Ekaterina Bogdanova: Non-vanishing of quantum geometric Whittaker coefficients

Let $\mathbb{k}=\mathbb{C}$ and $X$ a smooth projective connected curve over $\mathbb{k}$.

Let $G$ be a simple adjoint group, $\check{G}$ be its Langlansd dual, $\Lambda=\mathbb{X}_ {\ast}$ be the coweight lattice of $G$, and $I_ {c}$ be the set of simple roots.

Notation:

  • Let $\kappa = c_ {\kappa}\kappa_ {\mathrm{Killing}}$ for $\mathfrak{g}$, and similarly $c_ {\check{\kappa}}$.
  • $c_ {\kappa}\neq -\frac{1}{2}$.
  • $(\kappa-\kappa_ {\text{crit}})|_ {\mathfrak{t}}$ and $(\check{\kappa}-\check{\kappa}_ {\text{crit}})|_ {\check{\mathfrak{t}}=\mathfrak{t}^{\ast}}$ are dual symmetric bilinear form.

Assume that $c_ {\kappa}\in \mathbb{Q}$.

Conjecture 1. Quantum GLC predicts an equivalence $\mathscr{D}_ {\kappa}(\mathrm{Bun}_ {G})\xrightarrow[\cong]{\mathbb{L}_ {\kappa}} \mathscr{D}_ {-\check{\kappa}}(\mathrm{Bun}_ {\check{G}})$, where $\mathscr{D}_ {\kappa}(\mathrm{Bun}_ {G}) := \mathscr{D}_ {\text{det}^{c_ {\kappa}}}(\mathrm{Bun}_ {G})$.

For $\kappa = \kappa_ {\text{crit}}$ and $\check{\kappa}=\infty$.

Theorem 2. $\mathscr{D}_ {\kappa_ {\text{crit}}}(\mathrm{Bun}_ {G})\cong \operatorname{QCoh}(T^{\ast}_ {\operatorname{det}}\mathrm{Bun}_ {\check{G}})\cong \operatorname{QCoh}(\mathrm{LS}_ {\check{G}}(X))$ and $\text{det}$ is the determinant line bundle.

What properties fix $\mathbb{L}_ {\kappa}?$

  1. Hecke eigenproperty: $\mathbb{L}_ {\kappa}$ intertwines $\mathscr{D}_ {L^{+}G}(LG/L^{+}G)$-action with appropriate action on right hand side.

Remark 3. $c_ {\kappa}=\frac{p_ {1}}{p_ {2}}\notin\mathbb{Z}$, $\mathscr{D}_ {\kappa}(\mathrm{Gr}_ {G})$ is more degenerate. For $\check{\lambda}\in \mathbb{X}^{\ast}(T)=\check{\Lambda},$ $$\mathscr{D}_ {\kappa}(L^{+}G t^{\check{˛\lambda}})^{L^{+}G}\neq 0$$ if and only if $\kappa_ {\text{killing}}(\check{\lambda},-)\in p_ {2}\Lambda$.

  1. Compatibility with Whittaker coefficients

Remark 4. For $\mathbb{k}=\mathbb{F}_ {q}$, $K$ fraction field of $X$, we have $\mathrm{Bun}_ {G}(\mathbb{k})\cong G(K)\backslash G(\mathbb{A})/G(\mathbb{O}).$

There is a map $$\mathrm{coeff}: \operatorname{Fun}^{G(K)}(G(\mathbb{A})/G(\mathbb{O}))\rightarrow \operatorname{Fun}^{N(\mathbb{A},\chi)}(G(\mathbb{A})/G(\mathbb{O}))$$ defined by $$f\mapsto \int_ {N(\mathbb{k})\backslash N(\mathbb{A})} f(n\cdot -)\chi^{-1}(n)dn.$$

Theorem 5. The functor $\mathscr{D}_ {\kappa}(\mathrm{Bun}_ {G})^{\text{cusp}}\subset \mathscr{D}_ {\kappa}(\mathrm{Bun}_ {G}) \xrightarrow{\text{coeff}} D_ {\kappa}(\mathrm{Gr})^{LN,\psi}_ {\mathrm{Conf}}$ is conservative and compatible with $\operatorname{CT}$.

Quantum Whittaker coefficients (global)

Let $D$ be a $\check{\Lambda}^{+}$-valued divisor on $X$, diffine $\operatorname{coeff}_ {D}:\mathscr{D}_ {\kappa}(\mathrm{Bun}_ {G})\rightarrow \operatorname{Vect}$.

Definition 6. $\mathrm{Bun}_ {N}^{\omega(-D)}:= \mathrm{Bun}_ {B}\times_ {\mathrm{Bun}_ {T}}\{\omega(-D)\}$, where $\omega$ is the dualizing sheaf, and $\omega(-D)\in \mathrm{Bun}_ {T}$ characterized by for any $\lambda\in \mathbb{X}^{\ast}(T)$, $\lambda(\omega(-D)) = \omega(-\lambda(D)).$

One defines $\operatorname{coeff}_ {D}(\mathscr{F}):=C_ {\text{dR}}(p^{!}\mathscr{F}\otimes^{!}\psi_ {D}^{!}(\text{exp}))[\cdots]$.

Theorem 7. For any $0\neq \mathscr{F} \in \mathscr{D}_ {\kappa}(\mathrm{Bun}_ {G})^{\text{cusp}}$, there exists some $\check{\Lambda}^{+}$-valued $D$, such that $\operatorname{coeff}_ {D}(\mathscr{F})\neq 0.$

Sketch of Proof

Theorem 8 (Faltings-Ginzburg). $\operatorname{Nilp}\subseteq T^{\ast}\mathrm{Bun}_ {G}$ is Lagragian.

By argument of Faergeman-Raskin, we can reduce the theorem to nilpotent singular support case.

Theorem 9. For any $0\neq \mathscr{F} \in \mathrm{Shv}_ {\kappa,\text{Nilp}}(\mathrm{Bun}_ {G})^{\text{cusp}}$, there exists some $\check{\Lambda}^{+}$-valued $D$, such that $\operatorname{coeff}_ {D}(\mathscr{F})\neq 0.$

Theorem 10 (Nadler-Taylor, Faergeman-Raskin). Suppose that $T^{\ast}_ {\mathrm{Bun}_ {N}^{\omega(-D)}}\mathrm{Bun}_ {G}+d\psi_ {D}$ intersects $\Lambda$ transversally at a single smooth point $\{\lambda_ {D}\}$, then $\mathrm{coeff}_ {D}|_ {\mathrm{Shv}_ {\kappa,\Lambda}(\mathrm{Bun}_ {G})}$ is $t$-exact and commutes with Verdier duality, and $\mathrm{CC}(\mathscr{F})=\sum_ {\beta\in \operatorname{Irr}(\Lambda)}c_ {\beta,\mathscr{F}}(\beta)$ and we have $$\chi(\mathrm{coeff}_ {D,\kappa}(\mathscr{F}))=c_ {\beta_ {D},\mathscr{F}},$$ where $\mathscr{F}\in \mathrm{Shv}_ {\kappa,\Lambda}$ and $\beta_ {D}$ satisfies $\lambda_ {D}\in \beta_ {D}$.

$\mathrm{Nilp}^{\text{reg}}\hookrightarrow \mathrm{Nilp}\hookleftarrow \mathrm{Nilp}^{\mathrm{Irreg}}$.

Theorem 11 (Beilinson-Drinfeld). $\mathrm{Nilp}^{\text{reg}}=\bigcup_ {\check{\lambda}\in \check{\Lambda}, \text{s.t.}(\alpha, \check{\lambda})+(2g-2)\geq 0, \forall \alpha\text{ simple}}\mathrm{Nilp}_ {D}^{\text{reg},\check{\lambda}}$

Fact: $\mathrm{Nilp}^{\text{reg},\check{\lambda}}$ is smooth connected of dimension $\mathrm{Bun}_ {G}$.

Remark 12. The slice $\operatorname{Kos}_ {D}\cap \mathrm{Nilp}^{\text{irreg}} = \emptyset$.

Proposition 13. We have that $(\mathrm{Nilp}^{\text{reg}})^{\nless \text{deg}D-(2g-2)\check{\rho}}\cap \mathrm{Kos}_ {D}=\{f_ {D}^{\text{glob}}\}$.

Corollary 14. For $\mathscr{F}\in \mathrm{Shv}_ {\kappa,\mathrm{Nilp}^{\nless \text{deg}D-(2g-2)\check{\rho}}}(\mathrm{Bun}_ {G}),$ one has that $\text{coeff}_ {D,\kappa}(\mathscr{F})=c_ {\mathrm{Nilp}^{\text{deg}D-(2g-2)\check{\rho}},\mathscr{F}}.$

Claim 15. For any $\mathscr{F}\in \mathrm{Shv}_ {\kappa,\mathrm{Nilp}}(\mathrm{Bun}_ {G})^{\text{cusp}},$ then $\mathrm{SS}(\mathscr{F})\cap \mathrm{Nilp}^{\text{reg}}\neq \emptyset$.

Proof of Theorem 9:

For any $\mathscr{F}\in \mathrm{Shv}_ {\kappa,\mathrm{Nilp}}(\mathrm{Bun}_ {G})^{\text{cusp},\heartsuit},$ take $$\mathrm{Nilp}^{\alpha}\subseteq \mathrm{SS}(\mathscr{F})$$ with $\alpha$ minimal. Then $\mathrm{coeff}_ {\kappa,D_ {\alpha}}(\mathscr{F})\neq 0$, for any $D_ {\alpha}$ with $\text{deg}(D_ {\alpha})-(2g-2)\check{\rho}=\alpha$.