---- > [!theorem] Theorem. ([[on the weights of a representation]]) > Let $\mathfrak{g}$ be a ([[semisimple Lie algebra|semisimple]]) $\mathbb{C}$-[[Lie algebra]], $\mathfrak{t}$ [[Cartan subalgebra|CSA]], $V$ a finite-dimensional[^1] [[Lie algebra representation|representation]] of $\mathfrak{g}$. Denote by $(\Phi, E)$ the [[root system]] [[root system of a Lie algebra|corresponding to]] $\mathfrak{g}$, $E=\mathbb{R}\Phi$. Let $X \subset \mathfrak{t}^{*}$ be the associated [[root and weight lattice of a root system|weight lattice]]. > > **(A)** first observe that an element $\lambda \in \mathfrak{t}^{*}$ lies in $X$ if and only if $\lambda(h_{\alpha}) \in \mathbb{Z}$ for all $\alpha \in \Phi$[^5] (equivalently, all $\alpha \in \Delta$ for $\Delta$ a [[root basis]]). Indeed, [[root system of a Lie algebra|recall]] $\lambda(h_{\alpha})=\langle \lambda, \check \alpha \rangle$.[^6] > > The following discussion mimics/generalizes the [[classification of the irreps of sl2 over C]]. > > Since $\mathfrak{t}$ is [[abelian Lie algebra|abelian]] and every element of $\mathfrak{t}$ is [[semisimple element of a semisimple Lie algebra|semisimple]], we may [[on factorizing a Lie algebra representation into weight spaces|factorize]] $V$ into '[[simultaneously diagonalizable|simultaneous eigenspaces]]' $\begin{align} > \bigoplus_{\lambda \in \mathfrak{t}^{*}} V_{\lambda} , \text{ where } \\ > V_{\lambda}= \{ v \in V: t \cdot v =\lambda(v) &\text{ for all } t \in \mathfrak{t} \}. > \end{align}$ > We can make some basic observations about this decomposition, mimicking those [[special linear Lie subalgebra|in the]] $\mathfrak{sl}_{2}(\mathbb{C})$-case. Note that the [[Weyl group of a root system|Weyl group]] $W$ [[group action|acts on]] $X$.[^4] Denote by $\mathfrak{g}=\mathfrak{t} \oplus \bigoplus_{\alpha \in \Phi}\mathfrak{g}_{\alpha}$ the [[root space decomposition of a Lie algebra|root space decomposition]] of $\mathfrak{g}$. > > 1. **(Hopping right)** Let $\alpha \in \Phi$ and $e_{\alpha} \in \mathfrak{g}_{\alpha}-\{ 0 \}$ (recall any such $e_{\alpha}$ [[root spaces are one-dimensional|generates]] $\mathfrak{g}_{\alpha})$ . Then $e_{\alpha} \cdot V_{\lambda} \subset V_{\lambda+\alpha}$; > ![[Pasted image 20250516175342.png|500]] > 2. **(Justifying the term *weight* lattice)** If $V_{\lambda} \neq 0$, then $\lambda \in X$, i.e. $\lambda$ lies in the [[root and weight lattice of a root system|weight lattice]]; > 3. **(Higher-dimensional version of $\mathfrak{sl}_{2}$ result '$\lambda$ a weight $\implies -\lambda$ a weight)** $\text{dim }V_{\lambda}=\text{dim }V_{w(\lambda)}$ for all $w \in W$. > > > Therefore, the weights of $V$ lie in the weight lattice $X$ and are preserved by the [[Weyl group of a root system|Weyl group]]. > > > > [!definition] Definition. (Highest Weight Vector) > In the above notation, suppose that $v \in V_{\lambda}$ is nonzero and satisfies $e_{\alpha} \cdot v=0$ for all $\alpha \in \Delta$.[^3] Then $v \in V$ is called a **highest weight vector**; $\lambda \in \mathfrak{t}^{*}$ a **highest weight**. > > Compare to [[weight space for sl2(C)|the version for]] $\mathfrak{sl}_{2}$. There, the idea was that one could not 'hop any farther rightward'. Here, if we imagine each simple root as a 'dimension/direction', the idea is that one 'can't hop any more in any direction' in $\Phi$. > > Any finite-dimensional [[Lie algebra representation|representation]] $V$ has a highest weight vector.[^2] Moreover, every highest weight is [[dominant weight in a root system|dominant]].[^7] ^definition > [!note] Remark 1. > [[classification of the irreps of sl2 over C|In the sl2 case]], it was easy to see that an $\mathfrak{sl}_{2}$-[[irreducible Lie algebra representation|irrep]] has a unique highest weight, and that this weight uniquely determines the representation. *The same is true for general semisimple $\mathfrak{g}$*, though it takes more preparation to prove: [[the classification of complex irreducible semisimple Lie algebra representations]]. Ultimately, the conclusion is that for every [[dominant weight in a root system|dominant]] $\lambda \in X$, there is a unique irreducible $\mathfrak{g}$-[[Lie algebra representation|representation]] $V(\lambda)$ with highest weight $\lambda$. > > In light of this, there are two different contexts in which one has cause to visualize weights. One is as in the image in the original Theorem box, which visualizes weights for a *single* representation. The other is as in [[the classification of complex irreducible semisimple Lie algebra representations]], where one represents *all* representations at once on such a diagram, denoting each by their highest weight. ^note > [!note] Remark 2. > *Here is a departure from the $\mathfrak{sl}_{2}$ case.* In the $\mathfrak{g}=\mathfrak{sl}_{2}$ case, once you know the highest weight $n$, you know all the weights of the irrep and their multiplicity: they are $n,n-2,\dots,-n$ (or really $n\omega, (n-2)\mathbf{\omega}.., -n\omega$ for $\omega$ the fundamental weight), each with multiplicity 1. For a general irrep $V(\lambda)$ of a general [[semisimple Lie algebra|semisimple]] complex Lie algebra $\mathfrak{g}$, it takes a bit more work to find out the weights: [[reading off the weights (with some information on multiplicity) of an irreducible semisimple Lie algebra representation]]. Sometimes it can take a *lot* more work if one just starts with a general algebra and wants to find multiplicities: [[Weyl character formula]]. ^note [^1]: Being finite-dimensional has so far been something we've taken for granted. From now on we will care both about finite-dimensional and infinite-dimensional representations (it turns out the latter helps inform the former). There are a couple spots in this note where finite-dimensionality is explicitly used (e.g. existence of highest weight vectors). > [!note] Remark 3. > It should be emphasized: the results here are not merely analogues to the $\mathfrak{sl}_{2}$ case, they *follow from it*. ^note [^2]: Morally, the idea is that $V$ is finite-dimensional — if it had no highest weight vector then it would be composed of infinitely many nontrivial weight spaces. "Can't hop forever." Maybe there could be some more explicit justification here, though. [^3]: I.e., $v$ is a simultaneous eigenvector for the $\mathfrak{t}$-action killed by any positive root space generator. (We can equivalently say $\alpha \in \Phi^{+}$ instead of $\alpha \in \Delta \subset \Phi^{+}$.) > [!proof]- Proof. ([[on the weights of a representation]]) > ~ **1.** This is a simple computation, similar to showing $e \cdot V_{\lambda} \subset V_{\lambda+2}$ [[classification of the irreps of sl2 over C|in the]] $\mathfrak{sl}_{2}$ case. If $t \in \mathfrak{t}$ and $v \in V_{\lambda}$, then: $\begin{align} t \cdot (e_{\alpha} \cdot v)= e_{\alpha} \cdot (t \cdot v) + [t, e_{\alpha}] \cdot v&= e_{\alpha} \cdot \lambda(t)v + \alpha(t) e_{\alpha} \cdot v \\ & = (\lambda + \alpha)(t) (e_{\alpha} \cdot v). \end{align}$ The other two parts will follow from $\mathfrak{sl}_{2}$ theory. **2.** Let $\alpha \in \Phi$ and choose $e_{\alpha} \in \mathfrak{g}_{\alpha}$ and $e_{- \alpha} \in \mathfrak{g}_{-\alpha}$ such that $\{ e_{\alpha}, h_{\alpha}, e_{-\alpha} \}$ [[finding sl2-triples|form an]] $\mathfrak{sl}_{2}$-triple, [[submodule generated by a subset|spanning]] a [[Lie subalgebra|subalgebra]] $\mathfrak{m}_{\alpha} \subset \mathfrak{g}$ [[isomorphism|isomorphic]] to $\mathfrak{sl}_{2}$. Restricting the action $\mathfrak{g} \to \mathfrak{gl}(V)$ to $\mathfrak{m}_{\alpha}$ gives an $\mathfrak{sl}_{2}$-rep. The [[weight space for sl2(C)|weights of this representation]] [[classification of the irreps of sl2 over C|must be]] integers. In other words, $\lambda(h_{\alpha}) \in \mathbb{Z}$ for all $\lambda$ such that $V_{\lambda} \neq 0$. It follows from **(A)** that $\lambda \in X$. **3.** It is enough to show equality of dimensions for $w=w_{\alpha}$ a single [[reflection]], $\alpha \in \Phi$ (rather than a general composition of reflections). Restrict the action of $\mathfrak{g}$ to $\mathfrak{m}_{\alpha} \subset \mathfrak{g}$ [[any representation of sl2(C) is completely reducible|Decompose]] $V$ as a [[direct sum of representations|direct sum]] of $\mathfrak{sl}_2 \cong \mathfrak{m}_{\alpha}-$[[irreducible Lie algebra representation|irreps]] : $V |_{\mathfrak{m}_{\alpha}}= \bigoplus_{i} V^{(i)}$ where $V^{(i)}$ is an irreducible $\mathfrak{m}_{\alpha}-$representation. By $\mathfrak{sl}_{2}$ theory, each $V^{(i)}$ is one-dimensional, hence have a basis of $V_{\lambda}$ consisting of $v_{1},\dots,v_{n}$ with each $v_{i}$ in a distinct $V^{(j)}$. Then apply $e_{\alpha}$ or $e_{-\alpha}$ repeatedly and use the fact that $w_{\alpha}(\lambda)(h_{\alpha})=-\lambda(h_{\alpha})$. ---- #### [^4]: Indeed, $W$ acts on $E$ and any $w \in W$ preserves $\Phi$, [[reflection|so]] $\langle w(\lambda), \check \alpha \rangle \in \mathbb{Z}$ for all $\alpha \in \mathbb{Z}$ as a check-pairing of elements in a [[root system]]. [^5]: Recalling notation: [[root space decomposition of a Lie algebra|recall]] that [[killing form]] $\kappa$ restricted to $\mathfrak{t}$ is [[nondegenerate bilinear form|nondegenerate]], yielding an [[isomorphism]] $\mathfrak{t} \to \mathfrak{t}^{*}$, $t \mapsto \kappa(t, -)$, $t_{\alpha} \leftarrow_{|} \alpha$. [[finding sl2-triples|Then recall]] that $h_{\alpha}$ is defined as $\frac{2t_{\alpha}}{\kappa(t_{\alpha}, t_{\alpha})}$, given $\alpha \in \mathfrak{t}^{*}$. [^6]: $\lambda(h_{\alpha})=\lambda\left( \frac{2t_{\alpha}}{\kappa(t_{\alpha} , t_{\alpha})} \right)=\frac{2}{\kappa(t_\alpha, t_\alpha)} \lambda(t_{\alpha})$. Now [[root system of a Lie algebra|recall]] how the [[inner product]] on $E$ is defined by transferring $\kappa$ over the isomorphism $\mathfrak{t} \to \mathfrak{t}^{*}$ it induces: in particular, $\lambda(t_{\alpha})=(\lambda, \alpha)$, and similarly $\kappa(t_{\alpha}, t_{\alpha})=(\alpha, \alpha)$. Thus, $\lambda(h_{\alpha})=\frac{2(\lambda, \alpha)}{(\alpha, \alpha)}=\langle \lambda, \check \alpha \rangle$. This is interesting: something geometric (the scale factor of twice the projection of $\lambda$ onto $\text{span}(\alpha)$) equals something algebraic (the simultaneous eigenvalue $\lambda$ evaluated at $h_{\alpha}$). [^7]: For $\mathfrak{sl}_{2}$, this statement is just stating that [[classification of the irreps of sl2 over C|the highest weight]] is nonnegative. We can bootstrap from the $\mathfrak{sl}_{2}$ case, using that $\lambda(h_{\alpha})=\langle \lambda, \check{\alpha} \rangle$ for $\alpha \in \Delta$. A highest weight vector $v$ for $V$ must also be one when we view $V$ as an $\mathfrak{m}_{\alpha}$-representation (it's the same $e_{\alpha}$ acting in both scenarios). By $\mathfrak{sl}_{2}$ theory, $\lambda(h_{\alpha}) \in \mathbb{Z}_{\geq 0}$ (highest weight of an $\mathfrak{sl}_{2}$-representation is a nonnegative integer). Then we're done, since $\lambda(h_{\alpha})=\langle \lambda, \check \alpha \rangle$. ----- #### References > [!backlink] > ```dataview > TABLE rows.file.link as "Further Reading" > FROM [[]] > FLATTEN file.tags as Tag > WHERE Tag = "#definition" OR Tag = "#theorem" OR Tag = "#MOC" OR Tag = "#proposition" OR Tag = "#axiom" > GROUP BY Tag > ``` > [!frontlink] > ```dataview > TABLE rows.file.link as "Further Reading" > FROM outgoing([[]]) > FLATTEN file.tags as Tag > WHERE Tag = "#definition" OR Tag = "#theorem" OR Tag = "#MOC" OR Tag = "#proposition" OR Tag = "#axiom" > GROUP BY Tag > ```