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The classification of simple Lie algebras (over $\mathbb{C}$ or other sufficiently large field of characteristic 0) correlates these Lie algebras with the irreducible reduced root systems (in Bourbaki's language). Here there are at most two possible lengths of roots, relative to a euclidean metric arising from the nondegenerate Killing form restricted to a fixed but arbitrary Cartan subalgebra. In the "simply-laced" types ADE, all roots have equal length, whereas in other types BCFG there are both "long" and "short" roots.

At least since the early work of Dynkin, it has been conventional to say that all roots in types ADE are "long" (though it would be possible to say all are "short", or else just avoid the distinction here). For example, Dynkin's method for drawing his diagrams uses open circles for long simple roots, filled-in circles for short simple roots.

Is there a most natural rationale for the convention that all roots are long in types ADE?

One situation in which the convention seems handy involves the classification of the finitely many nilpotent orbits in the Lie algebra (which also goes back to Dynkin). Here one finds a unique minimal nonzero orbit, characterized as the set of root vectors corresponding to long roots (for any choice of Cartan subalgebra). But I'm uncertain about the main motivation behind the convention (and its history).

ADDED: I was thinking at first just about the algebraic theory of simple Lie algebras, which can be studied over any splitting field of characteristic 0 in terms of root systems and Weyl groups. Over $\mathbb{C}$ these arise classically as complexifications of various real Lie algebras belonging to Lie groups. In this direction the answer (resp. comment) by Andre (resp. Henrik) are both useful. The paper by Jeff Adams is a unification of previous case-by-case study, working in a traditional Lie group setting, whereas the KNR paper also brings in some more modern ideas as well as tools from algebraic geometry. I can accept such an answer but should wait a little longer in case someone else can suggest an answer in the purely algebraic setting I've sketched. (By the way, in his papers from about 1946 to 1950, Dynkin's convention for labelling vertices seems to have evolved. Maybe this was influenced by his classification of nilpotent orbits, including the description of the minimal nonzero orbit?)

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    $\begingroup$ I know from papers by J. Adams and G. Prasad that long roots have the property that their root subgroups detect nontriviality in coverings of simple Lie groups $\tilde{G}\rightarrow G$, and short root subgroups do not have this property (for Dynkin diagrams that have both). In ADE, all roots detect nontriviality in this way. I tried to turn this into an answer, but couldn't find an elegant formulation. $\endgroup$ – Henrik Winther Jan 14 '15 at 15:41
  • $\begingroup$ @Henrik: This is unfamiliar to me. Can you add a reference or two? $\endgroup$ – Jim Humphreys Jan 14 '15 at 20:16
  • $\begingroup$ @HenrikWinther: Since the induced maps between root groups for central isogenies are isomorphisms, what do you mean by "detect"? $\endgroup$ – user74230 Jan 14 '15 at 20:44
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    $\begingroup$ @JimHumphreys I'm refering to J. Adams "Non-Linear Covers of Real Groups", theorem 1.5 and 1.6. Here is a link: math.umd.edu/~jda/preprints/groupswithcovers.pdf $\endgroup$ – Henrik Winther Jan 14 '15 at 21:42
  • $\begingroup$ This is probably naïve, but could it just be because $A_n$ and $D_n$ can always be realised as sets of long roots, but not always as sets of short roots, in a non-simply laced root system? (I don't know $E_8$ well enough to know whether this holds for the $E$ series as well.) $\endgroup$ – LSpice Jan 14 '15 at 22:54
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Let us define a coroot $\alpha^\vee \in \mathfrak g$ to be `short' if the corresponding group homomorphism $SU(2)\to G$ generates $\pi_3(G)$ (the homomorphism has Dynkin index $1$).

With the above definition, in the ADE case, all coroots are short.

Dually, it is then reasonable to call all roots `long' in the ADE case.

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  • $\begingroup$ This is intriguing, but what are the precise conditions on $G$? It would help to see more details, maybe in a reference. $\endgroup$ – Jim Humphreys Jan 15 '15 at 13:19
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    $\begingroup$ This works under the assumption that $G$ is a real, compact, simply connected Lie group. You can easily get a similar statement in the world of complex reductive groups over $\mathbb C$ by replacing $SU(2)$ by $SL(2,\mathbb C)$. A discussion of the relation between Dynkin index and $\pi_3$ can be found in the Kumar-Narasimhan-Ramanathan paper "Infinite Grassmannians and moduli spaces of G-bundles". $\endgroup$ – André Henriques Jan 15 '15 at 17:53
  • $\begingroup$ Thanks for the details. The KNR paper is here: gdz.sub.uni-goettingen.de/dms/load/img/?PPN=GDZPPN00233979X $\endgroup$ – Jim Humphreys Jan 16 '15 at 14:18
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I like being able to say "the highest root is always long".

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  • $\begingroup$ This is close to my comment about the minimal nonzero nilpotent orbit (which some people just characterize as the orbit of a highest root vector). But it's almost as easy to say "the highest root is long if there are two root lengths". Is there a more compelling rationale for the convention in ADE types? My intuition is that "long" and "short" are meaningless when there is just one length; but the convention seems well established. $\endgroup$ – Jim Humphreys Jan 14 '15 at 20:24

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