⇤Math diary
May 28, 2023: Borel-Weil theory, quantizing spin, and spheres.
Back in entry 10, I talked a bit about the Borel-Bott-Weil theorem from the perspective of geometric quantization. Borel-Bott-Weil gives a systematic way to build all representations of $G$ as sections of $G$-invariant holomorphic line bundles over the flag manifold. The flag manifold is $G/B$ where $B$ is a borel subgroup of $G$. We construct this line bundle via a quotient construction:
\(L_\lambda = G \times_{\chi_\lambda} \mathbb{C}^\ast / B\) We start with a trivial line bundle on $G$, then mod out by the action of the Borel group $B$, which acts by multiplication on $G$ and by a holomorphic homomorphism $\chi_\lambda$ on $\mathbb{C}^\ast$. Every such homomorphism extends from a unique homomorphism of the maximal torus $T \subset B \subset G$ (This is not super hard, but non-obvious). Thus, possible holomorphic line bundles are classified by characters of $T$, which are elements of the root lattice in the lie algebra $\mathfrak{t}$. entry 10 motivated this thru constructing a symplectic form which agreed with the curvature of $L_\lambda$. In this post, I want to talk a bit about the proof, using standard constructive methods.
\[[e,f]=h \quad [h,e]=2e \quad [h,f]=-2f\]Theorem: Borel-Weil Consider the line bundle $L_\lambda$ on the flag manifold $G/B$ defined by an integral weight $\lambda$. Then, the group $G$ acts on the space of holomorphic sections $H^0(G/B,L_\lambda)$. This vector space is nonzero if and only if $\lambda$ is dominant. If so, this representation is the unique irreducible representation of $G$ with highest weight $\lambda$.
There are a lot of things to prove, but let me just explain why we need $\lambda$ to be dominant to have any nonzero sections. The idea is a central one in representation theory: Pick a single positive root $\alpha$, and look at the rank one algebra which it generates. This gives you a Lie algebra embedding $i_\alpha: \mathfrak{sl}_2 \to \mathfrak{g}$. The subalgebra $\mathfrak{sl_2}$ will have a positive “creation operator” $e$ which shifts weights by $\alpha$, a negative creation operator $f$ which shifts weights by $-\alpha$, and an energy operator $h$ which satisfies:
Exponentiating, this holomorphically embeds $SL(2,\mathbb{C})$ into $G$. Modding out by $B$ cuts out all the negative weight parts of the lie algebra. Since $SL(2,\mathbb{C})$ has one foot negative and one foot positive, $i_\alpha$ gives us a holomorphic embedding $SL(2,\mathbb{C})/B_{sl2} \hookrightarrow G/B$. However, the flag manifold $SL(2,\mathbb{C})/B_{sl2}$ is equal to its associated real quotient $SU(2)/U(1)$ where $U(1)$ is a maximal torus of $SU(2)$. This is equal to the variety of flags of one dimensional subspaces of $\mathbb{C}^2$, which is $\mathbb{C}P^1$. In other words, every positive root of $\mathfrak{g}$ gives a holomorphic embedding of the Riemann sphere $\mathbb{C}P^1$ into the flag manifold $G/B$.
Let’s recall the representation theory of $\mathfrak{sl}_2$. There are irreducible representations of every dimension $d+1$, which are nicely described as $S_d$, the space of degree $d$ polynomials in 2 variables. Algebraically, $S_d$ has highest weight $d$1 and is indeed irreducible. Geometrically, a homogenous polynomial of degree $d$ in 2 variables is a holomorphic section of the line bundle $\mathcal O (d)$ on $\mathbb{P}^1$, and the SU(2) action by rotation agrees with the algebraic description. This is Borel-Weil for $SU(2)$. Note that $\cal O(d)$ only has sections when $d$ is positive.
Now, we have an embedding $i_\alpha: SL(2,\mathbb{C}) \hookrightarrow G$ which induces a holomorphic embedding of flag manifolds (Denoted by $C_\alpha$) $i_\alpha: C_\alpha \cong \mathbb{P}^1 \hookrightarrow G/B$. if the line bundle $L_\lambda$ has global sections on $G/B$, it must surely have global sections on every $\mathbb{P}^1$. However, we can pull back $L_\lambda$ to a holomorphic section on $C_\alpha$, and its weight is the evaluation $\lambda(\alpha)$. For this to have sections on $\mathbb{C}_\alpha$ we need $\lambda(\alpha)>0$. Thus, if there are nonzero global sections, we need $\lambda(\alpha)>0$ for all positive roots $\alpha$, which by definition means $\lambda$ is dominant
I think of this as a very physicsy perspective, using creation and annihilation operators to understand the structure of representations of the group. We can frame things differently: Inside the flag manifold, we can always find holomorphic spheres fixed and rotated by some subgroup $SU(2)$ of $G$. To translate this into represenations, we quantize this classical system. The spheres quantize into representations of $SU(2)$ whose dimension depends on the sphere radius: this is the geometric quantization of a classical spin 1/2 system.
\[e^{i \theta} \to \begin{pmatrix} e^{i \theta} & 0 \\ 0 & e^{-i \theta} \end{pmatrix}\]This acts by weight 1 on $x$ and weight $-1$ on $y$. The highest weight vector in $S_d$ is $x^d$, which has weight $d$.
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call the variables $x,y$, and consider the maximal torus $U(1) \hookrightarrow SU(2)$ defined by ↩