Take the complex semisimple Lie algebra $\frak{sl}_{n+1}$, with space of dominant integral weights $P(\frak{sl}_{n+1})$. For $V(\lambda)$ the irreducible representation corresponding to $\lambda \in P(\frak{sl}_{n+1})$, for which $\lambda$'s does it happen that the degree zero weight space of $V(\lambda)$ is non-trivial? I am also interested in this question for the other series.
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4$\begingroup$ This happens exactly when lambda belongs to the coroot lattice. (Incidentally my phone keeps autocorrecting “coroot” to “corporate.”) $\endgroup$– Sam HopkinsJun 6, 2022 at 12:48
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3$\begingroup$ Based on the last couple questions you’ve asked, I think you could benefit from reading a standard treatment of the theory of simple Lie algebras and their representations, like Bourbaki or Humphreys. $\endgroup$– Sam HopkinsJun 6, 2022 at 12:57
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1$\begingroup$ @Sam Hopkins: In general, it is the root lattice, not the coroot lattice. $\endgroup$– Friedrich KnopJun 6, 2022 at 18:50
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$\begingroup$ @FriedrichKnop: whoops, you are of course right. (It is easy to get these things mixed up, which is why it is nice to go back to a standard source like Bourbaki...) $\endgroup$– Sam HopkinsJun 6, 2022 at 18:55
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