# highest weight representations inside tensor product

Let $G$ be a semisimple simply connected group over an algebraically closed field $k$ of characteristic zero, $B$ a Borel and $T$ a maximal torus. Let $\lambda,\mu,\nu$ be dominant characters of $T$. Let $V(\lambda)$ be the irreducible representation of highest weight $\lambda$.

If $\lambda=\mu+\nu$, then we know that $V(\lambda)$ is the direct factor of multiplicity one inside $V(\mu)\otimes V(\nu)$.

Is it possible to characterize $V(\lambda)$ inside this tensor product ?

• What kind of characterisation do you hope for? – Vít Tuček Jan 12 '15 at 9:02
• You seem to be assuming $k$ to have characteristic $0$, is this correct (in which case it would be a good idea to add it to the question). – Tobias Kildetoft Jan 12 '15 at 9:10
• for example there is a finite group acting on the tensor product wich eliminates everything but $V(\lambda)$. – prochet Jan 12 '15 at 10:27
• @prochet: I don't really understand what you mean by 'eliminate' in your comment. However, another possibility might be this: If $R\subset \mathsf{S}(\frak{g}^\ast)$ is the ring of $\mathrm{Ad}(G)$-invariant polynomials (for example, the Casimir would be an element of degree $2$), then $R$ acts as a commuting ring on $V(\mu)\otimes V(\nu)$, and you could, perhaps characterize $V(\mu{+}\nu)$ as the space of elements $s\in V(\mu)\otimes V(\nu)$ that satisfy $r(s) = h_{\mu{+}\nu}(r)s$ for the appropriate 'eigenvalue' homomorphism $h_{\mu{+}\nu}:R\to k$. – Robert Bryant Jan 12 '15 at 12:45
• How $R$ acts on this tensor product? – prochet Jan 12 '15 at 13:26

To expand my comment (and Robert Bryant's) a little further, the main point here is that tensor products of finite dimensional irreducible representations behave nicely. In the equivalent Lie algebra setting, the finite dimensional irreducibles are those whose highest weights are dominant integral in the dual of a Cartan subalgebra $\mathfrak{t}:= \mathrm{Lie} \;T$ of $\mathfrak{g} := \mathrm{Lie} \;G$, relative to some fixed choice of positive (or simple) roots.

In particular, $V(\mu) \otimes V(\nu)$ has highest weight $\lambda:= \mu + \nu$ with multiplicity 1. The complete reducibility of this tensor product depends mainly on the fact that the center $Z(\mathfrak{g})$ of the universal enveloping algebra of $\mathfrak{g}$ (generated by "Casimir operators") acts by distinct central characters on the distinct finite dimensional representations. In other words, the sum of all copies of a typical $V(\pi)$ in the tensor product decomposition is a single "eigenspace" for the action of the center, where the action is given by a "central character" $\chi_\pi$.

Since the summand $V(\lambda)$ occurs only once in the tensor product (the weight space for $\lambda$ being one dimensional), $V(\lambda)$ may be characterized as the set of all vectors on which $Z(\mathfrak{g})$ acts by the central character $\chi_\lambda$. (There may of course be many isomorphic summands of smaller highest weights $\pi$, where the center acts by the single character $\chi_\pi$.) In different language this is essentially Robert's comment.