# Should the Dynkin diagrams of types $A_1$ and $B_2$ be labelled $C_1$ and $C_2$?

The labels $A$--$G$ attached to connected Dynkin diagrams are of course arbitrary, the result of historical accidents. In order to avoid repetitions, the four infinite families $A_\ell, B_\ell, C_\ell, D_\ell$ usually have rank subscripts limited respectively by $\ell \geq 1, 2, 3, 4$ (where $A_1 = C_1, \;B_2 = C_2$, etc.).
But sometimes it seems more natural to emphasize the entire family $C_\ell$ with $\ell \geq 1$.

Here are two examples which arise independently in areas of Lie theory remote from each other:

(A) The noncompact Lie groups $Sp(2n,\mathbb{R})$ of rank $n \geq 1$ have infinite cyclic topological fundamental groups: the topology of such a group is that of a maximal compact subgroup $K$ (Iwasawa), which here has a 1-torus as factor. We include the extreme case $n=1$, where $Sp(2,\mathbb{R}) = SL(2,\mathbb{R})$ has as a maximal compact subgroup the 1-torus $SO(2)$. But for other simple Lie types the fundamental group is always finite. This has implications for the existence of a complex structure on the symmetric space $G/K$. It goes back to the work of E. Cartan and is recounted by Helgason (is his book the optimal modern source for the calculation of fundamental groups?).

(B) In the 1980s Henning Andersen studied, over an algebraically closed field of characteristic $p>0$, the extensions involving two irreducible modules for a simple algebraic group $G$ and for its Frobenius kernels $G_r$. He found for the latter that such a module can extend itself nontrivially precisely when $r=1$, $p=2$, and $G$ is of type $C_\ell$ for $\ell \geq 1$. (Irreducible representations of the first Frobenius kernel agree with Lie algebra representations coming from rational representations of $G$. But extensions in the two settings require more delicate analysis.)

Inspired by this definitive result, along with Ext computations for the finite Chevalley groups $SL(2,q)$ (Andersen-Jorgensen-Landrock), I constructed self-extensions for the groups $Sp(4,q)$ of type $C_2$ over the prime field when $p$ is odd. More technical work by Chris Bendel, Dan Nakano, and Cornelius Pillen concluded (except for small primes where the question is undecided) that self-extensions occur precisely in type $C_\ell$ for $\ell \geq 1$, over a prime field.

In (B), but not obviously in (A), an essential role is played by one special feature of type $C_\ell$ root systems which distinguishes them from all others:

(*) For $C_\ell$ ($\ell \geq 1$) some simple root is twice a fundamental weight $\varpi_i$.

To expand on this point, note that the highest root $\widetilde{\alpha}$ (the highest weight of the adjoint representation) is always a fundamental weight except in types $A_\ell$ and $C_\ell$: indeed, $\widetilde{\alpha} = 2 \varpi_1$ in type $C_\ell$ (for all $\ell \geq 1$) using Bourbaki's numbering and tables with $C_1 = A_1$.

It would of course be interesting to see further examples illustrating the exceptional status of type $C$. But my main question is:

Is there any unified explanation for the special role played by Dynkin types $C_\ell$ with $\ell \geq 1$ in examples such as (A) and (B)?

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A small comment on your first paragraph that is in the theme of the rest of your question: one might prefer to start the series with B3 and C2 (instead of B2 and C3) because the extended Dynkin diagram for B2=C2 is of the same form as the rest of the C-series, but looks different than for B3, B4, etc. – Skip Jan 17 '11 at 15:05
Another reason to start the series B at 3 and C at 2 (i.e., call the root system C2 instead of B2, even though both mean the same thing) is that the Dynkin index of the (split) simply connected group of type $C_n$---i.e., $Sp_{2n}$---is 1 for all $n$, but the Dynkin index of $B_n$---i.e., $Spin_{2n+1}$---is 2 for $n \ge 3$. – Skip Jan 19 '11 at 3:48
@Skip: can you please define what you mean by Dynkin index? – fherzig Oct 27 '12 at 21:01
I just wanted to ask precisely the same question but good old Jim Humphreys already asked it! – Bugs Bunny Oct 2 '14 at 8:10
@Skip: Your points are well taken. Usually $B_2, C_2$ are considered interchangeable, but the behavior of fundamental weights and the highest root makes $C_2$ a better label. – Jim Humphreys Jun 13 at 23:10