# Dual versions of “folding” symmetric ADE Dynkin diagrams?

Start with the Dynkin diagram of an irreducible root system, typically associated with a simple Lie algebra over $\mathbb{C}$ or a simple algebraic group. Most of the simply-laced ADE diagrams admit natural symmetries of order 2: type $A_r$ when $r \geq 2$, type $D_r$ when $r \geq 4$, and type $E_6$. In the case of type $D_4$, there is also a symmetry of order 3. There is an obvious way to "fold" each diagram by identifying those vertices which are interchanged by the symmetry. This yields Dynkin diagrams with two root lengths. For example, $E_6 \leadsto F_4$.

What I don't see is any general theory dictating in advance how to interpret this process.
In particular, does a pair of simple roots (vertices) interchanged by the symmetry in type $E_6$ lead to a long or a short simple root in type $F_4$? In effect, there are (Langlands) dual possibilities, switching long and short roots while also switching types $B_r$ and $C_r$. It may be difficult to formulate a single theory of folding and duality which makes all instances look natural, but it's at least reasonable to ask:

What is the origin of the notion of "folding" for Dynkin diagrams?

It's well-known that the ADE graphs are ubiquitous in various areas of mathematics; for an older survey see the paper by Hazewinkel et al. here. The usual convention is to regard all vertices of the graph as corresponding to long simple roots. In their examples, the most common type of folding occurs this way: $A_{2r-1} \leadsto C_r, \: D_{r+1} \leadsto B_r, \: D_4 \leadsto G_2, \: E_6 \leadsto F_4$, where multiple simple roots are always taken to short simple roots. Examples occur in various parts of Lie theory: classification of real forms of complex semisimple Lie algebras, classification of $k$-forms of reductive $k$-isotropic algebraic groups (Satake, Tits), etc.

But in the study of simple singularities of types ADE (work of Grothendieck, Brieskorn, Slodowy), adapted to other Lie types, the natural folding procedure is dual to the above one: $A_{2r-1} \leadsto B_r, \: D_{r+1} \leadsto C_r, \: D_4 \leadsto G_2, \: E_6 \leadsto F_4$, where now multiple simple roots lead to long simple roots.

My own motivation comes from related work on representations of Lie algebras of simple algebraic groups in prime characteristic. Here the most elusive non-restricted representations belong to nilpotent elements (or orbits). The case of regular nilpotents is easy, but already the subregular case is too difficult for current algebraic methods to complete in case of two root lengths. (For large enough primes, the deep work of Bezrukavnikov, Mirkovic, and Rumynin has pushed farther.)

It's instructive to think of (dual) folding in connection with Dynkin curves (Springer fibers in the subregular case) and the detailed results of J.C. Jantzen in Subregular nilpotent representations of Lie algebras in prime characteristic, Represent. Theory 3 (1999), 153-222, freely available here. His incomplete dimension calculations involving two root lengths would become transparent if the folding process could be rigorously justified in advance.

• My guess is that there is a basic duality going on here. If memory serves, in the case of folding for Lie algebras, one recovers precisely the subalgebra fixed by the symmetry group (Z/2 most of the time, permutations of 3 things in the $D_4 \to G_2$ case). Could the "dual" cases correspond to times when you recover the cofixed points of the symmetry, rather than the fixed points? – Theo Johnson-Freyd Nov 4 '12 at 17:05
• Why look at roots instead of coroots? You can see the $F_4$ embedded in $E_6$, for example, by seeing each simple coroot of $F_4$ as a sum of one or two coroots from $E_6$. When one group embeds in another compatibly with max tori, e.g. $F_4$ in $E_6$, the map is covariant from coroot lattice of the small group to coroot lattice of the large group. – Marty Nov 4 '12 at 19:28
• @Theo and Marty: I agree that there is a strong flavor of both roots and coroots in the ways these foldings arise, plus sometimes a clear rationale for taking fixed points in the Lie algebra (or group). Asking about origins of these ideas is just a step toward unifying them, which may or may not be feasible. My last two paragraphs indicate the kind of problem where I get stuck, even though the folding is somehow implicit in the eventual results on representation theory. – Jim Humphreys Nov 4 '12 at 23:22
• geometers study these sorts of foldings - although for them the diagrams have slightly different meaning. IIRC, in the J.Tits' book on buildings of spherical type folding of diagrams is mentioned. The book books.google.com/books/about/… by A.Pasini has a chapter on foldings. – Dima Pasechnik Nov 5 '12 at 3:08