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Suppose you are given a linear combination of points of $\mathbb{Z}^n$ which corresponds to the weight datum of a nontrivial representation $V$ of some simple Lie algebra $g$. Is it possible to reconstruct $g$ and $V$ from this data, without knowing the canonical form on the weight lattice? (It looks like there might be some problems with dual Lie algebras --- can you at least find the pair $g, g^{\vee}$?)

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By "linear combination of points of $\mathbb{Z}^n$" are you referring to the full set of weights of $V$ or some subset? –  ARupinski Mar 24 '11 at 21:29
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The fundamental representation of $\mathfrak{g}=\mathfrak{sl}_n$ and its dual cannot be distinguished in this way; the same applies to any non-self-dual representation of $\mathfrak{g}$ and its dual. Similarly, the defining representations of $\mathfrak{sp}_{2n}$ and $\mathfrak{o}_{2n}$ are indistinguishable, and these Lie algebras are not the Langlands duals of each other, because they are of types $C$ and $D.$ –  Victor Protsak Mar 24 '11 at 21:46
    
The original tag 'weights' is usually applied to other senses of the word, so I edited this and also added 'simple' to clarify which class of Lie algebras you are discussing. Aside from that, I agree with other comments that the question needs to be better focused since there are problems with the formulation. Some added motivation might help too, since the finite dimensional representations of simple Lie algebras (over the complex numbers) have been so well studied from many viewpoints. –  Jim Humphreys Mar 24 '11 at 22:05
    
Thanks Victor! These are indeed counterexamples to my (intended) question. –  Dmitry Vaintrob Jun 11 '12 at 10:20
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I'll write this as an answer rather than a comment to close the question. Victor Protsak's comment gives the original question a definitive answer of no.

The formulation of the original question is indeed fuzzy, as is my knowledge of representation theory. I was interested in characters of (not necessarily simple) representations of simple Lie algebras as formal combinations of points in the full root lattice up to automorphisms of the lattice (ignoring the quadratic form). The question comes from number theory: we were trying to efficiently reconstruct the Zariski closure of the image of a Galois representation knowing its Frobenius polynomials at primes of good reduction. I'll close the question, but if the experts know a good answer, I'd be curious to know how far the indeterminacy can go (i.e., say, is there some relationship between all simple representations of a given simple algebra whose collections of weights are related by automorphisms of the root lattice).

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