Gubir Dhillon: modular representation of affine Lie algebras

This is joint work with Ivan Losev.

Basic summary

  • in char $0$, highest representations of affine Lie algebras are fairly well understood.
  • for some particular tricky representations, ideas form geometric Langlands played a role in computing their characters (work of Feigin-Frenkel-Gaitsgory).
  • What about char $p$?
  • We will state a series of results (in progress) and conjectures but the basis fun new feature is that phenomenon seen by Feigin-Frenkel-Gaitsgory at critical level now appear at all levels.

Notation:

  • $\mathfrak{o}$: affine Lie algebras and category $\mathcal{O}$.
  • $G$ split reductive group over field $k$.
  • $F=k(\!(t)\!)$.
  • $G_ {F}$ loop group.
  • $\mathfrak{g}_ {F}=\mathfrak{g}\otimes_ {k}k(\!(t)\!)$ loop Lie algebra.

Consider $H$ action on $\mathbb{P}(V)$,

For $G_ {F}$ and $\mathfrak{g}_ {F}$, most representations require actual nontrivial central extension. The central extensions are classified $$\{0\rightarrow k\cdot 1\rightarrow \hat{\mathfrak{g}}\rightarrow \mathfrak{g}\rightarrow 0\}/\cong \cong (\operatorname{Sym}^{2}\mathfrak{g}^{\ast})^{G}.$$

Example 1. When $G$ is simple, $(\operatorname{Sym}^{2}\mathfrak{g}^{\ast})^{G}\cong k\cdot \textrm{Killing form}$.

Definition 2. Let $\mathfrak{g}_ {K}$ be the central extension corresponding to $\mathscr{K}$, and $$\hat{\mathfrak{g}}_ {K}\text{-mod}=\{\text{smooth representations on which }1\text{ acts by }\operatorname{id}\}.$$

Category $\mathcal{O}$:

Definition 3. Let $\widetilde {O}_ {\mathscr{K}}:=(\hat{\mathfrak{g}}_ {\mathscr{K}},\mathring{I})$-mod, i.e. modules for $\hat{\mathfrak{g}_ {\mathscr{K}}}$ on which the action of $\operatorname{Lie}(\mathring{I})$ is integrated to $\mathring{I}$.

Example 4. $T\cong I/\mathring{I}\leftarrow I \rightarrow G(\!(t)\!)$. Any module $M$ for $\operatorname{Lie}(T)$, then the parabolic induction $$\operatorname{pInd}(M)=\operatorname{Ind}_ {(\operatorname{Lie}(I),\mathring{I})}^{(\hat{\mathfrak{g}}_ {\mathscr{K}},\mathring{I})}\circ \text{res}_ {\operatorname{Lie}(T)}^{(\operatorname{Lie}(I),\mathring{I})}M.$$

For $\lambda\in \mathfrak{t}^{\ast}$, one view $k_ {\lambda}$ as a 1-dimensional $\mathfrak{t}$-module, where $\mathfrak{t}=\operatorname{Lie}(T)$ and $k_ {\lambda}=k$ as plain vector spaces.

Then one constructs $M_ {\lambda}=\operatorname{pInd}(k_ {\lambda})$,

How do you write down elements in $M_ {\lambda}$?

Example 5. One write

where $\langle \lambda \rangle$ is highest weight.

To discuss characters, i.e. $M_ {\lambda} = \oplus \text{f.d. weight spaces }$, one should track

  1. $\mathfrak{t}$-grading,
  2. $\#$ powers of $t^{-1}$ (lopp rotation of $\mathbb{G}_ {m}$).

In char $p$, $\mathfrak{t}$ thinks $\langle \lambda \rangle$ and $f^{p}\langle \lambda \rangle$ have the same weight, so use full $T$ not just $\operatorname{Lie}(T)=\mathfrak{t}$.

If $\lambda\in \mathfrak{t}^{\ast}\backslash \mathbb{X}^{\ast}$, really $\int$ not action of $\mathfrak{t}$ but a shift of it.

Definition 6. $\mathcal{O}_ {\mathscr{K}}=(\hat{\mathfrak{g}}_ {\mathscr{K}},(I,\chi))\text{-mod}^{\mathbb{G}_ {m}^{\text{rot}},\text{weak}}$, where $\chi\in \mathfrak{t}^{\ast}\backslash \mathbb{X}^{\ast}$.

One has Verma modules $\mathfrak{m}_ {\lambda}$ in $\mathcal{O}_ {\mathscr{K}}$ and $!$ simple quotients $L_ {\lambda}$.

Remark 7. Concretely, integrating $\mathring{I}$ action gives divided powers $\operatorname{Lie}(\mathring{I})$ ($e,e^{p}/p!,e^{p^{2}}/p^{2}!,\dots$).

Theorem 8 (Kac-Kazhdan). Assume that $\text{char}(k)=0$ and $\mathscr{K}\neq \mathscr{K}_ {c}:=-\frac{1}{2}\mathscr{K}_ {\text{Killing}}$. Then given generic $\lambda\in\mathfrak{t}^{\ast}$, one has that $M_ {\lambda}$ is irreducible.

So in this case, $$\operatorname{ch}L_ {\lambda}=\operatorname{ch}M_ {\lambda}=\frac{e^{\lambda}}{\prod_ {\alpha\in \Phi_ {\text{aff}}^{+}}(1-e^{-\alpha})}.$$ More concretely, the denominator is $$\prod_ {\alpha_ {f}\in \Phi_ {\text{fin}}^{-},n\geq 0}(1-q^{n}e^{\alpha_ {f}})\cdot\prod_ {\alpha_ {f}\in \Phi_ {\text{fin}}^{+},n>0}(1-q^{n}e^{\alpha_ {f}})\times \prod_ {n>0}(1-q^{n})^{\text{dim}\mathfrak{t}},$$ where $q$ is the energy grading, the third term is character of $\operatorname{Sym}(\mathfrak{t}(\!(t)\!)/\mathfrak{t}[\![t]\!]).$

Remark 9. Similar statement in char 0 and char $p$. However, false for $\hat{\mathfrak{g}}$ in char $p$.

Conjecture 10 (Kac-Kazhdan, theorem due to Hayashi, Rocha-Candi-Wallach, Feign-Frenkel). For generic $\lambda$, at $\mathscr{K}=\mathscr{K}_ {c}$, $$\text{ch}(L_ {\lambda}) = \frac{e^{\lambda}}{\prod_ {\Phi_ {\text{fin}}^{-}}(-)\prod_ {\Phi_ {\text{fin}}^{+}}(-)},$$ i.e. the third term is gone.

Theorem 11 (Feign-Frenkel). The center $$Z(U(\hat{\mathfrak{g}}_ {\mathscr{K}_ {c}}))\cong \operatorname{Fun}(\operatorname{Op}_ {\check{G}}(\mathbb{D}^{\times})).$$

For non-critical level, the center is $Z\cong k$.

Theorem 12 (Dhillon-Losev, in progress). Suppose that $(\mathscr{K},\lambda)$ is generic. Then the character $$\operatorname{ch}(L_ {\lambda}) = \frac{e^{\lambda}}{\prod_ {\Phi_ {\text{fin}}^{+}}\prod_ {\Phi_ {\text{fin}}^{-}}\prod_ {n>0,(p,n)=1}(1-q^{n})^{\text{dim}\mathfrak{t}}},$$ i.e. the third term eats every $p$-th element in $\mathfrak{t}_ {F}/\mathfrak{t}_ {\mathbb{O}}.$

Theorem 13 (Dhillon-Losev). For $\mathscr{K}$ generic, $U(\hat{\mathfrak{g}}_ {\mathscr{K}})^{G(\!(t)\!)} = \operatorname{Fun}(\operatorname{Op}_ {\check{G}}(\mathbb{D}^{\times,(1)}))$, where $(1)$ means Frob twist.

Remark 14. For $G$: $\mathrm{U}(\mathfrak{g}_ {\mathbb{F}_ {p}})^{G}\xleftarrow {-\otimes_ {\mathbb{Z}}\mathbb{F}_ {p}} \mathrm{U}(\mathfrak{g}_ {\mathbb{Z}})^{G}\xrightarrow{-\otimes_ {\mathbb{Z}}\mathbb{Q}}\mathrm{U}(\mathfrak{g}_ {\mathbb{Q}})^{G}$. But this is not the case for affine $\hat{\mathfrak{g}}_ {\mathscr{K}}$.

Let $\mathscr{K}$ be the level for $G$, and $\check{\mathscr{K}}$ the dual level for $\check{G}$. Then $$\hat{\mathfrak{g}}_ {\mathscr{K}}\text{-mod}^{G_ {\mathbb{O}}}\cong \mathscr{D}\text{-mod}_ {\mathscr{K}^{\vee}}(\check{N}_ {F,\psi}\backslash \check{G}_ {F}/\check{G}_ {\mathbb{O}}).$$ This is true in char 0 (local quantum geometric Langlands), and in char $p$ case, this should be some thing for an Azumaya algebra on twsited cotangent bundle $T^{\ast}(\check{N}_ {F,\psi}\backslash \check{G}_ {F}/\check{G}_ {\mathbb{O}})^{(1)}$, and corresponds to $\operatorname{Op}_ {\check{G},\check{\mathscr{K}}^{p}-\mathscr{K}}(\mathbb{D}^{(1)})$.

Conjecture 15. Let $\mathscr{K}$ and $\mathscr{K}^{\prime}$ be two non-critical integral level. Then there is an equivalence of categories $$ \hat{\mathfrak{g}}_ {\mathscr{K}}\text{-mod}^{G_ {\mathbb{O}}}\cong \hat{\mathfrak{g}}_ {\mathscr{K}^{\prime}}\text{-mod}^{G_ {\mathbb{O}}}.$$