⇤Math diary
May 20, 2023: Borel-Bott-Weil and analytic methods on Lie group representation theory
I spent the last week playing Zelda, oops. Let me try to get back into things. I talked to potential advisor again, and decided a more precise plan for my summer. In particular, there’s going to be more geometric representation theory than I expected. in honor of this development, let me talk a bit about some analytic methods in the representation theory of lie groups. All of these ideas are very natural using the perspective of quantum mechanics.
Borel-Bott-Weil and the orbit method
The progenitor of essentially all of geometric representation theory is the Borel-Bott-Weil theorem, which gives an explicit, geometric way to construct irreducible representation. To fix the notation, let $G$ be a lie group, and $V$ an irrep. The idea, as presented by Witten, is that $V$ is a vector space of quantum states with symmetry $G$. What is the corresponding classical system, a kahler manifold $(X,\omega)$ with a hamiltonian $G$-action whose quantization gives $V$? Using kahler geometric quantization, $X$ must carry a holomorphic line bundle $L$, with curvature $\omega$, such that the vector space of holomorphic sections is $H^0(X,L) = V$.
The answer comes from the coadjoint orbits of $G$ on $\mathfrak{g}^\ast$. The dual of the lie algebra $\mathfrak{g}^\ast$ carries a natural symplectic form born from the lie bracket, and the natural action of $G$ preserves the symplectic form. On an orbit $\cal{O}$ of $G$, the symplectic structure of $\mathfrak{g}^\ast$ restricts to one on $\cal{O}$. (Note, in particular, that we are not taking a symplectic quotient). The stabilizer of a generic point of $\mathfrak{g}^\ast$ is essentially the centralizer of a generic point $g\in G$. For a simple group $G$, this solely consists of the maximal torus $T$ containing $g$. A generic orbit $\cal{O}$ locally looks like $G/T$, and in fact is globally $G/T$. This is the Flag manifold. Orbits through different points in $\mathfrak{g}^\ast$ have different symplectic structures, and thus quantize to different represetations.
We can explictly construct the line bundles with curvature equal to the symplectic form. First, take the intersection of $\mathcal{O}$ with a fixed Cartan subalgebra $\mathfrak{t}^\ast$, which intersects in a unique point $\lambda$ in the dominant Weil chamber. Call this orbit $\mathcal{O}_{\lambda}$ Exponentiating $\lambda$ gives us a one dimensional representation $\chi_\lambda$ of $T$. Form a trivial line bundle $G \times \mathbb{C}$, then quotient by the action of $T$, by multiplication on $G$ and by $\chi_\lambda$ on $\mathbb{C}$. This gives a holomorphic line bundle $L_\lambda$ on $G/T$. However, the curvature of $L_\lambda$ must be integral, representing the degree of $L_\lambda$. So, it can only equal the symplectic form when the symplectic form itself is integral. This restricts the value of $\lambda$ to the root lattice inside $\mathfrak{t}^\ast$. Since $G$ acts holomorphically on $G/T$ and hence on $L_\lambda$, it acts on the space of holomorphic sections of $L_\lambda$. This brings us to the borel-weil theorem:
Theorem: Borel-Weil Let $\mathcal{O_\lambda} = (G/T,\omega_\lambda)$ be $G/T$ with symplectic structure inherited from a co-adjoint orbit thru $\lambda$ in $\mathfrak{g}^\ast$. There is a line bundle $L_\lambda$ with curvature $\omega_\lambda$. If $\lambda$ is a dominant, integral weight, then $H^0(G/T,L_\lambda)$ is the irrep of $G$ with highest weight $\lambda$.
From this point we can apply Kodira embedding theorem, to embed $G/T$ into projectivization of the space of holomorphic sections $\mathbb{P}H^0(G/T,L_\lambda) = \mathbb{P}V$. In fact, $G/T$ is the $G$ orbit of a point in $\mathbb{P}V$ using the representation of $G$ on $V$, and $L_\lambda$ is the pullback of the canonical line bundle on $\mathbb{P}V$. Using this representation, it is easier to see that $H^0(G/T,L_\lambda)$ is really isomorphic to $V$.
The full Borel-Bott-Weil theorem extends this to higher cohomology groups. The number of weyl group flips needed to make a weight $\lambda$ dominant gives you the only nonvanishing degree of $H^\ast(G/T,L_\lambda)$.
Let’s step back a bit: we have built a symmetric space $G/T$ whose quantization gives us any representation we want. Now we can just work with complex geometry of line bundles to understand representation theory! The only way to really understand this is through detailed examples, the simplest being $SU(2)$ and its flag manifold $SU(2)/U(1) = S^2$. In quantum mechanics language, this is called the Bloch sphere, and it tells you how to quantize 2-qubit systems.
Building on borel-bott-weil
For a representation $\rho$, we are interested in its trace, known as the character $\chi_\rho: G \to \mathbb{C}$, $\chi_\rho(g) = \mathrm{Tr } \rho(g)$. In a sense, this contains most of the useful information about a representation. Weyl has a formula for the character of the heigest weight represntation $\rho_\lambda$ in terms of a sum over weyl group elements. But how do we understand this geometrically? Well, the trace of an operator is the projection of the operator onto the subspace spanned by the identity operator. We realized the represenation as the action of $G$ on holomorphic sections of $L_\lambda$ over $G/T$. The identity operator on $L_\lambda$ preserves the base point, so the trace has to be computable purely from points in $G/T$ which are fixed by $G$. These points are in bijection with elements of the Weyl group, so the trace becomes a sum over elements of the Weyl group, just like the Weyl characters formula. To get the explicit formula, we use the Atiyah-Bott-Lefshetz fixed point formula, which is a homological tool for computing number of fixed points. There is a good treatment in Pressley-Segal’s book on loop groups, chapter 14.2.
Stretching the quantum analogy a step too far, this sort of localization is remiscent of the Desutermaat-Heckman theorem, which stipulates an equality between semiclassical and full quantum theorires, when the energy function generates a $U(1)$ action. The atiyah-bott-lefshetz formula is essentially a semiclassical approximation of the quantum path integral, computed from the action of $G$ on the germ of $L_\lambda$ at the fixed points. On the other hand, the character encodes the full action of $G$ on the space of quantum states. The weyl charecter formula bridges between the classical and quantum versions of these theories.
Finally, we can look at spectral theory of the laplacian on $G$, and compute the an eigenbasis of $L^2(G)$. This is our physics-y approach to trying to extend fourier series to arbitary locally compact groups. This decomposition matches exactly with the decomposition of $L^2(G)$ into irreps by Peter-Weyl theorem. The heat kernel can be exactly described on $G$, see THE HEAT EQUATION ON A COMPACT LIE GROUP. The formula for the heat kernel is a small modification of the dedikend eta function. These formula give a new proof for an old identity of Macdonald.
We could try to quantize a different way. Instead of using the Kahler polarization, we can use the more standard cotangent bundle polarization, which turns the vector space from holomorphic sections to $L^2$ sections of of $T^\ast G$. Carefully tracing this with the holomorphic polarization above, gives us that $L^2(G) = \bigoplus_\lambda V_\lambda \otimes V_\lambda^\ast$, summing over all irreducible representations $V_\lambda$. That is, square integrable functions on a compact group contain every irriducible representation exactly once. This is the famous Peter-Weyl theorem, and here we are saying it is a manifestation of the equality of two different quantizations. See The double Gelfand-Cetlin system, invariance of polarization, and the Peter-Weyl theorem.
also see this paper by guilleman-sternberg: The Gelfand-Cetlin System and Quantization of the Complex Flag Manifolds