Highest Weight Theory for Complex Semisimple Lie Algebras

Assume the Axiom of Choice. Let $\mathfrak{g}$ be a finite-dimensional complex semisimple Lie algebra with Cartan subalgebra $\mathfrak{h}$, root set $\mathrm{\Phi }=\mathrm{\Phi }(\mathfrak{g},\mathfrak{h})$ and Killing form $B$ (Root and root space, Killing form). Let $E:={\mathrm{span}}_{\mathbb{R}}\mathrm{\Phi }\subseteq {\mathfrak{h}}^{\ast },{\mathfrak{h}}_{\mathbb{R}}:={\mathrm{span}}_{\mathbb{R}}\{h_{\alpha }:\alpha \in \mathrm{\Phi }\}$ be the real span of the roots and the real span of the coroots (Coroot of a Lie-algebra root). Then:

(i) every $\lambda \in E$ is real valued on ${\mathfrak{h}}_{\mathbb{R}}$, and restriction is a linear isomorphism $E\to {\mathrm{Hom}}_{\mathbb{R}}({\mathfrak{h}}_{\mathbb{R}},\mathbb{R})$;

(ii) ${\mathfrak{h}}_{\mathbb{R}}$ is a real form of $\mathfrak{h}$, that is $\mathfrak{h}={\mathfrak{h}}_{\mathbb{R}}\oplus i{\mathfrak{h}}_{\mathbb{R}}$, and the Killing form restricted to ${\mathfrak{h}}_{\mathbb{R}}$ is positive definite;

(iii) the formula $(\lambda ,\mu ):=B(H_{\lambda },H_{\mu })(\lambda ,\mu \in E),$ where $H_{\lambda }\in {\mathfrak{h}}_{\mathbb{R}}$ is the vector with $B(H_{\lambda },H)=\lambda (H)$ for all $H\in {\mathfrak{h}}_{\mathbb{R}}$ (Killing-dual vector of a root), defines a positive definite inner product on $E$, and for all roots $\alpha ,\beta$ $\frac{2(\beta ,\alpha )}{(\alpha ,\alpha )}=\beta (h_{\alpha })\in \mathbb{Z};$

(iv) with this inner product, $(E,\mathrm{\Phi })$ is a reduced crystallographic Euclidean root system in the sense of Reduced crystallographic Euclidean root system, and for every root $\alpha$ the abstract reflection $x\mapsto x-\frac{2(x,\alpha )}{(\alpha ,\alpha )}\alpha$ of that definition agrees with the reflection $s_{\alpha }(\lambda )=\lambda -\lambda (h_{\alpha })\alpha$ of Root reflection defined by a coroot;

(v) if $\mathrm{\Delta }=\{{\alpha }_1,\dots ,{\alpha }_r\}$ is the base of a positive system of $(E,\mathrm{\Phi })$ (Positive systems and simple roots), then $\mathrm{\Delta }$ is a basis of $E$ and $\{h_{{\alpha }_1},\dots ,h_{{\alpha }_r}\}$ is a basis of ${\mathfrak{h}}_{\mathbb{R}}$, hence also a basis of $\mathfrak{h}$ over $\mathbb{C}$.

Let $\mathfrak{g}$ be a complex semisimple Lie algebra with a fixed Cartan subalgebra $\mathfrak{h}$ (Cartan subalgebra), and let $V$ be a representation of $\mathfrak{g}$ (Representations of Lie algebras), written $H\cdot v$ for $\rho (H)(v)$. No finite-dimensionality of $V$ is assumed here.

For $\mu \in {\mathfrak{h}}^{\ast }$ the weight space of $\mu$ is $V_{\mu }=\{v\in V:H\cdot v=\mu (H)v\text{ for every }H\in \mathfrak{h}\};$ this is a linear subspace of $V$ (Linear subspace of a vector space), being the intersection of the kernels of the endomorphisms $\rho (H)-\mu (H){\mathrm{id}}_V$. A weight of $V$ is a functional $\mu \in {\mathfrak{h}}^{\ast }$ with $V_{\mu }\mathrm{\ne }0$, and a nonzero $v\in V_{\mu }$ is a weight vector of weight $\mu$. The zero functional is allowed as a weight; $V_0$ is the space of vectors fixed by $\mathfrak{h}$.

Distinct weight spaces are independent. Indeed, if $v_1+\cdots +v_m=0$ with $0\mathrm{\ne }{v}_j\in V_{{\mu }_j}$ and pairwise distinct functionals ${\mu }_1,\dots ,{\mu }_m$, choose $H\in \mathfrak{h}$ with the scalars ${\mu }_j(H)$ pairwise distinct; this is possible because the finitely many sets $\{H\in \mathfrak{h}:({\mu }_i-{\mu }_j)(H)=0\}$ are proper subspaces of $\mathfrak{h}$ (the functionals ${\mu }_i-{\mu }_j$ are nonzero for $i\mathrm{\ne }j$) and a finite union of proper subspaces of a vector space over the infinite field $\mathbb{C}$ is proper. Then the $v_j$ are eigenvectors of $\rho (H)$ for the pairwise distinct eigenvalues ${\mu }_j(H)$ and hence are linearly independent (Eigenvectors belonging to pairwise distinct eigenvalues are linearly independent), forcing $v_1=\cdots =v_m=0$, a contradiction. Thus the sum ${\sum }_{\mu }{V}_{\mu }$ is direct.

If $V$ is finite-dimensional, then $V$ has only finitely many weights: the nonzero weight spaces form a direct sum inside $V$, so their number is at most $\dim V$.

Assume the Axiom of Choice. Let $\mathfrak{g}$ be a finite-dimensional complex semisimple Lie algebra with Cartan subalgebra $\mathfrak{h}$, and let $V$ be a finite-dimensional representation of $\mathfrak{g}$ (Representations of Lie algebras). Then $V$ is the direct sum of its weight spaces for $\mathfrak{h}$ (Weight and weight space): $V={⨁}_{\mu \in {\mathfrak{h}}^{\ast }}{V}_{\mu },$ and only finitely many of the spaces $V_{\mu }$ are nonzero.

Assume the Axiom of Choice. Let $\mathfrak{g}$ be a finite-dimensional complex semisimple Lie algebra with Cartan subalgebra $\mathfrak{h}$ and root system $\mathrm{\Phi }$, choose a positive system ${\mathrm{\Phi }}^{+}$ with base $\mathrm{\Delta }=\{{\alpha }_1,\dots ,{\alpha }_r\}$ (Positive systems and simple roots), let $V$ be a finite-dimensional representation of $\mathfrak{g}$, and let $W$ be the Weyl group of $\mathrm{\Phi }$, acting on ${\mathfrak{h}}^{\ast }$ by complex-linear extension of its action on $E={\mathrm{span}}_{\mathbb{R}}\mathrm{\Phi }$ (Weyl group, The roots form a reduced crystallographic Euclidean root system). Then for every simple root ${\alpha }_i$ and every $\mu \in {\mathfrak{h}}^{\ast }$ $\dim V_{\mu }=\dim V_{s_i(\mu )},s_i(\mu )=\mu -\mu (h_{{\alpha }_i}){\alpha }_i,$ and consequently $\dim V_{w\mu }=\dim V_{\mu }$ for every $w\in W$ and every $\mu \in {\mathfrak{h}}^{\ast }$ (Weight and weight space).

Let $\mathfrak{g}$ be a finite-dimensional complex semisimple Lie algebra with Cartan subalgebra $\mathfrak{h}$, let $V$ be a representation of $\mathfrak{g}$, let $\alpha$ be a root and let ${\mathfrak{g}}_{\alpha }$ be its root space (Root and root space). If $x\in {\mathfrak{g}}_{\alpha }$ and $v\in V_{\mu }$ for some $\mu \in {\mathfrak{h}}^{\ast }$, then $x\cdot v\in V_{\mu +\alpha }$ (Weight and weight space); in particular $x\cdot v=0$ is allowed.

Assume the Axiom of Choice. Let $\mathfrak{g}$ be a finite-dimensional complex semisimple Lie algebra with Cartan subalgebra $\mathfrak{h}$ and root set $\mathrm{\Phi }$, and let ${\mathrm{\Phi }}^{+}$ be a positive system of the root system with base $\mathrm{\Delta }=\{{\alpha }_1,\dots ,{\alpha }_r\}$ of simple roots (Positive systems and simple roots, The roots form a reduced crystallographic Euclidean root system). Write ${\mathrm{\Phi }}^{-}=-{\mathrm{\Phi }}^{+}$ for the negative roots. Define ${\mathfrak{n}}^{+}={\sum }_{\alpha \in {\mathrm{\Phi }}^{+}}{\mathfrak{g}}_{\alpha },{\mathfrak{n}}^{-}={\sum }_{\alpha \in {\mathrm{\Phi }}^{-}}{\mathfrak{g}}_{\alpha },\mathfrak{b}=\mathfrak{h}\oplus {\mathfrak{n}}^{+}\mathrm{.}$ Then ${\mathfrak{n}}^{+}$ and ${\mathfrak{n}}^{-}$ are called the positive and negative nilpotent subalgebras and $\mathfrak{b}$ the Borel subalgebra attached to ${\mathrm{\Phi }}^{+}$. Here the term Borel uses this root-space construction as its defining convention, as in Knapp, Chapter V §7. The definition depends only on the set ${\mathrm{\Phi }}^{+}$ and not on any enumeration of it, because each sum is the span of a fixed set of subspaces.

These are Lie subalgebras. Let $\alpha ,\beta \in {\mathrm{\Phi }}^{+}$ and $x\in {\mathfrak{g}}_{\alpha }$, $y\in {\mathfrak{g}}_{\beta }$. By Brackets of root spaces, $[x,y]\in {\mathfrak{g}}_{\alpha +\beta }$, and ${\mathfrak{g}}_{\alpha +\beta }=0$ unless $\alpha +\beta$ is a root. When $\alpha +\beta$ is a root, write $\alpha ={\sum }_i{m}_i{\alpha }_i$ and $\beta ={\sum }_i{n}_i{\alpha }_i$ with nonnegative integral coefficients, as Simple roots form a signed integral basis permits; then $\alpha +\beta ={\sum }_i(m_i+n_i){\alpha }_i$ has nonnegative integral coefficients and is nonzero, so by the same theorem it is a positive root. Hence $[{\mathfrak{n}}^{+},{\mathfrak{n}}^{+}]\subseteq {\mathfrak{n}}^{+}$, and ${\mathfrak{n}}^{+}$ is a subalgebra; the same argument with signs reversed gives $[{\mathfrak{n}}^{-},{\mathfrak{n}}^{-}]\subseteq {\mathfrak{n}}^{-}$. Finally $[\mathfrak{h},\mathfrak{h}]=0$ and $[\mathfrak{h},{\mathfrak{g}}_{\alpha }]\subseteq {\mathfrak{g}}_{\alpha }$ for every root $\alpha$, so $[\mathfrak{h},{\mathfrak{n}}^{+}]\subseteq {\mathfrak{n}}^{+}$ and $\mathfrak{b}$ is closed under the bracket (Lie subalgebras, ideals, and center).

The subalgebras ${\mathfrak{n}}^{\pm }$ are nilpotent. For a positive root $\gamma ={\sum }_i{n}_i{\alpha }_i$ put $\mathrm{ht}(\gamma )={\sum }_i{n}_i$ and, for $k\ge 1$, $F_k=\mathrm{span}\{{\mathfrak{g}}_{\gamma }:\gamma \in {\mathrm{\Phi }}^{+},\mathrm{ht}(\gamma )\ge k\},$ the span being $0$ when no such root exists; by Root-space decomposition we have ${\mathfrak{n}}^{+}=F_1$, and $\mathrm{ht}(\gamma )\ge 1$ for every positive root because $\gamma \mathrm{\ne }0$ has nonnegative integral coefficients. If $\alpha ,\gamma \in {\mathrm{\Phi }}^{+}$ and $\alpha +\gamma$ is a root, then $\mathrm{ht}(\alpha +\gamma )=\mathrm{ht}(\alpha )+\mathrm{ht}(\gamma )$ by uniqueness of the simple-root coefficients, so $[{\mathfrak{n}}^{+},F_k]\subseteq F_{k+1}$. Induction gives ${\gamma }_k({\mathfrak{n}}^{+})\subseteq F_k$ for the lower central series (Lower central series and nilpotent Lie algebras). If ${\mathrm{\Phi }}^{+}$ is empty, then ${\mathfrak{n}}^{+}=0$ is nilpotent. Otherwise the finite nonempty set ${\mathrm{\Phi }}^{+}$ has a maximal height $H$, so $F_{H+1}=0$ and hence ${\gamma }_{H+1}({\mathfrak{n}}^{+})=0$: the algebra ${\mathfrak{n}}^{+}$ is nilpotent. The negative case is identical, with heights of the positive roots $-\gamma$ for $\gamma \in {\mathrm{\Phi }}^{-}$, since ${\mathrm{\Phi }}^{-}=-{\mathrm{\Phi }}^{+}$. Thus the terms "positive and negative nilpotent subalgebras" are justified.

Assume the Axiom of Choice. Let $\mathfrak{g}$ be a finite-dimensional complex semisimple Lie algebra with Cartan subalgebra $\mathfrak{h}$ and a chosen positive system, and let ${\mathfrak{n}}^{\pm }$ and $\mathfrak{b}=\mathfrak{h}\oplus {\mathfrak{n}}^{+}$ be as in Positive and negative nilpotent subalgebras and the Borel. Then:

(i) $\mathfrak{g}={\mathfrak{n}}^{-}\oplus \mathfrak{h}\oplus {\mathfrak{n}}^{+}$ is a direct sum of vector spaces;

(ii) ${\mathfrak{n}}^{+}$ and ${\mathfrak{n}}^{-}$ are nilpotent Lie subalgebras and $\mathfrak{b}$ is a solvable Lie subalgebra in which ${\mathfrak{n}}^{+}$ is an ideal, so that $\mathfrak{b}$ is the semidirect sum $\mathfrak{h}⋉{\mathfrak{n}}^{+}$.

Assume the Axiom of Choice. Let $\mathfrak{g}$ be a finite-dimensional complex semisimple Lie algebra with Cartan subalgebra $\mathfrak{h}$ and root system $\mathrm{\Phi }$, and let ${\mathrm{\Phi }}^{+}$ be a chosen positive system with base $\mathrm{\Delta }=\{{\alpha }_1,\dots ,{\alpha }_r\}$ of simple roots (Positive systems and simple roots, The roots form a reduced crystallographic Euclidean root system). By Simple roots form a signed integral basis $\mathrm{\Delta }$ is a basis of $E={\mathrm{span}}_{\mathbb{R}}\mathrm{\Phi }$, so every element of $E$ has unique real coefficients in this basis; the root lattice $Q$ consists of the integral combinations of $\mathrm{\Delta }$ (Root, coroot, weight, and coweight lattices). Put $Q_{+}:=\{{\sum }_{i=1}^r{n}_i{\alpha }_i:n_i\in {\mathbb{Z}}_{\ge 0}\}\subseteq Q\mathrm{.}$

For $\lambda ,\mu \in {\mathfrak{h}}^{\ast }$ write $\mu \le \lambda$ when $\lambda -\mu \in Q_{+}$; in words, when $\lambda -\mu$ is a nonnegative integral combination of the chosen simple roots. This is the root order on ${\mathfrak{h}}^{\ast }$. In particular $\mu \le \lambda$ forces $\lambda -\mu \in E$, so comparable functionals lie in the same affine coset of $E$ in ${\mathfrak{h}}^{\ast }$; neither functional need itself lie in $E$.

The root order is a partial order. Reflexivity holds with $n_i=0$. Antisymmetry: if $\mu \le \lambda$ and $\lambda \le \mu$, then ${\sum }_i(m_i+n_i){\alpha }_i=0$ with $m_i,n_i\ge 0$, and linear independence of the simple roots forces $m_i+n_i=0$, so $m_i=n_i=0$ and $\lambda =\mu$. Transitivity: if $\nu \le \mu$ and $\mu \le \lambda$, then $\lambda -\nu =(\lambda -\mu )+(\mu -\nu )$ is a sum of two elements of $Q_{+}$, hence lies in $Q_{+}$ and $\nu \le \lambda$. Thus $\le$ is a partial order on ${\mathfrak{h}}^{\ast }$; restricted to any set of weights it is a partial order on that set, and every comparison chain of weights is finite whenever the weight set is finite, because a strict increase adds a nonzero element of $Q_{+}$.

Assume the Axiom of Choice. Let $\mathfrak{g}$ be a finite-dimensional complex semisimple Lie algebra with Cartan subalgebra $\mathfrak{h}$, a chosen positive system with nilpotent subalgebra ${\mathfrak{n}}^{+}$ (Positive and negative nilpotent subalgebras and the Borel), and let $V$ be a representation of $\mathfrak{g}$. A highest weight vector of $V$ is a nonzero vector $v\in V_{\lambda }$, for some $\lambda \in {\mathfrak{h}}^{\ast }$, with ${\mathfrak{n}}^{+}\cdot v=0$ (Weight and weight space); the functional $\lambda$ is then called the weight of $v$. A highest weight module of highest weight $\lambda$ is a representation $V$ generated, as a $\mathfrak{g}$-module, by a highest weight vector of weight $\lambda$: the smallest subrepresentation of $V$ containing $v$ is $V$ itself.

By Lie representations are U(g)-modules the action of $\mathfrak{g}$ extends uniquely to a unital action of $U(\mathfrak{g})$, so the subrepresentation generated by $v$ is exactly $U(\mathfrak{g})v$; this is the content of the word "generated" above and makes the notion independent of any choice of generators. A highest weight vector $v$ satisfies $H\cdot v=\lambda (H)v$ for all $H\in \mathfrak{h}$ by the definition of $V_{\lambda }$. The zero representation is not a highest weight module, since a highest weight vector is required to be nonzero.

Assume the Axiom of Choice. Let $\mathfrak{g}$ be a finite-dimensional complex semisimple Lie algebra with Cartan subalgebra $\mathfrak{h}$ and a chosen positive system, and let $V$ be a representation generated by a highest weight vector $v$ of weight $\lambda$ (Highest-weight vectors and modules). Then $V=U({\mathfrak{n}}^{-})v=U({\mathfrak{n}}^{-})\mathbb{C}v,$ every weight $\mu$ of $V$ is of the form $\mu =\lambda -{\sum }_i{n}_i{\alpha }_i$ with $n_i\in {\mathbb{Z}}_{\ge 0}$, so that $\mu \le \lambda$ in the root order (Root order on weights), and $V_{\lambda }=\mathbb{C}v\mathrm{.}$