If two Verma modules have the same set of composition factors, must they be the same/isomorphic?
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$\begingroup$ This question may be subtle, requiring for example methods of Jantzen developed in his Springer Lecture Notes in Mathemartics 750 (1979). What motivates the question itself? (The answer is of couorse no if the requirement that both modules are Verma modules is dropped, using the 'duality' functor introduced by BGG in the category $\mathcal{O}$. But a Verma module with repeated composition factors and a quotient module with lower multipicities may be hard to distinguish.) $\endgroup$– Jim HumphreysDec 23, 2018 at 19:27
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3$\begingroup$ Jim, the question is about distinguishing Verma modules themselves, not their quotients or more complicated modules, and so the answer is quite straightforward. (A nontrivial quotient of a Verma module cannot be a Verma module, because homomorphisms between Verma modules are injective.) $\endgroup$– Victor ProtsakDec 24, 2018 at 4:35
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$\begingroup$ @Victor: Sorry for misreading this question! It's apparently not at research-level. (Concerning the "same/isomorphic" choice, it depends on how you define "Verma module".) $\endgroup$– Jim HumphreysDec 24, 2018 at 16:51
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1 Answer
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The answer is affirmative. It is enough to provide a way to uniquely determine the highest weight $\lambda$ a Verma module $M(\lambda)$ from the composition factors of $M(\lambda)$. Every composition factor of $M(\lambda)$ is a simple highest weight module $L(\mu)$ with $\mu\leq\lambda$ in the dominant order, and $L(\lambda)$ is a quotient of $M(\lambda)$. It follows that the set of highest weights of the composition factors of $M(\lambda)$ has a unique maximal element $\lambda$.
answered Dec 24, 2018 at 4:23