2023-07-26

Declaration: I was lost in abstract definitions for too long. I will now compute examples, examples, and examples. Not only because abstraction makes me sleepy, but also examples makes my hands dirty and my mind happy.

• Reading Ginzburg's paper
• Heisenberg algebra and Weyl algebra (just names)
• Writing my own paper

So, I got curious about Geometric Langlands. I have heard about the two words, and also used it myself, but I don't know exactly what I am talking about. There are some equivalence of categories, which is a nice way to package my ignorance. There is a hard side, and an easy side. The hard side involves certain constructible sheaves on $Bun_G$ with some singular support in the global Nilpotent cone, the easy side involves some local system on the same curve but valued in the dual Langlands group.

Is the abelian case known?

What is an example of this? Some sample curve, and sample group?

Under some nice cases, the local Hecke eigensheaf in $D^bSh(Bun_G(X))$ is also a global Hecke eigensheaf. How to think about $Bun_G$ and sheaves on it? A point in $Bun_G$ is a $G$-bundle over $X$, but we are not just thinking about one G-bundle, and we are not moving one G-bundle to another. So, what is Hecke transformation?

Well, this is just like matrix and vectors. We use vector to store numbers on a bunch of points, and we use matrix to recombine them. If we replace the finite set of points by some 'continuous space', then what we do is integral transform on space of functions (or distributions) on that space. Now, we categorify. Still over a finite collection of points, we can consider a bunch of vector spaces. We want to output a new set of vector spaces. How to build a new vector space?

• there is something called pushforward. What does that mean? It is like taking limit of some diagram (does taking projective limit commute with taking $Hom(X, -)$? limit is the universal receiver of Hom-to). For example, taking intersection of two subspaces is taking 'limit'. This is like compressing data, data cleaning.
• Taking colimit, this is like collecting data. You read-in input vector spaces on each site, and combines them. And you solve equation in the big vector space.
• Now, back to the 'Hecke modification'. The ambient space is just the broadest, most tolerant space, just a pair of $G$-bundles, isomorphic away from one point. Say, we fix that point. Now, we are going to make some modifications. We put some IC sheaves on those orbit closures. Strange, strange, there is not much choices involved in IC sheaves. Just like, there are not much choices in $Rep(G^\vee)$, they are just direct sums of irreducible representations. Well, they have interesting tensor product!

How to think about semi-simple category? Other than label the irreducible objects, there is not much left to do. Now, if we have monoidal structure, then we can say, we have a 'semi-ring', that is, we have summation, multiplications. Again, not very interesting. So, you are saying that, the Satake category is not interesting? In a sense, no. And the K-theory of that category is just the representation ring.

However, there is something a bit non-trivial here. If you take the K-theory of the Satake category, we are supposed to get constructible functions on these strata, say over a finite field. Then, we just do summation (since finite field, finite set, summations is good enough. The size may depend on $q$, but that is OK).

We can do 'bi-invariant' functions, which is as good as constructible functions. What is equivariant BM homology? I don't know. Can I say 'structure sheaf on some orbit closure'? It might be a singular space, still OK do so? Sure. OK, the only stupid thing that I know is about orbit closure. I think the good thing to do is to take 'IC extension' of perverse (coherent) sheaves from whatever is natural on the smooth open orbit itself. So I don't really know what is the BM homology.

The Heisenberg algebra $h_n$ is a Lie algebra (possibly infinite dimensional), generated by $p_i, q_i$ and a central element $c$, given by $[p_i, q_j] = c \delta_{ij}$.

The Weyl algebra is the universal enveloping algebra $U(h_n) / (c=1)$.

Just take the 1-dimensional case $n=1$. There is another set of generators, take $a = p-iq, a^\dagger = p+iq$, then $[a,a^\dagger]=2ic$, maybe after you do some normalization, you can make $[a, a^\dagger]=1$.

What is Stone-Von Neumann theorem? It says, for $c\neq 0$, there is only one unitary irreducible reprensentation of $h_1$, upto isomorphism. (read Folland, analysis on Phase space).

• You can either do $L^2(\R)$, and do Harmonic oscillator's eigenstates;
• or you can do coherent state, $a = \d_z, a^\dagger = z \cdot $, and the inner product is using the Bargmann-Fock metric.

Let $G=GL_n$, a linear algebraic group.

The claim is, $G$-equivariant sheaf of a point is stupid, just vect; but $G$-equivariant coherent sheaf on a point is very clever. Why?

First off, we can consider the definition. Let $X$ be a space, we have $p, a: G \times X \to X$. An equivariant sheaf is a sheaf $F$ on $X$, together with the data of an isomorphism $\phi: p^*F \to a^*F$. Now, if $F$ is just a sheaf, then, we just have sheaf pullback, $p^{-1}(F)$ and $a^{-1}(F)$, and if $X$ is a point, then we have $p^{-1}(F), a^{-1}(F)$ just being constant sheaf, and the isomorphism is just the same as automorphism. If $F$ is pulled back as a coherent sheaf, then we tensor with the $O_G$, structure sheaf of upstairs. For $F$ a represenation of $G$, we can have algebraic map $G \to Aut(V)$, which is just $Aut(O_G \otimes V)$.

So, no luck? How about $I$-equivariant perverse sheaf on $G(K)/I$? Of course, everything maps.