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--- blog:2023-07-26 [2023/07/26 18:21] – [Ginzburg] pzhou
+++ blog:2023-07-26 [2023/07/27 05:59] (current) – [2023-07-26] pzhou
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 ====== 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
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 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.
+===== Heisenberg Algebra, Weyl algebra, Fock module =====
+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.
+===== $G$-equivariant sheaf vs equivariant coherent sheaf=====
+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.