Algebraic Analysis notes Lecture 1 (7 Jan 2019)

This quarter I’m taking a course called “Algebraic Analysis”. It’s being taught by Carl Mautner, who does rep theory and algebraic geometry here at UCR. I’ve had a bunch of classes with him before, and they were always fun. These are the notes from the first class. It was sort of running through the motivation from analysis, plus an overview of the rest of the quarter. He said this is the first and last time we were going to do analysis in this class.

What is Algebraic Analysis?

“analysis using ring theory and homological algebra”
Suppose we want to study a linear PDE on an open subset of $X \subseteq \mathbb C^n$

here u is some unknown function, f analytic on X. P is an element of D, the (non-commutative) ring of differential operators on X.

Let $\mathcal O$ be the commutative ring of analytic functions on X. Then $\mathcal O$ is naturally a left D-module, where D acts by differentiation.

Saito: Associate to the PDE Pu=0 the left D-module M := D/DP. Then

Note: $Hom_D(D, \mathcal O) \rightarrow \mathcal O$ by $\phi \mapsto \phi(1)$ is an isomorphism. Then under this identification

because $Pf = P\phi(1) = \phi P (1) = \phi (P1) = \phi(P) = 0$ . Hence $Hom_D(M, \mathcal O)$ is the group of holomorphism solutions to our PDE!

Note: the same is true if we replace $\mathcal O$ with any other flavor of functions that admit a natural action of D, e.g. smooth functions or Schwartz distributions.

More generally, given a system of linear PDE’s of n unknown functions $u_1, \dots, u_n$ :

$\sum_{j=1}^n P_{i,j} u_j =0$

for i=1,…,k, $P_{i,j} \in D$ . Consider the left D-module M defined as the cokernel:

where $\psi (Q_1, \dots, Q_k) = (\sum^k Q_{i_1} P_i, \dots, \sum^k Q_i P_{i_n}$ . Then similar to before, we have $Hom_D(M, \mathcal O)$ is the space of solutions to the family of equations.

So studying linear PDE’s is equivalent to studying the functor $Hom_D(-, \mathcal O)$ from the category of left D-modules admiting finite presentation to the category $\mathbb C-mod$ .

Equations can have local solutions that don’t extend… so we study sheaves. Instead of $\mathcal O$ , we study $\mathcal O_x$ sheaf of functions, and instead of D, we study $D_x$ sheaf of diff ops.

Our main objects of study then become left D_x-modules, i.e. coherent D_x-modules, and the functor $\mathscr Hom_{D_x} (-, \mathcal O_x)$ sending left coherent D_x modules to $\underline{\mathbb C}_x -mod$ .

Remark: We can let X be any complex manifold.

The space of holomorphic solutions to any ODE is finite dimensional. In order to get finite dimensionality in higher dimensions, we have to restrict to a “nice” class of D_x-modules, called holonomic.

In ODE’s, one has a good class of equations that have “mild singularities”, called regular singularities. By analogy, one defines regular holonomic D_x-modules. The functor $\mathscr Hom_{D_x} (-, \mathcal O_x)$ is not exact, so we want to study derived functors $R\mathscr Hom_{D_x} (-, \mathcal O_x)$ .

So really in this class we’ll study $R\mathscr Hom (-, \mathcal O_x)$ restricted to regular holonomic D_x-modules, mapping into $D^b(\underline{\mathcal C}_x-mod)$ .

Theorem [Riemann-Hilbert]
This functor is fully faithful and it has essential image in $D^b(\underline{\mathbb C}_x-mod)$ , an abelian category called the category of perverse sheaves.

Connection to Representation Theory

Let G be a semisimple $\mathbb C$ -algebraic group (e.g. SL_n, SO_n, Sp_n, … , E_8), let $\mathfrak g = Lie(G)$ .

Question [Verma]: how can we determine the characters of infinite dimensional highest weight modules of $\mathfrak g$ ?

[Kazhdan-Lusztig]: conjecture answer in terms of geometry of Schubert varieties, flag varieties, G/B where B is the Borel subgroup (sorry the notes get a little fuzzy here). Key idea: G acts on G/B, which gives
$\mathfrak g \rightarrow Vect(G/B)$ , which gives
$U(\mathfrak g) \rightarrow \Gamma(G/B, D_{G/B})$ .