Highest weights of the restriction of an irreducible representation of a simple group to a Levi subgroup - MathOverflow

Highest weights of the restriction of an irreducible representation of a simple group to a Levi subgroup

2010-08-16T21:15:22Z

Let $G$ be a simple Lie group over ${\mathbb C}$, $P \subset G$ a parabolic subgroup, and $L \subset P$ its Levi subgroup. Let $\lambda$ be a $G$-dominant weight and $V_G^\lambda$ an irreducible representation of $G$ with highest weight $\lambda$. I am interested in restrictions on highest weights of irreducible components of the restriction $$ (V_G^\lambda)_{|L}. $$

In the simplest case, when $P = B$ is the Borel subgroup, $L = T$ is the maximal torus, and there is a well-known restriction --- the weights of $V_G^\lambda$ all lie in $$ Conv(\lbrace w\lambda \rbrace_{w \in W}), $$ where $W$ is the Weyl group. I would be happy to know something of the same sort for arbitrary parabolic subgroup.

Answer by Jim Humphreys for Highest weights of the restriction of an irreducible representation of a simple group to a Levi subgroup

2010-08-16T22:07:34Z

The standard classical question concerns multiplicities of the irreducible representations of $L$ (or its derived group) in the restriction: these are given by branching rules. This is complicated to work out in detail but is treated in many textbooks and other sources. I'm not sure exactly how much information you are asking for. The original $W$ -invariant set of weights relative to $T \subset L$ is unchanged, but the original representation decomposes into a direct sum of irreducibles for $L$ , with the subgroup $W_L \subset W$ acting on the separate weight diagrams within the convex region you describe. (It's easy to picture this in restricting from rank 2 to rank 1, for example.)

By the way, your $G$ -dominant weight means a weight of $T$ which is dominant relative to a fixed Borel subgroup $B$ for which $T \subset B \subset P$ .

Answer by Allen Knutson for Highest weights of the restriction of an irreducible representation of a simple group to a Levi subgroup

2010-08-17T14:51:12Z

This is not a complete answer, but maybe it will be of use.

You're asking which $L$-high weights $\mu$ occur in the $G$-irrep $V_\lambda$. Let me say that $\mu$ occurs classically if for some $N>0$, $N\mu$ occurs in $V_{N\lambda}$.

The set of such $\mu$ form a rational polytope lying inside $L$'s positive Weyl chamber.
The vertices of this polytope strictly inside $L$'s chamber ("regular vertices") are exactly those of the form $w\cdot \lambda$ that are lucky enough to be in there.
The vertices of this polytope lying on $L$'s Weyl walls are very likely to be very complicated. In particular they may not be integral weights of $L$. As I recall this already happens for $GL(3) \supset GL(2)\times GL(1)$.

Parts 1 & 3 apply to any branching problem (and much further). Part 2 is special to your case that $L$ has the same rank as $G$ (I'm not actually using that it's a Levi).

If all you want is an upper bound, as your comment to Jim suggests, then that's easy: the $L$-high weights that can occur are a subset of the $T$-weights that occur, which you already described. Probably you want something better than that though. In principle it wouldn't be too hard to figure out the local structure of your polytope nearby the regular vertices, but I expect that not all facets contain regular vertices.

Littelmann describes (in the case of a Levi) the highest weights that occur and their multiplicities: one looks at all the Littelmann paths for the irrep $V_\lambda$ that lie entirely inside the closed $L$-chamber.