the following sources:

"involution d'opposition" at end of 2.1 in Borel / Tits: Groupes réductifs Publications Mathématiques de l'IHÉS, Tome 27 (1965), pp. 55-151.

"opposition involution" in 1.5.1 (page 36 = 41 of the pdf file) of Tits' article in the Boulder proceedings (1965)

"opposition involution" in 20.10 of Milne's Course Notes on reductive groups

as well as various hits at google (in particular an article by T. A. Springer from 1987, The classification of involutions of simple algebraic groups, at the end of 1.6) and right here on MSE, user Moishe Kohan in a comment to Finite-dimensional modules of the lie algebra $\frak{so}(n)$.

I strongly suspect this term is also used in T. A. Springer's textbook on Linear Algebraic Groups, which unfortunately I do not have access to right now.

Full disclosure: I also called it "opposition involution" in 4.3.1 of my thesis, so I hope this use continues.

Torsten Schoeneberg
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