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Let $G$ be a compact Lie group with Lie algebra $\mathfrak{g}$, and suppose $\mathfrak{g}$ is semisimple.

An integral weight for $G$ is an element $\lambda \in \mathfrak{t}^*$ with $\lambda(H_\alpha) \in \mathbb{Z}$ for each coroot $H_\alpha$, where $\alpha$ is a positive root of $G$.

Let $T$ be a maximal torus in $G$. The set of all integral weights for $G$ forms a lattice in $\mathfrak{t}^*$, and the theorem of the highest weight tells us that dominant integral weights $\lambda$ are in one-to-one correspondence with irreducible representations $\pi_\lambda$ of $G$ of highest weight $\lambda$. We also know that the weights that occur with non-zero multiplicity in $\pi_\lambda$ all lie inside the convex hull of the points $w(\lambda)$ for $w$ in the Weyl group $W(G)$, and that any weight occurring with positive multiplicity can be written as $\lambda$ plus an integer combination of the roots of $G$.

I'm wondering what happens if we consider a subgroup $H \subset G$ of the same rank which is not semisimple.

Am I correct that the set of integral weights of $H$ is no longer necessarily a lattice in $\mathfrak{t}^*$? It seems so, since $H$ has fewer roots than $G$ so there are fewer constraints of the form $\lambda(H_\alpha) \in \mathbb{Z}$.

If so, then let $\mu$ be an integral weight for $H$ but not for $G$.

My question (finally) is:

  • Can $\mu$ be the highest weight of some irreducible representation of $H$? Must it?
  • If it is, are the other weights appearing in that representation necessarily $\mu$ plus an integer combination of roots of $H$?
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