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The question involved here is natural and very classical, but I'm unsure what has been formally stated and proved in the literature. The only approach I know involves assembling facts that apparently weren't all known until the 1970s.

Start with a simple Lie algebra $\mathfrak{g}$ over $\mathbb{C}$, along with some Cartan subalgebra $\mathfrak{h}$ and the resulting weight lattice $X$ and root sublattice $X_r$ inside $\mathfrak{h}^*$ (Bourbaki's $P \supset Q$). Fix simple roots $\alpha_1, \dots, \alpha_\ell$ and corresponding fundamental dominant weights $\varpi_1, \dots, \varpi_\ell$. [The symbol \varpi gives an old version of the handwritten letter pi, for "poids".] Finite dimensional simple modules $L(\lambda)$ are then parametrized by dominant weights $\lambda$ with $L(0)$ the trivial module, while arbitrary finite dimensional modules are direct sums of these. Everything here just depends up to isomorphism on $\mathfrak{g}$, by standard conjugacy theorems.

Now the natural problem is: Find the smallest nontrivial (hence faithful) finite dimensional modules. By complete reducibility, it's enough to consider simple modules. A detailed answer requires the Killing-Cartan classification of simple Lie algebras, but there is a classification-free result:

All weights of $L(\lambda)$ lie in a single coset of $X_r$ in $X$, and these weights consist of Weyl group orbits of various dominant $\mu \leq \lambda$ in the standard ordering (possibly with multiplicity $>1$ if $\mu < \lambda$).

(1) Fix a coset not containing 0. Then there is a unique smallest nontrivial $L(\lambda)$ corresponding to a "minuscule" $\lambda$ (always one of the fundamental weights), with weights consisting of the Weyl group orbit of $\lambda$.

(2) Fix the coset containing 0. Then there is a unique smallest nontrivial $L(\lambda)$, whose weights consist of 0 together with the Weyl group orbit of $\lambda$. Here $\lambda^\vee$ is the highest root in the dual root system.

Is this written down and proved somewhere?

The history might also be interesting (or just messy). Once this result is in hand, it's not difficult to fill in the details for each simple Lie algebra, but much of that is found only in exercises of books by Bourbaki or me.

REFERENCES: As I commented to Sasha Premet, I've only seen fragments of the story in earlier textbooks. There is a short note by H. Freudenthal in Proc. Amer. Math. Soc. 7 (1956), 175-176, concerning occurrence of the 0 weight; but this is superseded by later results. For general background and details on (1) and (2), I'll refer to [B1] = Bourbaki, Chap. 6 (1968), my book [H] = GTM 9 (1972), [B2] = Bourbaki, Chap. 8 (1975).

[B1]: $\S1$, Exer. 24, and $\S4$, Exer. 15.

[H]: Exer. 13.10 (just using root system axioms) and Prop. 21.3 (using basic representation theory), Exer. 21.3. Also Exer. 13.13 ("minimal nonzero" = "minuscule").

[B2]: $\S7.2$, $\S7.3$, and $\S7$, Exer. 22.

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Something a little weird. The set of cosets $X/X_r$ is $Z(G)^*$, where $G$ is the simply-connected group. The set of minuscule representations plus the trivial representation corresponds to the pointiest corners of the Weyl alcove, which in turn correspond to $Z(G)$. So your (1) seems to be asserting a bijection between $Z(G)$ and its dual. Right? –  Allen Knutson Oct 10 '12 at 13:09
    
Yes, this involves the whole story about affine Weyl groups relative to the root system and its dual: see [B1] $\S6.2$ and $\S4.9$ of my book on Coxeter groups along with the papers by Verma and Iwahori-Matsumoto. Everything here is connected to everything else, but for the representation theory it's hard to find a conceptual pathway through the maze. –  Jim Humphreys Oct 15 '12 at 22:01
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1 Answer

up vote 4 down vote accepted

In Bourbaki, Ch. VIII, $\S$ 7, Sect. 2, one can find the notion of an $\mbox{$R$-saturated set}$, and Corollary to Prop. 4 in that section proves that for every $R$-saturated set $\mathcal X$ there is a finite dimensional $\mathfrak g$-module whose set of weights coincides with $\mathcal X$. Prop. 6 in the next sections proves that the smallest $R$-saturated sets have the form $W.\lambda$ where $\lambda$ is minuscule. This answers Question (1). For Question (2) we take for $\mathcal X$ the union of $0$ and the set of all short roots in $R$. It is easy to see that this is an $R$-saturated set as verifying this reduces to root systems of rank 2 where everything is clear (even in type ${\rm G}_2$). Applying the above-mentioned Corollary we get a $\mathfrak g$-module with desired properties. If $\beta$ is the dominant short root then, by construction, our set will coincide with the set of weigths of $L(\beta)$ as that set cannot be any smaller.

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Thanks for assembling these details, which I admit I deliberately omitted in hopes of finding somewhere a more explicit formulation. I've added to my already long question a summary of the bits and pieces I've found in older literature. –  Jim Humphreys Oct 3 '12 at 12:16
    
Perhaps it is also interesting to comment on the above subquestion of Jim, namely to find a smallest faithful finite dimensional module for a given Lie algebra - which always exists by Ado's theorem (that is, a faithful module of minimal dimension). This has been studied in connection with fundamental groups of affinely flat manifolds (Milnor, Margulis, Abels, Soifer etc.). For complex reductive Lie algebras this minimal dimension has been determined. –  Dietrich Burde Mar 27 '13 at 12:31
    
@Sasha: I'm still looking for a more explicit representation-theoretic statement in the literature but did find your answer useful. –  Jim Humphreys Apr 28 '13 at 22:12
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