⇤Math diary
Jun 3, 2023: Borel-Weil theory mark 3
In entries 10 and 16, I talked about the Borel-Weil theorem, which describes representations of lie groups using line bundles on their flag manifolds. I’m steadily undestanding just how much Borel-Weil underlies geometric representation theory. I keep coming back to it, each time understanding a little better. Let me give another crack at explaining Borel-Weil, eventually I’ll converge to a good exposition.
Borel-Weil theory
If group theory is the study of symmetries, then Lie theory is the study of continuous symmetries. Take for example $SO(3)$: We can think of this as the group of 3D rotation matricies, or alternatively as the symmetries of the sphere with its standard round metric. This lets us study the group using spherical geometry. For example, the commutator of two rotation matrices is the parralel transport of a unit tangent vector. To see this explained using zelda, see my twitter thread.
There are many different spaces $X$ with symmetry group $G$. If we assume that $G$ acts transitively on $X$ (sends each point to every other point) and homogeneously (Acts on each point in the same way), then $X$ is totally determined by the subgroup $G_x$ of $G$ which fixes a given point $x$. That is, $X = G/G_x$, and we can equivalently talk about the subgroup $P\subset G$ instead of $X$. We define $P$ to be a parabolic subgroup if $G/P$ is an algebraic variety, the zero locus of some polynomial. Fun fact, every parabolic subgroup $P$ contains a maximal solvable subgroup $B \subset G$ (think the upper triangular matrices). Moreover, every one of these subgroups is isomorphic and related by conjugation. We call $B$ the Borel subgroup, and it is the smallest possible parabolic subgroup, contained in all other parabolic subgroups. Therefore, $G/B$ is the largest compact variety acted on transitively by $G$, the Generalized flag manifold! Anything which can be captured thru varieties with symmetry $G$, can be captured with $G/B$.
Now we wish to use the action of $G$ on $G/B$ to study representations of $G$. Though this action is nonlinear, we can linearize it with a classic trick: pass to spaces of functions. For example, if $G$ acts on itself by multiplication, then it acts linearly on $L^2(G)$. This $G$ action decomposes into irreducible representations: By the Peter-Weyl theorem, every irreducible representation appears exactly once. Turning a nonlinear operation into a linear operator on functions is used in dynamics, where the philosophy is called “Koopmanism”. It originates from Koopman–von Neumann classical mechanics, A Hilbert space approach to classical mechanics. We can think of the $G$ action on $G/B$ as classical mechanics, and the induced representation on $L^2(G/B)$ as quantum mechanics. (See entry 10).
To make this representation finite dimensional, we limit to a finite dimensional subset of functions. The cleanest way to ensure finite dimensionality is to choose functions solving an elliptic PDE, which is symmetric under the action of $G$. Here we take the solutions to the Cauchy-Riemann equations on $G/B$, holomorphic functions, which are preserved since the action of $G$ on $G/B$ is holomorphic. Unfortunately, $G/B$ is compact, so the only holomorphic functions are constant and the associated representation is trivial. We can instead try twisted holomorphic functions, replacing the space of functions with spaces of sections of a line bundle. Turns out, See entry 16, $G$-invariant line bundles are classified by characters of the maximal torus $T \subset B \subset G$. When this character $\lambda$ is a dominant, integral weight, then the global sections of the associated line bundle $L_\lambda$ on $G/B$ is $H^0(G/B,L_\lambda)$, an irreducible representation of $G$ with highest weight $\lambda$. This is the Borel-Weil theorem.