Xinchun Ma: Cherednik Algebras and Hilbert Schemes

  • $G$ semi-simple Lie group with Lie algebra $\mathfrak{g}$ over $\mathbb{C}$,
  • $\mathfrak{h}$ Cartan acted by Weyl group $W$.

Theorem 1 (Levasseur-Stafford). We have an isomorphism $$(D(\mathfrak{g})/D(g)\operatorname{ad}\mathfrak{g})^{G}\xrightarrow{\cong}D[\mathfrak{h}]^{W}.$$

This can be understood as a quantization of Chevalley isomorphism.

Consider $G=\operatorname{SL}_ {n}$ and $V=\mathbb{C}^{n}$, and $\operatorname{GL}_ {n}$ acts on $\mathfrak{g}\times V$, one gets $\tau: \mathfrak{gl}_ {n}\rightarrow D(\mathfrak{g}\times V)$. For any $c\in \mathbb{C}$, set $\tau_ {c}:=\tau-c\operatorname{tr}$.

Then one has $$(D(\mathfrak{g}\times V)/D(\mathfrak{g}\times V)\tau_ {c}(\mathfrak{gl}_ {n}))^{\operatorname{GL}_ {n}}\cong eH_ {c}e.$$

Definition 2. The rational Cherednik algebra $H_ {c}$ is the subalgebra of $D(\mathfrak{h}_ {\text{reg}})\rtimes W$ generated by $W, x_ {i}-x_ {i+1}, y_ {i}-y_ {i+1}$ for $i=1,\dots, n-1$, where $y_ {i} = \frac{\partial}{\partial x_ {i}} - c\sum_ {j\neq i}\frac{1-(ij)}{x_ {i}-x_ {j}}$.

Example 3. Take $c=0$, then $H_ {0} = D(\mathfrak{h})\rtimes W$ and $e= \frac{1}{|W|}\sum_ {w\in W}w$. Then $e H_ {0} e = D(\mathfrak{h})^{W}$.

$eH_ {c}e$: spherical RCA.

Let $\mathscr{O}(H_ {c})$ be the category $\mathscr{O}$ of $H_ {c}$. For $\tau$ a partition of $n$, one has $\mathscr{O}(H_ {c})\ni\Delta_ {c}(\tau)\cong H_ {c}\otimes_ {S\mathfrak{h}\rtimes W}\pi_ {\tau}\twoheadrightarrow L_ {c}(\pi)$ simple, where $\pi_ {\tau}\in \operatorname{IrrRep}(S_ {n})$.

Theorem 4. Only when $c=\frac{m}{n}$, where $m,n$ are coprime, $H_ {c}$ has finite dimensional representations. When $c= \frac{m}{n}$ with $m>0$, the only simple finite dimensional module is $L_ {c}(\text{triv})$.

Definition 5 (Etingof-Gorsky-Losev). We call $M\in \mathscr{O}(H_ {c})$ of minimal support if there is no subset of $\text{supp}(M)\subseteq \mathfrak{h}$ of smaller dimension that is the support of some $N\in \mathscr{O}(H_ {c})$.

Theorem 6 (Wilcox). When $c=\frac{m}{n}$ and $m>0$, the simple minimal support modules are $L_ {c}(n_ {0}\tau)$, where $d=\operatorname{gcd}(m,n)$ and $n_ {0}=\frac{n}{d}$, $\tau$ a partition of $d$, which is of support $W\mathfrak{h}_ {d}$, where $\mathfrak{h}_ {d}=\{x_ {1}=\cdots = x_ {n_ {0}}, \dots, x_ {(d-1)n_ {0}+1}=\cdots\}$.

Set $T_ {m,n}:=\{x^{m}=y^{n}\}\cap S_ {\epsilon}^{3}$. For example, $T_ {4,2}$ is 2 linked unknots.

The element $h=\frac{1}{2}\sum_ {i=1}^{n}(x_ {i}y_ {i}+y_ {i}x_ {i})$ acts on $L_ {c}(\lambda)=\oplus_ {k\in \mathbb{Z}}L_ {c}(\lambda)(k)$, and $\text{ch}_ {q}(L_ {c}(\lambda)) = \sum_ {k}q^{k}\text{dim}L_ {c}(\lambda)(k)$.

Theorem 7 (Gorsky-Oblomkov-Rasmussen-Shende). For $m>0$, $$\sum_ {u \vdash d}\text{dim}\pi_ {u} \sum_ {i}a^{2i}\text{ch}_ {q}(\operatorname{Hom}_ {S_ {n}}(\wedge^{i}\mathfrak{h},L_ {c}(n_ {0}u)))=\operatorname{Hom} \operatorname{FLY}_ {a,q}(T_ {m,n}).$$

Conjecture 8. There exists a filtration on $L_ {c}(n_ {0}u)$ such that $$\sum_ {u \vdash d}\text{dim}\pi_ {u} \sum_ {i}a^{2i}\text{ch}_ {q,t}(\operatorname{Hom}_ {S_ {n}}(\wedge^{i}\mathfrak{h},\text{gr}^{F}L_ {c}(n_ {0}u)))= \text{dim}_ {a,q,t}\operatorname{HHH}(T_ {m,n}).$$

Theorem 9 (Ma). Conjecture holds if $(m,n)=1$ with respect to $F^{\text{Hodge}}$ ($F^{\text{ind}}= F^{\text{Hodge}}=F^{\text{alg}}$).

Remark 10. There is $m,n$-symmetry and $q,t$-symmetry.

Proof strategy

Construct $\mathscr{F}_ {c}\in \operatorname{Coh}^{\mathbb{C}^{\times}\times \mathbb{C}^{\times}}(\operatorname{Hilb}^{n}(\mathbb{C}^{2}))$ whose equivariant $K$-theory class computes LHS.

Hamiltonion reduction (Gan-Ginzburg) \begin{align} \mathbb{H}_ {c}: (D(\mathfrak{g}\times V),\mathfrak{gl}_ {n})\text{-Mod} &\rightarrow eH_ {c}e\text{-mod}, \\ M & \mapsto M^{\tau_ {c}(\mathfrak{gl}_ {n})},\\ ? & \mapsto eL_ {c}(n_ {0}u). \end{align} Consider $\mathfrak{h}_ {d}^{\circ} = \{\begin{pmatrix} x_ {1} & & & & \\ & x_ {1} & & & \\ & & \ddots & & \\ & & & x_ {2} & \\ & & & & \ddots \end{pmatrix}: x_ {i}\neq x_ {j}, i\neq j\}$. Then $Y_ {d}=\mathfrak{g}_ {\text{reg}}\times_ {\mathfrak{h}/W}\mathfrak{h}_ {d}^{\circ}$ and $\pi_ {1}(Y_ {d})=\operatorname{Br}_ {d}\times \mathbb{Z}/n_ {0}\mathbb{Z},$ where $n_ {0}=n/d$.

Take $\mathscr{L}_ {c}$ to be local system on $Y_ {d}$ corresponding to $c\in (\mathbb{Z}/n_ {0}\mathbb{Z})^{\ast}$ and let $M_ {c}$ be the minimal extension of $\mathscr{L}_ {c}$ to $\mathfrak{g}\times \mathfrak{h}_ {d}$. Let $p:\mathfrak{g}\times \mathfrak{h}_ {d}\rightarrow \mathfrak{g}$ and $p_ {\dagger}M = \oplus_ {\tau\vdash d}\pi_ {\tau}\otimes N_ {c}(\tau)$, for $\pi_ {\tau}\in \operatorname{Irr}(S_ {n})$.

Example 11. For $d=n$, $Y_ {d}=\widetilde {\mathfrak{g}}_ {\text{rs}} $ and $M_ {c}$ HC $\mathscr{D}$-module, and $p_ {\dagger}M$ is the Springer $\mathscr{D}$-module in the sense of Hotta-Kashiwara. When $d=1$, $M_ {c}$ is cuspidal character $\mathscr{D}$-module studied by Lusztig.

Theorem 12 (Calaque-Enriques-Etingof). One has $\mathbb{H}_ {c}(N_ {c}(\tau)) = eL_ {c}(n_ {0}\tau)$.

  • $M_ {c}$: Hodge module $F^{H}$.
  • $\pi_ {\ast}\text{gr}^{H}M_ {c}$ is supported on $\{[x,y]=0\}$ ,where $\pi: T^{\ast}(\mathfrak{g}\times \mathfrak{h}_ {d})\rightarrow T^{\ast}\mathfrak{g}$. This gives $\mathscr{F}_ {c}\in \operatorname{Coh}(\operatorname{Hilb}^{n}(\mathbb{C}^{2}))$, where $\operatorname{Hilb}^{n}(\mathbb{C}^{2})=\{(x,y,v): x,y\in \mathfrak{g}, v\in V, [x,y]=0, \mathbb{C}[x,y]v=V\}/\operatorname{GL}_ {n}$.

What is $[\mathscr{F}_ {c}]\in K^{\mathbb{C}^{\times}\times \mathbb{C}^{\times}}(\operatorname{Hilb}(\mathbb{C}^{2}))$?

Elliptic Hall algebra $$ A= \frac{\mathbb{C}(q,t)\langle P_ {m,n},(m,n)\in \mathbb{Z}^{2}-\{0\} \rangle}{[P_ {m,n},P_ {m^{\prime},n^{\prime}}]=P_ {m+m^{\prime},n+n^{\prime}}+\cdots}.$$

Then $A$ acts on $\oplus_ {n} K^{\mathbb{C}^{\times}\times \mathbb{C}^{\times}}(\operatorname{Hilb}^{n}(\mathbb{C}^{2}))\otimes_ {\mathbb{C}[q^{\pm},t^{\pm}]}\mathbb{C}(q,t)\cong \mathbb{C}(q,t)[z_ {1}z_ {2}\cdots]^{S_ {\infty}}.$

Theorem 13. $\oplus_ {u\vdash d}\text{dim}\pi_ {u}\text{ch}_ {q,t}(L_ {c}(n_ {0}u))\sim K[\mathscr{F}_ {c}]=P_ {m_ {0},n_ {0}}^{d}\cdot 1\sim \operatorname{HHH}(T_ {m,n})$ when $(m,n)=1$ (by Mellit, when $m>0$ conjectured by Wilson), where $m_ {0}=m/d$ and $n_ {0}=n/d$.

THe case $\mathscr{F_ {c}}=P$ and $c=0$ is proved by Ginzburg.

What is $\pi_ {\ast}\text{gr}^{H}M_ {c}$?

Assume that $(m,n)=1$. Consider $\mathfrak{m}\supseteq \mathfrak{b}\supseteq \mathfrak{n}$, and then $$G/B\xleftarrow {q}G\times^{B}(\mathfrak{m}\times \mathfrak{n})\leftarrow \mathfrak{Z}^{\mathrm{DG}}$$ and $p:G\times^{B}(\mathfrak{m}\times \mathfrak{n})\rightarrow T^{\ast}\mathfrak{g}$, and $\mathfrak{Z}^{\mathrm{DG}}:= G/B\times_ {G\times^{B}\mathfrak{b},[-,-]}G\times^{B}(\mathfrak{m}\times \mathfrak{n})$.

Then $\pi_ {\ast}\text{gr}^{H}M_ {c}= p_ {\ast}(q^{\ast}\mathscr{L}_ {\lambda_ {c}}\otimes\mathscr{O}_ {\mathfrak{Z}^{\mathrm{DG}}})$, $\lambda_ {c}=(\lceil c\rceil,\lceil 2c\rceil-\lceil c\rceil,\dots)$.

For general $m>0$, take blocks.

Convolution product of coprime case.