on the center of a Lie group

Question

I'm trying to set straight my various pieces of knowledge about the center of a compact Lie group, and I'm running in circles...

First some definitions:

• Let $G$ be compact, simple, and simply connected Lie group, with Dynkin diagram $\Gamma$.
• Let $\{\alpha_i\}_{i=1...n}$ be the simple roots of $G$; they form by definition the vertex set of $\Gamma$
(sometimes, I'll write just $i$ instead of $\alpha_i$ for a vertex of $\Gamma$).
• Let $\Gamma^e=\Gamma\cup\{\alpha_0\}$ be the extended Dynkin diagram, where one adds the lowest root to $\Gamma$.
• Let $Z(\Gamma^e)$ be the subset of $\Gamma^e$ defined as the orbit of $\alpha_0$ under the automorphism group of $\Gamma^e$.
• Let $Z(G)$ be the center of $G$, which is also the quotient $\Lambda_{coweight}/\Lambda_{coroot}$.
• Let $\omega^\vee_i\in\Lambda_{coweight}$ be the fundamental coweights, defined by $\alpha_i(\omega^\vee_j)=\delta_{ij}$.
Let us also agree that, by convention, $\omega^\vee_0:=0$.

Then my question:

Is it true (and if so, where can I find a proof of this fact) that $$\{\omega^\vee_i \,|\, i\in Z(\Gamma^e)\}$$ forms a complete set of representatives of $\Lambda_{coweight}/\Lambda_{coroot}$.

Answer 1

You probably won't find exactly this formulation in any of the sources, which differ considerably in style, notation, motivation. As Dave comments, highest weights of minuscule representations (which correspond here to coweights) play a key role in the combinatorics. Searching MO for "minuscule weight" will return some entries with references for example to Bourbaki, for instance here.

The connections with affine Weyl groups and extended Dynkin diagrams go back quite a ways, but one of the earliest coherent accounts is given in the first section of a classic 1965 paper by Iwahori and Matsumoto here. This was adapted by Verma in his contribution to the proceedings of the Budapest summer school on Lie groups (1971), published some years later. There is a similar account at the end of Chapter 4 in my 1990 textbook Reflection Groups and Coxeter Groups, with references. Both Iwahori-Matsumoto and Verma work with automorphisms of the extended Dynkin diagram in the spirit of the question here, whereas Bourbaki deals more with minuscule weights. In any case these weights are indexed by certain vertices of the Dynkin diagram and provide coset representatives for all nontrivial cosets of the root lattice in the weight lattice. The quotient group is isomorphic to the center of $G$.

In the context of compact Lie groups and their representations, related material on the center (less explicitly formulated than you want) occurs in Chapter V.7 of the text by Brocker and tom Dieck Representations of Compact Lie Groups, Springer GTM 98, 1985.

One other remark on your terminology: when you refer at the end to "complete set of representatives" I guess you are implicitly describing the group often denoted $\Omega$ in the cited literature. Its order is what Bourbaki calls the "index of connection" $f$, the index of the root lattice in the weight lattice (or the dual formulation). This group is implicitly the same as the center of the compact Lie group, which doesn't figure explicitly enough in your question.

Answer 2

I have found a reference that contains the exact statement which I wanted: It is Theorem 3, on page 14 of this paper of Stephen Sawin.

The vertices of the Weyl alcove which are in bijection with $Z(G)$ are called sharp corners (for self-explanatory reasons). The other corners are called dull corners.

Answer 3

You may find complete answer to your question in Wallach, Harmonic Analysis on homogeneous spaces, chapter 4, best regards jorge