Joakim Faergeman: Singular support for $G$-categories and applications $W$-algebras

Plan:

  1. Define singular support for $G$-categories.
  2. Whittaker coeffecients for character sheaves
  3. Applications to $W$-algebras

Singular support for $G$-categories

Let $G$ be a connected reductive group over $\mathbb{k}=\overline{\mathbb{k}}$ with characteristic $0$. Fix $(-,-)\in S^{2}\mathfrak{g}^{\ast}$ non-degenerate paring.

Let $(D(G),\ast)$ be the unbounded dg-category of $\mathscr{D}$-modules on $G$.

Definition 1. A $G$-category is a dualizable category $\mathscr{C}$ with an action $\mathscr{D}(G)\otimes \mathscr{C}\rightarrow \mathscr{C}$.

Recall that for $V\in \operatorname{Rep}(G)$, one has a map \begin{align} \operatorname{End}(V) &\xrightarrow{\mathrm{MC}} \operatorname{Fun}(G), \\ \varphi &\mapsto (g\mapsto \operatorname{tr}(\varphi\circ g)).
\end{align}

Similarly, given any $G$-category $\mathscr{C}$, one can construct \begin{align} \operatorname{End}(\mathscr{C}) &\xrightarrow{\mathrm{MC}} \mathscr{D}(G) \\ \varphi &\mapsto (\text{fiber over $g$ is $\text{tr}(\varphi\circ g)$})
\end{align} where one uses categorical trace.

Given $Z\subseteq \mathfrak{g}^{\ast}$ that is closed conical and $\operatorname{Ad}$-invariant, one can define $$\mathscr{D}_ {Z}(G)=\{\mathscr{F}\in \mathscr{D}(G): \operatorname{SS}(\mathscr{F})\subseteq G\times Z\subseteq T^{\ast}G\}.$$

Definition 2 (categorical singular support). Given any $G$-category $\mathscr{C}$, we say $\operatorname{SS}(\mathscr{C})\subseteq Z\subseteq \mathfrak{g}^{\ast}$, if $\operatorname{MC}: \operatorname{End}(\mathscr{C})\rightarrow \mathscr{D}(G)$ factors through $\mathscr{D}_ {Z}(G)$.

This is analogous to wave front cycle of $p$-adic representations.

Example 3.

  1. For $\mathscr{C}=\mathscr{D}(G)$, $\operatorname{SS}(\mathscr{C})=\mathfrak{g}^{\ast}$.
  2. For $\mathscr{C}=\langle \text{constant sheaves} \rangle$, then $\operatorname{SS}(\mathscr{C})=\{0\}$.
  3. For $\mathscr{C}=\mathscr{D}(G/B)$, then $\operatorname{SS}(\mathscr{C})=\mathcal{N}\subseteq \mathfrak{g}^{\ast}$.
  4. More generally, if $G$ acts on $X$ such that $\mu: T^{\ast}X\rightarrow \mathfrak{g}^{\ast}$ factors through $\mathcal{N}\subset \mathfrak{g}^{\ast}$, then $\operatorname{SS}(\mathscr{D}(X)) = \overline{\operatorname{Im}(\mathcal{N})}$ (quite subtle).
  5. For $\mathscr{C} = \{M\in \mathfrak{g}\text{-Mod}_ {0}: V(\mathrm{U}\mathfrak{g}/\operatorname{Ann}_ {\mathrm{U}\mathfrak{g}}(M))\subseteq \overline{\mathbb{O}}\}$, one has that $\operatorname{SS}(\mathscr{C}) = \overline{\mathbb{O}}_ {\text{sp}}=\bigcup\text{closure of special nilpotent orbits contained in $\overline{\mathbb{O}}$}.$

Definition 4. We sat $\mathscr{C}$ is nilpotent if $\operatorname{SS}(\mathscr{C})\subseteq \mathcal{N}$.

This is the main case of interest.

Our first goal is to characterize $\operatorname{SS}(\mathscr{C})$ in terms of vanishing of its Whittaker models.

Whittaker models

Given $e\in \mathbb{O}\subseteq \mathcal{N}$, extend it to $\mathfrak{sl}_ {2}$-triple $\{e,f,h\}\subset\mathfrak{g}$, then

  • $\mathfrak{g}= \bigoplus_ {i\in \mathbb{Z}}\mathfrak{g}(i)$ grading by $h$,
  • $\psi_ {e}:=(e,-)\in\mathfrak{g}^{\ast}$,
  • $u_ {e} = \bigoplus_ {i\leq -2}\mathfrak{g}(i)\oplus \mathfrak{l}$, where $\mathfrak{l}\subset \mathfrak{g}(-1)$ Lagragian.
  • $U_ {e}\subset G$ subgroup such that $\operatorname{Lie}(U_ {e})=u_ {e}$.
  • Let $\Psi: U_ {e}\rightarrow \mathbb{A}^{1}$ be a character.

We can view $\operatorname{Vect}$ as $\mathscr{D}(U_ {e})$-module via $\mathscr{D}(U_ {e})\rightarrow \operatorname{Vect}$ defined by $\mathscr{F}\mapsto C_ {\text{dR}}^{\bullet}(U_ {e},\mathscr{F}\otimes^{!}\Psi_ {e}^{!}\operatorname{exp}).$

Given $G$-category $\mathscr{C}$, one defines $$\operatorname{Whit}_ {\mathbb{O}}(\mathscr{C}):= \mathscr{C}^{U_ {e},\Psi_ {e}} = \operatorname{Hom}_ {\mathscr{D}(N)}(\operatorname{Vect}_ {\Psi_ {e}},\mathscr{C}).$$

Theorem 5 (Dhillon-F.). Lett $\mathscr{C}$ be a nilpotent $G$-category.

  1. If $\operatorname{O}$ is not contained in $\operatorname{Whit}_ {\mathbb{O}}(\mathscr{C})=0$.
  2. If $\mathbb{O}\subset \operatorname{SS}(\mathscr{C})$ is maximal, then $\operatorname{Whit}(\mathscr{C})\neq 0$.

This is analogous to results in $p$-adic representation theory.

Whittaker coeffecients of character sheaves

Consider $G/G:=G/\operatorname{Ad}G$. Then $T^{\ast}(G/G) = \{(g,x)\in G\times \mathfrak{g}^{\ast}: \operatorname{Ad}_ {g}(x)=0\}/G.$

Let $\Lambda:=T^{\ast}(G/G)\times_ {\mathfrak{g}^{\ast}}\mathcal{N}$.

For $\mathbb{O}$ nilpotent orbit, we define $\Lambda_ {\overline{\mathbb{O}}}:=\Lambda\times_ {\mathcal{N}}\overline{\mathbb{O}}$.

Then $\Lambda_ {\overline{\mathbb{O}}}\subseteq T^{\ast}(G/G)$ is Lagragian but not connected in general.

Let $\Lambda_ {\mathbb{O},0}\subset \Lambda_ {\overline{\mathbb{O}}}$ be the component containing $(1,\mathbb{O})$.

Let $\mathscr{D}_ {\Lambda}(G/G)$ be the category of character sheaves. Given $\mathscr{F}\in \mathscr{D}_ {\Lambda_ {\overline{\mathbb{O}}}}(G/G)$, write $c_ {\mathbb{O},\mathscr{F}}\in \mathbb{Z}$ multiplicity of $\Lambda_ {\mathbb{O},0}$ in characteristic cycle of $\mathscr{F}$.

Our second goal is to access $c_ {\mathbb{O},\mathscr{F}}$ via Whittaker coeffecients.

Define Whittaker coeffecient $$\operatorname{coeff}_ {\mathbb{O}}: \mathscr{D}(G/G) \rightarrow \operatorname{Vect}$$ via $$\operatorname{coeff}_ {\mathbb{O}}(\mathscr{F}):= C_ {\text{dR}}(U_ {e}/U_ {e}, \Psi_ {e}^{!}(\operatorname{exp})\otimes^{!}p^{!}\mathscr{F}),$$ where

$$\mathbb{A}^{1}\xleftarrow {\Psi_ {e}} U_ {e}/U_ {e}\xrightarrow{p} G/G. $$

Theorem 6 (Dhillon-F.).

  1. $\operatorname{coeff}_ {\mathbb{O}}: \mathscr{D}_ {\Lambda_ {\overline{\mathbb{O}}}}(G/G)\rightarrow \operatorname{Vect}$ is $t$-exact,
  2. $\chi(\operatorname{coeff}(\mathscr{F})) = c_ {\mathbb{O}}(\mathscr{F})$.

Remark 7. The functor $\operatorname{coeff}_ {\mathbb{O}}$ is the microstalk at $(1,e)\in \Lambda_ {\overline{\mathbb{O}}}$.

Applications to $W$-algebras

Given $e\in \mathbb{O}$, we have the associated $W$-algebra $\mathscr{W}_ {e}$ defined as follows. Set $$u_ {\psi}:=\{x-\psi_ {e}(x): x\in u_ {e}\}.$$ Then $$\mathscr{W}_ {e} = (\mathrm{U}\mathfrak{g}/\mathrm{U}\mathfrak{g}\cdot u_ {\Psi})^{\operatorname{ad}(u_ {e})}.$$ Alternatively,
$$\mathscr{W}_ {e} = \operatorname{End}_ {\mathrm{U}\mathfrak{g}}(\operatorname{ind}_ {u_ {e}}^{\mathfrak{g}}\psi_ {e}).$$

Then $Z(\mathrm{U}\mathfrak{g})\rightarrow \mathscr{W}_ {e}$ is an isomorphism onto $Z(\mathscr{W}_ {e})$.

Set

$$\mathscr{W}_ {e,0}:=\mathscr{W}_ {e}\otimes_ {Z(\mathrm{U}\mathfrak{g})}\mathbb{k}.$$

Then there are equivalences of categories

$$\mathscr{W}_ {e}\text{-mod}\cong \mathfrak{g}\text{-mod}_ {0}^{U_ {e},\Psi_ {e}}\cong \operatorname{Whit}_ {\mathbb{O}}(\mathscr{D}(G/B)).$$

Let $\mathscr{W}_ {e,0}\text{-mod}^{\text{fin}}\subseteq \mathscr{W}_ {e,0}\text{-mod}$ subcategory of finite dimensional modules.

Conjecture 8 (Theorem due to Losev-Ostrik and Bezrukavnikov-Losev). There is an isomorphism of $\mathscr{W}$-representations and $\mathscr{A}_ {\mathbb{O}}$-representations $$K_ {0}(\mathscr{W}_ {e,0}\text{-mod}^{\text{fin}})\cong H^{\text{top}}(\mathscr{B}_ {e})\otimes_ {W}[c],$$ where

  • $\mathscr{B}_ {e}$ is Springer fiber,
  • $\mathscr{A}_ {\mathbb{O}}$ stabilzer of the $\mathfrak{sl}_ {2}$-triple,
  • $[c]\subset \mathbb{k}[W]$ two sided cell module corresponding to $\mathbb{O}$.

Categorical trace

Let $\mathscr{C}$ be a dualizable category, i.e. there exists $\mathscr{C}^{\vee}$ and $u: \operatorname{Vect}\rightarrow \mathscr{C}\otimes \mathscr{C}^{\vee}$
and $e: \mathscr{C}^{\vee}\otimes \mathscr{C}\rightarrow \operatorname{Vect}$ satisfying triangle identities.

Define $\mathbf{HH}_ {\ast}(\mathscr{C})\in \operatorname{Vect}$ as $u\circ \text{ev}:\operatorname{Vect}\rightarrow \operatorname{Vect}$.

There is a $S^{1}$-action on $\mathbf{HH}_ {\ast}(\mathscr{C})$ and one can form $\mathbf{HP}_ {\ast}(\mathscr{C}).$

For us, cateogrical traces provides strong functoriality.

Example 9.

  • $\mathscr{C} = \mathscr{D}(B\backslash G/B)$, then $\mathbf{HH}_ {\ast}(\mathscr{C}) = \mathbb{k}[W]\otimes \operatorname{Sym}(\mathfrak{t}^{\ast}[-1]\oplus \mathfrak{t}^{\ast}[-2])$.
  • For $\mathscr{C}$ a $G$-category, define $\chi_ {\mathscr{C}}\in \mathscr{D}(G/G)$ defined as the image of $\text{id}_ {\mathscr{C}}$ under $\operatorname{MC}: \operatorname{End}(\mathscr{C})\rightarrow \mathscr{D}(G)$. Then $$\mathbf{HH}_ {\ast}(\operatorname{Whit}_ {\mathbb{O}}(\mathscr{C})) = \operatorname{coeff}_ {\mathbb{O}}(\chi_ {\mathscr{C}}).$$
  • $\mathscr{C}=\mathscr{D}(G/B)$, then $\chi_ {\mathscr{C}}=\widetilde {\operatorname{Spr}} $ Grothendieck-Springer sheaf. Then $$\mathbf{HH}_ {\ast}(\operatorname{Whit}_ {\mathbb{O}}(\mathscr{D}(G/B))) = \operatorname{coeff}_ {\mathbb{O}}(\widetilde {\operatorname{Spr}} ),$$ the left hand side is ssame as $\mathbf{HH}_ {\ast}(\mathscr{W}_ {e,0})$ and the right hand side is $H^{\ast}(\mathscr{B}_ {e})[2\text{dim}\mathscr{B}_ {e}]$, this recovers Etingof-Schedler 2010.

Consider $\mathscr{D}_ {\overline{\mathbb{O}}}(B\backslash G/B)\subseteq \mathscr{D}(B\backslash G/B)$, then $$K_ {0}(\mathscr{D}_ {\overline{\mathbb{O}}}(B\backslash G/B)) = [c]$$ as $W$-representations, $$\mathbb{k}[W] = K_ {0}(\mathscr{D}(B\backslash G/B)).$$

Let $\mathscr{D}(G/B)^{\overline{\mathbb{O}}}\subset\mathscr{D}(G/B)$ be the biggest $G$-stable subcategory, such that $$\operatorname{SS}(\mathscr{D}(G/B)^{\overline{\mathbb{O}}})\subseteq \overline{\mathbb{O}}.$$

Main observation:

Beilinson-Bernstein localization $\mathscr{W}_ {e,0}\text{-mod}\cong \operatorname{Whit}_ {\mathbb{O}} (\mathscr{D}(G/B))$ restricts to $$\mathscr{W}_ {e,0}\text{-mod}^{\text{fin}}\cong \operatorname{Whit}_ {\mathbb{O}} (\mathscr{D}(G/B)^{\overline{\mathbb{O}}}).$$

Then take chern class of $K_ {0}(\mathscr{W}_ {e,0}\text{-mod}^{\text{fin}})\rightarrow K_ {0}(\mathscr{W}_ {e,0})\text{-mod}$, one obtains that $\mathbf{HH}_ {0}(\mathscr{W}_ {e,0}\text{-mod}) = H^{\text{top}}(\mathscr{B}_ {e})$ and compose with $$i^{L}:\mathscr{W}_ {e,0}\text{-mod}\rightarrow \mathscr{W}_ {e,0}\text{-mod}^{\text{fin}},$$ one gets $$K_ {0}(\mathscr{W}_ {e,0}\text{-mod}^{\text{fin}})\hookrightarrow H^{\text{top}}(\mathscr{B}_ {e})\rightarrow \mathbf{HP}_ {0}(\mathscr{W}_ {e,0}\text{-mod}^{\text{fin}})$$ and this composition is an isomorphism. In fact, $$K_ {0}(\mathscr{W}_ {e,0}\text{-mod}^{\text{fin}})\hookrightarrow H^{\text{top}}(\mathscr{B}_ {e})\otimes_ {W}[c].$$

Faergemen