The action of the center on the extended Dynkin diagram

Question

Let $R$ be an irreducible root system with a basis $\Pi$. We obtain the Dynkin diagram $D$ and the extended Dynkin diagram ${\widetilde{D}}$ of $R$ with respect to $\Pi$. Let $Q^\vee\subset P^\vee$ denote the coroot lattice and the coweight lattice, respectively. It is known that the finite abelian group $P^\vee/Q^\vee$ (which is isomorphic to the center of the corresponding simply connected compact Lie group) acts on ${\widetilde{D}}$.

I am looking for references where the action of $P^\vee/Q^\vee$ on ${\widetilde{D}}$ is described in detail (preferably with examples, say for a root system of type $D_n$).

Answer 1

The question is perhaps best answered not in the context of Lie theory but in the related setting of affine Weyl groups, where an irreducible root system in Bourbaki's sense leads to an extended Dynkin diagram and corresponding Coxeter group. Here the isomorphic finite abelian groups $P/Q$ and $P^\vee/Q^\vee$ may be interpreted in Lie theory as fundamental groups of related adjoint compact Lie groups or as centers of their simply connected covers. But this interpretation may not shed much light on the question asked.

Probably the earliest detailed reference is the first part of the 1965 IHES paper by Iwahori and Matsumoto, which is usually (though not at the moment) available online through numdam.org: Iwahori-Matsumoto. In particular, they work out the complicated details for each Lie type, though this requires a large amount of notation. Here as elsewhere in the literature (Bourbaki for instance), notation varies a lot but is essential for understanding the situation.

Another (harder-to-locate) source, inspired by Iwahori-Matsumoto but using somewhat different notation due to the suggested applications in modular representation theory, is the first part of the conference write-up by D.N. Verma. This was published only in 1975 but reflects some of the talks given at the 1971 Budapest summer school on Lie groups: Verma. One advantage of Verma's exposition is that he gives a thorough account of how the finite abelian group acts on the extended Dynkin diagram, though without examples.

In both of these accounts the focus is on an affine Weyl group (a Coxeter group) along with a usually larger extended affine Weyl group. The resulting finite quotient group $\Omega$ is isomorphic to the above fundamental group but has the merit of being an explicit group of affine transformations which preserves the fundamental alcove (or simplex). The closure of this alcove is a fundamental domain for the affine Weyl group, and the vertices are in bijection with the set of simple reflections along with 0, or equivalently with the $\ell+1$ generators of the group (where $\ell$ is the rank of the underlying root system). While $\Omega$ preserves this alcove, it permutes the $\ell+1$ vertices in a way described in both papers cited. This realizes the action of $\Omega$ on the vertices of the extended Dynkin diagram. Here the elements of $\Omega$ are in natural bijection with the minuscule fundamental weights (along with 0): such weights correspond to simple roots having coefficient 1 in the expression of the highest root and come up frequently in representation theory.

ADDED: Concerning type $D_n$, the summary in Iwahori-Matsumoto (see page 19) naturally varies for $n$ even or odd (where the structure of $\Omega$ differs), but the permutation of vertices in each case is fairly natural even though hard to guess in advance.

Answer 2

Using the correspondences $K/Ad\ K \cong T/W \cong (\mathfrak t/Q^\vee)/W \cong \mathfrak t/(Q^\vee \rtimes W) = \mathfrak t/\hat W =: A$, you can think about the Weyl alcove $A$ as parametrizing the conjugacy classes in the simply connected compact Lie group $K$. (Note that the penultimate equality only holds for $K$ simply connected.)

$Z(K)$ obviously acts on the space of conjugacy classes by multiplication, and since $Z(K) \leq T$ we can pick logarithms of $Z(K)$ inside $\mathfrak t$ to implement this action by translation on $\mathfrak t$, and therefore act on $A$ by rigid motions.

The vertices (and facets) of $A$ correspond to the nodes of the affine Dynkin diagram, so a rigid-motion action (hence angle-preserving) gives an action on the diagram.

It's not obvious to me why (other than the classification) an affine Dynkin diaram automorphism is determined by where the "central" vertices go (those in the orbit of the affine vertex), but it's easily checked. EDIT: As Jim Humphreys mentions, those vertices are the ones corresponding to minuscule fundamental representations.

So now it's enough, given an element $z$ of $Z(K)$, to figure out which is the corresponding central vertex of $A$. I guess that amounts to picking a logarithm $X \in \mathfrak t$ with $\exp(X)=z$, and seeing where $X$ goes under the folding-up map $\mathfrak t \to A$. I'm a little afraid that this final step is exactly your question and that this may be an unsatisfying answer; maybe someone can provide a better one.

I believe the canonical reference for these ideas is Borel-de Siebenthal.