Zhiwei Yun: Hitchin Moduli Spaces and Wildly Ramified Geometric Langlands

The talk is based on [Bezrukavnikov-Boixeda Alvarez-McBreen-Yun].

Talk 1: Homogeneous Affine Springer Fibers

Let $\mathfrak{g}\supseteq \mathcal{N}\ni e$. We have $\mathfrak{sl}_ {2}$-triple $(e,h,f)$ ($[h,e]=2e,$ $\operatorname{Ad}(h(s))e = s^{2}e$) and we construct $$S_ {e}^{\mathfrak{g}} = e+ \mathfrak{g}^{f}$$ transverse slice to the adjoint orbit of $e$.

$$S_ {e}: = S_ {e}^{\mathfrak{g}}\cap \mathcal{N}$$ is singular in general.

Note that $\pi_ {e}$ is a symplectic resolution and $\widetilde {S}_ {e} $ is a smooth symplectic variety.

There is a conic structure: $\mathbb{C}^{\times}$ acts on $S_ {e}$ contracting to $\{e\}$, via $h:\mathbb{G}_ {m}\rightarrow G$, i.e. $s\in \mathbb{C}^{\times}$ acts by $s^{2}\cdot \operatorname{Ad}(h(s^{-1}))$.

gives constructible quantization $\operatorname{Sh}((U,\psi_ {e})\backslash \mathcal{B}),$ where $\mathcal{B}=G/B$.

Affine Analogue

We replace $\mathfrak{g}$ with $L\mathfrak{g}=\mathfrak{g}\otimes_ {\mathbb{C}}\mathbb{C}(\!(t)\!)$ and replace nilpotent elements with topologically nilpotent elements. Let $F= \mathbb{C}(\!(t)\!)$.

Assume that $\gamma\in L\mathfrak{g}$ is topologically nilpotent and regular semi-simple (i.e. $\lambda_ {i}\neq \lambda_ {j}$ for $i\neq j$ in Example 1).

What is the analogue of $\mathfrak{sl}_ {2}$-triple $(\gamma,?,?)$?

We still let $h: \mathbb{G}_ {m}\rightarrow LG$ and $\operatorname{Ad}(h(s))\gamma = s^{?\neq 0} \gamma$. Compute the characteristic polynomial one sees that $\text{char}(\gamma)\neq \text{char}(s^{?}\cdot \gamma)$. This is impossible. Instead, let $h: \mathbb{G}_ {m}\rightarrow LG\rtimes \mathbb{G}_ {m}^{\text{rot}}$, where $\mathbb{G}_ {m}^{\text{rot}}$ scales $t\in \mathbb{C}(\!(t)\!)$.

Construct homogeneous elements

Let $T\subseteq G$.

Let $h=(\lambda,m): \mathbb{G}_ {m}\rightarrow T\times \mathbb{G}_ {m}^{\text{rot}}$ where $\lambda\in \mathbb{X}_ {\ast}(T)$ and $m\in \mathbb{N}$.

Then $\operatorname{Ad}(h)$ gives $\mathbb{G}_ {m}$ action on $L\mathfrak{g}$, and provides a decomposition $L\mathfrak{g}=\bigoplus_ {i\in\mathbb{Z}}(L\mathfrak{g})(i)$, where $\gamma\in (L\mathfrak{g})(d)$ satisfies $\operatorname{Ad}(s^{\lambda})\text{rot}(s^{m})\gamma = s^{d}\gamma$ and if regular semi-simple, it is homogeneous of slope $\frac{d}{m}$.

Fact (Reeder-Yu): there exist homogeneous elements of slope $\nu=\frac{d}{m}$ if and only if $m$ is the order of a regular element in $W$.

Consider $W$ acts on $\mathfrak{h}$ by $\mathbb{C}$-reflection representation.

Fact: $\{\text{regular elements in }W\}/\sim^{\text{conj}}$ is classified by their orders.

For $\gamma$ homogeneous of slope $\frac{d}{m}$, $T_ {\gamma}:=C_ {LG}(\gamma)$ maximal torus in $G/F$.

Then $$\{\text{maximal tori in }G/F\}/_ {\sim^{G(F)}}\leftrightarrow H^{1}(F,W)\leftrightarrow\text{conjugacy classes in }W$$ We get $T_ {\gamma}\leftrightarrow [w]$ regular order $m$.

Take $\mathfrak{sl}_ {n}$, $h= (\rho^{\vee}, n):\mathbb{G}_ {m}\rightarrow T_ {\text{ad}}\times \mathbb{G}_ {m}^{\text{rot}}$, $$L\mathfrak{g}(1) = \oplus_ {i=0}^{n-1}(L\mathfrak{g})_ {\alpha_ {i}},$$ where $\{\alpha_ {0}=1-\theta, \alpha_ {1},\dots, \alpha_ {n-1}\}$ affine simple roots. Then $$(L\mathfrak{g})_ {\alpha_ {0}}= t\mathfrak{g}_ {-\theta}.$$

Then $\gamma\in L\mathfrak{g}(1)$ non-zero in each $(L\mathfrak{g})_ {\alpha_ {i}}$.

For general $G$, replace $n$ with the Coxeter number. Then one get a regluar element of slope $\frac{1}{\text{Coxeter number}}.$

For general slope $\nu = \frac{d}{m}$, one repeats the construction $(\rho^{\vee},m): \mathbb{G}_ {m}\rightarrow T_ {\text{ad}}\times \mathbb{G}_ {m}^{\text{rot}}$ and $\gamma\in (L\mathfrak{g})(d)$. If $m$ is regular, then $(L\mathfrak{g})(d)$ contains a regular semi-simple element and any such gives homogeneous element of slope $\nu$.

Affine Springer Fibers

This is studied by Kazhdan-Lusztig in late 80s.

Now assume that $\gamma$ is homogeneous of slope $\nu=\frac{d}{m}>0$ (which implies that $\gamma$ is topologically nilpotent).
$$\mathrm{FL}_ {\gamma}:=\{gI\in LG/I: \operatorname{Ad}(g^{-1})\gamma\in \operatorname{Lie}(I^{+})\}.$$

Compare $\mathcal{B}_ {e} = \{gB\in G/B: \operatorname{Ad}(g^{-1})e\in n_B\}.$

Fact: $\mathrm{FL}_ {\gamma}$ is finite-dimensional. One has $$\text{dim}(\mathrm{FL}_ {\gamma}) = \frac{\nu\cdot |\Phi| - \text{dim}(\mathfrak{h}/\mathfrak{h}^{w})}{2}.$$

The definition of affine Springer fibers make sense for any topologically nilpotent $\gamma$ but when it is homogeneous, there is a $\mathbb{G}_ {m}$ action on $\mathrm{FL}_ {\gamma}$, by $$\operatorname{Ad}(s^{\rho^{\vee}})\text{rot}(s^{m}),$$ which one can check preserves $\mathrm{FL}_ {\gamma}$.

The picture

has affine analogue

and $\mathbb{C}^{\times}$ contracts the Hitchin base $\mathscr{A}_ {\gamma}$ to $\{a_ {\gamma}\}.$

Geometry of $\mathrm{FL}_ {\gamma}$

1. For slope $\frac{1}{\text{Coxeter number}}$, we get $\mathrm{FL}_ {\gamma}=\ast$ a single point ($G$ simply connected).
2. $\mathfrak{sl}_ {2}$, slope $1$, $\gamma=\gamma_ {0}\cdot t$ (e.g. $\gamma = \begin{pmatrix} t & 0\\ 0 & -t \end{pmatrix}$), we have $\infty$-chain of $\mathbb{P}^{1}$.
3. $\mathfrak{sl}_ {2}$, slope $\frac{3}{2}$, $\begin{pmatrix} 0 & t\\ t^{2} & 0 \end{pmatrix}$ and $\mathrm{FL}_ {\gamma}\cong\mathbb{P}^{1}\vee \mathbb{P}^{1}$.
4. $\mathfrak{sl}_ {3}$, slope $\frac{2}{3}$, $\mathrm{FL}_ {\gamma}=\mathbb{P}^{1}\vee \mathbb{P}^{1}\vee \mathbb{P}^{1}$.
5. Bernstein’s example: $\operatorname{Sp}_ {6}$, $\gamma$ slope $\frac{1}{2}$, $\gamma=\begin{pmatrix} 0 & A \\ tB & 0 \end{pmatrix}$ with $A,B\in \operatorname{Sym}^{2}(\mathbb{C}^{3})$ generic, then $(\mathrm{FL}_ {\gamma})^{\mathbb{G}_ {m}}$ in general is a disjoint union of smooth projective varieties (Hessenberg variety). In this example, one of the Hessenberg variety is isomorphic to elliptic curve.

Talk 2: Hitchin Moduli and Their Quantization

Take $\gamma$ a homogeneous element in $L\mathfrak{g}$ with slope $\nu = \frac{d}{m}$. Hichin fiber $M_ {\gamma}$ will turn out to be smooth symplectic, containing $\operatorname{FL}_ {\gamma}$ as a largrangian.

Take $e,f,h$ a $\mathfrak{sl}_ {2}$-triple and $\mathfrak{g} = \oplus_ {i\in \mathbb{Z}} \mathfrak{g}(i) $ grading by $h$. We have $e\in \mathfrak{g}(2)$. The centralizer $C_ {G}(e)\subseteq G(\geq 0)$. Then $G(\leq -2)$ acts on $e+\mathfrak{g}(\leq 0)$ freely.

Then $S_ {e}^{\mathfrak{g}}\cong G(\leq -2)\backslash (e+\mathfrak{g}(\leq 0))$. We have a Cartesian square

Now we move to affine setting.

Consider $L\mathfrak{g}=\oplus_ {i\in \mathbb{Z}}L\mathfrak{g}(i)$ grading by $(\check{\rho},m): \mathbb{G}_ {m}\rightarrow T_ {\text{ad}}\times \mathbb{G}_ {m}^{\text{rot}}.$ Take $\gamma\in L\mathfrak{g}(d)$ ($\sim$ Moy-Prasad group), then $L\mathfrak{g}(\leq -d)$ acts on $\gamma+L\mathfrak{g}(\leq 0)$.

There is an analogy:

Set $T_ {\gamma}=C_ {LG}(\gamma)$ loop group of a maximal torus of $G/F$. Then $L\mathfrak{g}\supseteq \mathfrak{t}_ {\gamma}=\hat{\oplus}_ {i}\mathfrak{t}_ {\gamma}(i).$

Then $K_ {\gamma}:=LG(\leq -d) T_ {\gamma}(< 0)$ acts on $\gamma+ L\mathfrak{g}(\leq 0)$. The subalgebra $L\mathfrak{g}(\leq 0)$ is parahoric subalgebra of $L_ {\infty}\mathfrak{g}$.

Think of (complete in $i\rightarrow -\infty$) $K_ {\gamma,\infty}\subseteq L_ {\infty}G=G(\!(t^{-1})\!)$.

Consider $\operatorname{Bun}_ {G}(I_ {0},K_ {\gamma,\infty})$ moduli stack of $G$-bundles on $\mathbb{P}^{1}$ with Iwahori level at $0$ and $K_ {\gamma,\infty}$ level at $\infty$. Concretely, given a $G$-bundle $\mathscr{E}$, Iwahori level at 0 means choosing a full falg of $\mathscr{E}|_ {0}$.

A higgs bundle $(\mathscr{E},\varphi)$ on $\mathbb{P}^{1}$ is

• a $G$-bundle $\mathscr{E}$ on $\mathbb{P}^{1}$,
• $\varphi$ a section of $\operatorname{Ad}(\mathscr{E})\otimes \omega_ {\mathbb{P}^{1}}$ (may have poles).

Then $M_ {\gamma}$ classifies $(\mathscr{E},\varphi)$, where

• $\mathscr{E}\in \mathrm{Bun}_ {G}(I_ {0}, K_ {\gamma,\infty})$,
• $\varphi$: a rational section of $\operatorname{Ad}(\mathscr{E})\otimes \omega_ {\mathbb{P}^{1}}$ such that
  • regular over $\operatorname{P}^{1}-\{0,\infty\}$,
  • at $0$, simple pole, and $\operatorname{res}_ {0}(\varphi)\in n(\mathscr{E}_ {0,B})$ (strictly upper triangular with respect to the full flag), where $\mathscr{E}_ {0,B}$ is the $B$-reduction of $\mathscr{E}_ {0}$.
  • at $\infty$, choose a trivialization of $\mathscr{E}|_ {D_ {\infty}}$ with $K_ {\gamma,\infty}$-level under which requre that $\varphi|_ {D_ {\infty}}\in -(\gamma+(L\mathfrak{g})(\leq 0))\frac{dt^{-1}}{t^{-1}}$.

The condition is motivated by the fiber product

There is another construction using Hamontonian reduction $$ G(\leq -2)\backslash e+\mathfrak{g}(\leq 0) \cong U_ {e}\backslash e+u_ {e}^{\perp},$$ amd $G(\leq -2)\subseteq U_ {e}$ is largrangian. Then $$\widetilde {S}_ {e} = T^{\ast}\mathcal{B}/\!/_ {e} U_ {e} $$ where $e\in u_ {e}^{\ast}$.

where

• $J_ {\gamma}:=LG(\lesssim -\frac{d}{2})T_\gamma(<0)$,
• the left column acts on the right column,
• $LG(< -\frac{d}{2})\subseteq LG(\lesssim -\frac{d}{2})\subseteq LG(\leq -\frac{d}{2})$, the first inclusion is largrangian modulo $\mathfrak{t}_ {\gamma}(-\frac{d}{2})$, and the quotient $L\mathfrak{g}(-\frac{d}{2})$ has alterating form $\langle \gamma,[-,-] \rangle$ with kernel $\mathfrak{t}_ {\gamma}(-\frac{d}{2})$.

One define $M_ {\gamma}^{\prime}$ similarly to $M_ {\gamma}$: replace $K_ {\gamma,\infty}$ with $J_ {\gamma,\infty}$ and $\varphi\in (\gamma+(\operatorname{Lie}J_ {\gamma})^{\perp})\frac{dt^{-1}}{t^{-1}}$. Then $$M_ {\gamma}\cong M_ {\gamma}^{\prime}\cong T^{\ast}\operatorname{FL}/\!/_ {\gamma}\,J_ {\gamma},$$ where $J_ {\gamma}$ acts on $\operatorname{FL}=LG/I_ {0}$ and $\gamma\in (\operatorname{Lie}J_ {\gamma})^{\ast}$.

Hitchin base

$\mathscr{A}_ {\gamma}$ affine space parametrizing all possible characteristic polynomial of $\varphi$ from $M_ {\gamma}$. Assume that $(f_ {1},\dots, f_ {r})$ generators of $\mathbb{C}[\mathfrak{g}]^{G}$ of degree $d_ {1},\dots,d_ {r}$. Then $f_ {i}(\varphi)$ is rational section of $\omega_ {\mathbb{P}^{1}}^{\otimes d_ {i}}$.

We have $$f_ {i}(\varphi) = f_ {i}(\gamma)+(\text{deg in $t$ $\leq (d_ {i}-1)\nu$ no constant term}) $$ and $f_ {i}(\gamma) = (\ast)\cdot t^{d_ {i}\nu}$ if $d_ {i}\nu\in \mathbb{Z}$ and 0 otherwise.

We have $\mathscr{A}_ {\gamma}\hookrightarrow (\mathfrak{g}/\!/G)(\!(t)\!)$ and $$\operatorname{dim}\mathscr{A}_ {\gamma} = \frac{1}{2}\operatorname{dim}M_ {\gamma}=\text{dim}\mathrm{FL}_ {\gamma}=\frac{1}{2}(|\Phi|\nu-\operatorname{dim}(\mathfrak{h}/\mathfrak{h}^{w})).$$

Talk 3: Betti Moduli Spaces, Wildly Ramified Geometric Langlands

Recall last time

We have $M_ {\gamma}\cong T^{\ast}\operatorname{FL}/\!/_ {\gamma} J_ {\gamma} \,\text{“}\cong\text{”}\, T^{\ast}(\operatorname{Bun}_ {G}(I_ {0},(J_ {\gamma,\infty},\gamma)))$.

Let $J_ {\gamma,\infty}^{\mathrm{Ker}}:=\operatorname{Ker}(\gamma: J_ {\gamma,\infty}\rightarrow \mathbb{G_ {a}})$, then $$\operatorname{Bun}_ {G}(I_ {0}, J_ {\gamma,\infty}^{\mathrm{Ker}})\rightarrow \operatorname{Bun}_ {G}(I_ {0},J_ {\gamma,\infty})$$ is a $\mathbb{G}_ {a}$-torsor, and $(T^{\ast}\operatorname{Bun}_ {G}(I_ {0},J_ {\gamma,\infty}^{\mathrm{Ker}}))/\!/_ {1} \mathbb{G}_ {a}$ constructible quantization of $M_ {\gamma}$, and one can consider “$\operatorname{Sh}(\operatorname{Bun}_ {G}(I_ {0},(J_ {\gamma,\infty},\gamma)))$”.

Over a space in characteristic $p$ and work with $\overline{\mathbb{Q}_ {\ell}}$-sheaves, one can consider $$\operatorname{Sh}_ {\mathbb{G}_ {a},\mathrm{AS}}(\operatorname{Bun}_ {G}(I_ {0},J_ {\gamma,\infty}^{\mathrm{Ker}})),$$

and over a space in characteristic 0, one can consider $$\mathscr{D}\text{-}\operatorname{mod}_ {\mathbb{G}_ {a},\mathrm{exp}}(\operatorname{Bun}_ {G}(I_ {0},J_ {\gamma,\infty}^{\mathrm{Ker}})).$$

In general, one can use Kirillov model introduced by Gaitsgory: for a space $X$ acted by $\mathbb{G}_ {a}\rtimes \mathbb{G}_ {m}$, one can consider $$\operatorname{Kir}(X)=\operatorname{Sh}_ {\mathbb{G}_ {m}}(X)/\operatorname{Sh}_ {\mathbb{G}_ {a}\rtimes \mathbb{G}_ {m}}(X).$$

Non-abelian companions of $M_ {\gamma}$ (Simpson correspondence)

Recall that $K_ {\gamma,\infty} = LG(\leq -d)T_ {\gamma}(<0)$.

de Rham moduli

We define $M_ {\gamma}^{\text{dR}}$ to classify $(\mathscr{E},\nabla)$, where

• $\mathscr{E}\in \operatorname{Bun}_ {G}(I_ {0},K_ {\gamma,\infty}),$
• $\nabla \text{ a $G$-connection}$,

such that

• $\nabla|_ {D_ {0}}\text{ simple pole, }\operatorname{res}_ {0}(\nabla)\in n(\mathscr{E}_ {0,B}),$
• $\nabla|_ {D_ {\infty}}\in d+ (\gamma+(L\mathfrak{g})(\leq 0))\frac{dt^{-1}}{t^{-1}}$.

Then $M_ {\gamma}^{\text{dR}}$ is smooth symplectic algebraic space.

There is a $\mathbb{G}_ {m}$-action on $M_ {\gamma}^{\text{Hod}}$ contracting to $\operatorname{FL}_ {\gamma}\subseteq M_ {\gamma}^{\text{Dol}}$.

Betti moduli space

The Betti moduli space classifies $G$-local system on $\mathbb{P}^{1}\backslash \{0,\infty\} + $ Stokes data at $\infty$.

Let $\beta\in B_ {W}^{+} = \{s_ {1},\dots,s_ {r} : \text{braided relation}\}$, and let $\beta= s_ {i_ {1}}\cdots s_ {i_ {n}}$ be a reduced expression. Then $\ell(\beta)=n$.

Let $M(\beta) = \{B_ {0}\xrightarrow{s_ {i_ {1}}}B_ {1}\xrightarrow{s_ {i_ {2}}}\cdots\xrightarrow{s_ {i_ {n}}}B_ {n}={}^{g}B_ {0}\}/\operatorname{Ad}(G)$, where $g$ is the topological monodromy $g\in G/\operatorname{Ad}G$ and is part of tehe data. This is related to

• Lusztig’s character sheaves,
• Shende-Trevmann-Zazlow,
• Minh-Tam Trinh’s thesis.

Alternative definition:

$M(\beta)= \{E_ {0},\dots, E_ {n}\text{ $B$-torsors }, E_ {0}\xrightarrow{s_ {i_ {1}}}\cdots \xrightarrow{s_ {i_ {n}}}E_ {n}\xrightarrow[\cong]{\text{iso of $B$-torsors}} E_ {0},\}$, where $E_ {0}\times^{B}G\xrightarrow{\cong}E_ {1}\times^{B}G$ isomorphism of $G$-torsors.

Let $B\twoheadrightarrow T=$universal Cartan, and $w=$image of $\beta$ in $W$.

We have a correspondence $$T/\operatorname{Ad}_ {w}T\xleftarrow {\text{formal monodromy}}M(\beta)\xrightarrow{\text{top monodromy}}G/\operatorname{Ad}(G). $$

Let $\gamma$ be homogeneous with slope $\nu = \frac{d}{m}$. Let $[w]$ be a regular conjugacy class in $W$ of order $m$. Assume that $\gamma$ is elliptic. Then one can construct $$\beta_ {\gamma}=\beta_ {\nu} = \dot{w}\dot{w}\cdots \dot{w}\in B_ {w}^{+}$$ $d$-many products, where $\dot{w}$ is a minimal length representative of $[w]$.

Then we have a Cartesian square

Wild ramified geometric Langlands conjecture

Let $\gamma$ be homogeneous as before with slop $\nu.$ We can consider $M_ {\check{G}, \check{\gamma}}^{\text{Betti}},$ where $\check{\gamma}$ has the same slope $\nu.$

Then the conjecture is that there is a fully faithful embedding

$$\operatorname{Sh}_ {\operatorname{FL}_ {\gamma}}(\operatorname{Bun}_ {G}(I_ {0},(J_ {\gamma,\infty})))\hookrightarrow \operatorname{IndCoh}(M_ {\check{G}, \check{\gamma}}^{\text{Betti}}),$$ where the subsript on the left hand side means singular support.

Evidence: The cases $\nu=1$ and $\nu=\frac{1}{m}$ are work in progress. In the case $\nu=1$, the essential image may be $\operatorname{Coh}_ {B^{\vee}}^{T^{\vee}}(\widetilde {\mathcal{N}} )$.