# Distinguished dominant integral weight related to a branching problem

Let $$G$$ be a simple compact connected Lie group and let $$K$$ be a connected closed subgroup of $$G$$. Let $$\widehat G$$ and $$\widehat K$$ denote the corresponding unitary duals, that is, the (equivalence classes of) irreducible representations. By the Highest Weight Theorem, elements in $$\widehat G$$ are in correspondence with elements in the set $$P^+(G)$$ of $$G$$-integral dominant weights. For $$\lambda\in P^+(G)$$, we will denote by $$\pi_\lambda$$ the irreducible representation of $$G$$ with highest weight $$\lambda$$.

Given $$\tau\in\widehat K$$, we define $$\widehat G_\tau=\{\pi\in\widehat G: \textrm{Hom}_K(\pi,\tau)\neq0\}$$. In other words, $$\pi$$ is in $$\widehat G_\tau$$ if and only if $$\tau$$ appears in the decomposition in irreducible representations of the restriction of $$\pi|_K$$ (the restriction of $$\pi$$ to $$K$$). Elements in $$\widehat G_\tau$$ are called $$\tau$$-spherical representations.

I wish to know if there is $$\omega\in P^+(G)$$, with $$\omega\neq0$$, satisfying the following property:

if $$\lambda\in P^+(G)$$ satisfies $$\pi_\lambda\in\widehat G_\tau$$, then $$\pi_{\lambda+\omega}\in\widehat G_\tau$$.

Here I list some examples coming from compact symmetric spaces of real rank one. In each case I used the obvious inclusion $$K\subset G$$ and standard conventions on the root system of $$\mathfrak g$$:

1. $$G$$ arbitrary and $$K=\{1\}$$, $$\omega =$$ any choice in $$P^+(G)$$.
2. $$G=SU(2)$$ and $$K=T$$ (maximal torus), $$\omega = 2\varepsilon_1$$ (twice the first fundamental weight).
3. $$G=SO(n+1)$$ and $$K=SO(n)$$, $$\omega=\varepsilon_1$$ (the first fundamental weight).
4. $$G=SU(n+1)$$ and $$K=S(U(n)\times U(1))$$, $$\omega=\varepsilon_1-\varepsilon_{n+1}$$ (the first plus the last fundamental weights).
5. $$G=Sp(n+1)$$ and $$K=Sp(n)$$, $$\omega=\varepsilon_1+\varepsilon_2$$ (the second fundamental weight).
6. $$G=SO(2n)$$ and $$K=T$$ (max. torus), $$\omega= 2\varepsilon_1$$ (twice the first fundamental weight).

Each of the above examples follows by the explicit branching law from $$G$$ to $$K$$, which is (of course) not available for arbitrary $$K$$. Clearly, in each case, any positive multiple of the indicated $$\omega$$ also works.

In case the answer is affirmative, it follows that $$\widehat G_\tau$$ can always be written as a union of subsets of the form $$\{\pi_{\lambda+k\omega}: k\geq0 \}$$ for some $$\lambda\in P^+(G)$$. (The union is not assumed finite or either disjoint.) This is the result I am looking for.

For $$G=SO(2n)$$. Does a positive multiple of $$\varepsilon_1$$ satisfy the above property for any closed subgroup $$K$$ of $$G$$?

Either a proof or a counterexample would help me to understand this problem. I picked $$SO(2n)$$ because is my favorite, but we may replace $$(SO(2n),\varepsilon_1)$$ in the question by $$(SO(2n+1),\varepsilon_1)$$, $$(SU(n+1),\varepsilon_1-\varepsilon_{n+1})$$, or $$(Sp(n+1),\varepsilon_1+\varepsilon_2)$$.

• You probably meant $K=T$, the maximal torus of $G$, in examples 1 and 2. If $K=\{1\}$ then any representation of $G$ is spherical, so the property trivially holds for any $\omega$. Also, you should exclude the dominant weight $\omega=0$, again, the property holds trivially. Apr 11 '19 at 5:19
• Although I had written what I wanted, it was confusing and your idea is better. I have just edited. Thanks! Apr 11 '19 at 10:04

It is true that there exists such a $$\omega$$. Indeed, $$every$$ $$\omega \neq 0$$ works. Fix a representation $$\pi _{\omega}$$ which has a nonzero $$K$$ fixed vector, with $$\omega \neq 0$$. One can replace $$G,K$$ by their complexifications. Fix a Borel subgroup $$B$$ of $$G(\mathbb C)$$ and view the vectors in the representations $$\pi _{\lambda}$$ as algebraic sections $$\phi (x)$$ of a homogeneous line bundle $${\mathcal L}_{\lambda}$$ on $$G/B$$ (the Borel-Weil theorem) . Fix a section $$\phi (x)$$ whose $$K$$ translates generate a representation of $$K(\mathbb C)$$ isomorphic to $$\tau$$.
If $$\psi (x)$$ is the non-zero $$K$$ fixed vector in $$\pi _{\omega}$$ viewed as a section of $${\mathcal L}_{\omega}$$, then the product $$\phi (x)\psi (x)$$ is a section of $${\mathcal L}_{\lambda +\omega}$$. But the $$K$$ translates of the product $$\phi \psi$$ generate a representation also isomorphic to $$\tau$$ since $$\psi$$ is $$K$$ invariant. Hence the restriction of $$\pi _{\lambda +\omega}$$ to $$K$$ contains $$\tau$$.
• So, the highest weight of any non-trivial representation in $\widehat G_{1_K}$ works ($1_K=$ trivial representation of $K$). In particular, the set of highest weight of elements in $\widehat G_{1_K}$ is a monoid ($0$ is in it, and the sum of any elements in it is again in it). Is all this right? In the affirmative case, thanks! Apr 12 '19 at 11:24
• In fact, the situation is controlled by an affine semigroup $A_{G,K}$ contained in $P(G)^{+}\times P(K)^{+}$ formed by the pairs $(\lambda,\tau)$ for which the restriction of the simple $G$-module with highest weight $\lambda$ to $K$ contains the simple $K$-module with highest weight $\tau$. This branching semigroup provides a multigrading on the branching algebra of $K$-highest weight vectors in $R(G/U)$. See the work on branching algebras by Roger Howe and others. Apr 13 '19 at 5:16
• @Venkataramana: of course, your proof works. I was pointing out the additional structure present, related to the monoid OP asked about, and commented on references. Viewing all line bundles $\mathcal{L}_\lambda$ simultaneously by considering $R(G/U)$ with the grading coming from the right $T$-action instead of individual spaces $\Gamma(G/B,\mathcal{L}_\lambda)$ streamlines the description and the proof. Apr 13 '19 at 15:48