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Jun 9, 2023: Bellinson-Bernstein localization
Bellinson-Bernstein localization
universal enveloping algebra
When studying lie groups, it is essentially just as good to study their infinitesimal linearization, the Lie algebra. Thinking about Lie groups as symmetries of spaces, their lie algebra represents infinitesimal symmetries: Vector fields whose flows generate the action of the Lie group. On functions, the lie algebra acts as the directional derivative in the direction of the associated vector field. The lie algebra turns into differential operators, whose commutator agrees with the lie bracket. But we can apply these operators multiple times, and get higher order differential operators. The space of all operators which we can get from repeatedly applying the lie algebra $\mathfrak{g}$ is called the universal enveloping algebra $\mathcal{U}(\mathfrak{g})$. $\mathcal{U}(\mathfrak{g})$ is the tensor algebra over $\mathfrak{g}$, modulo the relation where swapping two elements results in their commutator: $e_i\otimes e_j - e_j \otimes e_i = [e_i,e_j]$. Specifically, there is an isomorphism between the universal enveloping algebra $\mathcal{U}(\mathfrak{g})$ and the algebra of left-invariant differential operators (of any order) on the associated Lie group $G$. Unlike the Lie bracket of vector fields, the algebra of differential operators with operation composition of operators is associatvie. The universal enveloping algebra is the universal associative algebra whose commutator agrees with the Lie bracket. More pressingly, the representations of a Lie algebra agree exactly with those of its universal enveloping algebra.
If we choose a basis $e_i$ of $\mathfrak{g}$, then by the Poincare-Birkoff-Witt theorem, $\mathcal{U}(\mathfrak{g})$ has basis $e_0^{n_0}e_1^{n_1}\dots {e_{\dim \mathfrak{g}}}^{n_{\dim \mathfrak{g}}}$ for natural numbers $n_i$. That is, we can always use the commutators between basis elements to swap adjacent elements, and move them to a standard order. The real content of the theorem is that all such elements are linearly dependent.
$\cal D$-modules
Following the philosophy of entries 17, a representation of the lie algebra is some finite dimensional subspace of functions (or sections of some bundle) which is invariant under the first order differential operators corresponding to the lie algebra. By the discussion above, this is a representation of the universal enveloping algebra, which acts by differential operators of any order. Luckily, differential operators are local.
Instead of defining our representation with a vector space of global allowed functions, we can just specify the allowed functions locally, for neighborhoods of any point. The resulting global functions are those which obey all local rules. For example, global holomorphic functions are those which obey the Cauchy-Riemann equations at every point. The vector space of global holomorphic functions on a compact manifold is 1-dimensional, so these form the trivial representation. Conversely, the trivial representation is uniquely specified by the Cauchy-Riemann equations. Think about it like a traffic engineer. Instead of the keeping a ledger of all possible routes through manifold city and choosing a subset of legal ones, we specify the traffic laws for each intersection. We can check if any given route is legal in a much more efficient way. To find all possible legal paths, start at one point and go in all the directions, following the intersection rules as they decree, and throw out the routes which wind up running into themselves.
This new perspective brings us to sheaves. Namely, we build representations as spaces of global sections of sheaves. For example, the trivial representation of $G$ is the global sections of the structure sheaf $\cal O$ of the flag manifold $G/B$, where $B$ is some Borel subgroup. For each integral weight $\lambda$, we obtain a twisted sheaf $\cal{O}\lambda$ which locally looks like the structure sheaf but globally picks up a character of $B$. This is essentially a line bundle. Since $\cal O _\lambda$ is invariant under $G$, The space of global sections $H^0(G/B,\cal O\lambda)$ inherits a linear action by $G$. As described in entry 17, when $\lambda$ is dominant this representation is irreducible with highest weight $\lambda$. This is the Borel-Weil theorem.
sheafic machinery does much more, because it’s not restricted to line bundles. All just need to be able to differentiate our sections, and we can build representations of $\mathcal{U}(\mathfrak{g})$. In sheafville, since differential operators are local, they themselves form a sheaf $\cal D$. If a sheaf $\cal F$ is acted on by differential operators, then for any open $U$ we have a representation of $\Gamma(U,\cal D)$ (differential operators over $U$) on $\Gamma(U,\cal F)$ (sections of $\cal F$ over $U$). We call this a sheaf of modules of $\mathcal{D}$: More succinctly, a $\cal D$-module. For any $\cal D$-module $\cal F$ over the flag manifold $G/B$, the global sections $\Gamma(G/B,\cal F)$ form a representation of $\Gamma(G/B,\cal \cal D)$, hence of $\cal U(\mathfrak{g})$, hence of $\mathfrak{g}$.
The map $\mathcal{U}(\mathfrak{g}) \to \text{Diff}(G/B)$ has kernel, which consists of all elements with nonzero central character. We have an isomorphism $\mathcal{U}_0(\mathfrak{g}) \cong \text{Diff}(G/B)$., where $\cal U_0(\mathfrak{g})$ is everything in the universal enveloping algebra for where the Casimir acts trivially. In particular, every representation of $\mathfrak{g}$ where the Casimir acts trivially is the global sections of a $\cal D$-module. This is called Bellinson-Bernstein localization. This can be extended to all representations of $\mathfrak{g}$ by incorporating something called twisted $\cal D$ modules.
BB-localization gives a geometric representation of every representation of $\mathfrak{g}$ using $\cal D$-modules on the flag manifold. This includes finite dimensional ones (as in Borel-Weil), infintie dimensional ones (constant sheafs on the big schubert cell), other weird representations or even ones we haven’t discovered yet. Useful little critters.
For more information, see these notes by David Nadler, transcribed by Jacakson Van Dyke.