Dmitry Kubrak: cohomology of BG via derived geometric Satake

Motivation

Let $G$ be a split reductive group over $\mathbb{Z}$.

From $G$, one can form $H_ {\text{sing}}^{\ast}(BG(\mathbb{C}),\mathbb{k})$ for a field $\mathbb{k}$.

Proposition 1 (Borel). There is an isomorphism $H_ {\text{sing}}^{\ast}(BG(\mathbb{C}),\mathbb{Q})\cong (\operatorname{Sym}(\mathfrak{g}^{\ast}_ {\mathbb{Q}}[-2]))^{G_ {\mathbb{Q}}}.$

Proof (with $\mathbb{C}$-coefficients).

Using Hodge theory. By Grothendieck, $R\Gamma_ {\text{Sing}}(BG(\mathbb{C}),\mathbb{C})\cong R\Gamma_ {\text{dR}}(BG/\mathbb{C})$. Then one forms $$R\Gamma_ {\text{Hodge}}(BG/\mathbb{C}) = \oplus_ {q\geq 0}(BG_ {\mathbb{C}},\wedge^{q}\mathbb{L}_ {BG/\mathbb{C}}[-q]),$$ and then one uses $\mathbb{L}_ {BG}\cong \mathfrak{g}^{\ast}[-1]$. The Hodge-de Rham spectral sequence degenerates and gives the isomorphism.

Question: is there a similar formula for $H^{\ast}_ {\text{sing}}(BG(\mathbb{C}),\mathbb{F}_ {p})?$

Proposition 2 (Borel-Totaro). Let $p$ be a non-torsion prime for $G$ (i.e. $H^{\ast}_ {\text{sing}}(G(\mathbb{C}),\mathbb{Z})$ is $p$-torsion free). Then $H_ {\text{sing}}^{\ast}(BG(\mathbb{C}), \mathbb{F}_ {p})\cong (\operatorname{Sym}\mathfrak{g}_ {\mathbb{F}_ {p}}^{\ast})^{G_ {\mathbb{F}_ {p}}}$, which is a polynomial ring in even generators and no higher cohomology.

Example 3. For $G=\operatorname{GL}_ {n}$, one has $H^{\ast}_ {\text{Sing}}(B\operatorname{GL}_ {n},\mathbb{F}_ {p})\cong \mathbb{F}_ {q}[c_ {1},\dots,c_ {n}]$, where $\text{deg}(c_ {i})=2i$ are chern classes.

Question: what happens for torsion primes?

Example 4.

  1. $H_ {\text{sing}}^{\ast}(\operatorname{BSO}_ {n},\mathbb{F}_ {2})\cong \mathbb{F}_ {2}[w_ {2},w_ {3},\dots,w_ {n}]$, where $\text{deg}w_ {i}=i$ are Stiefel-Whitney classes.
  2. For $\operatorname{PGL}_ {n}$, torsion primes are primes $p$ that divide $n$. Then $H_ {\text{sing}}^{\ast}(\operatorname{BGL}_ {n},\mathbb{F}_ {p})$ is not known in general.

Naive attempt: check whether the same formula works

Note that $(\operatorname{Sym}_ {\mathbb{F}_ {q}}\mathfrak{g}_ {\mathbb{F}_ {p}}^{\ast})^{h G_ {\mathbb{F}_ {p}}} \cong R\Gamma_ {\text{Hodge}}(BG_ {\mathbb{F}_ {p}}).$

Totaro computed that $$\text{dim}H^{32}_ {\text{Sing}}(B\operatorname{Spin}_ {11},\mathbb{F}_ {2})<\text{dim}H_ {\text{dR}}^{32}(B\operatorname{Spin}_ {11}).$$

Theorem 5 (Kubrak-Prikhodko). $\text{dim}H_ {\text{dR}}^{k}(BG_ {\mathbb{F}_ {p}})\geq \text{dim}H_ {\text{sing}}^{\ast}(BG(\mathbb{C}),\mathbb{F}_ {p})$ and the difference is controlled by $H^{n+1}_ {\triangle/p}(BG)[u]$ (an $\mathbb{F}_ {p}[\![u]\!]$-module).

Less naive attempt: derived Satake

Expectation: there is a fully faithful embedding ($\mathbb{E}_ {3}$-monoidal) $$D^{b}_ {\text{constr}}(G(\mathbb{O})\backslash \operatorname{Gr}_ {G},\mathbb{F}_ {p})\hookrightarrow \operatorname{QCoh}(\operatorname{Map}(S^{2},B\check{G}_ {\mathbb{F}_ {p}})),$$ where the target is QCoh (IndCoh) on a derived stack $\operatorname{Map}(S^{2},B\check{G}_ {\mathbb{F}_ {p}})\cong (e\times_ {\check{G}} e)/\check{G}.$

Note that there is a embedding $BG\cong BG(\mathbb{O}) \xhookrightarrow{i} G(\mathbb{O})\backslash \operatorname{Gr}_ {G}.$ Then $$R\operatorname{Hom}(i_ {\ast}\mathbb{F}_ {p},i_ {\ast}F_ {p})\cong \operatorname{R}\Gamma_ {\text{sing}}(BG(\mathbb{C}),\mathbb{F}_ {p}).$$ Let $B=\mathscr{O}((e\times_ {\check{G}} e)/\check{G})$. Then $\pi_ {\ast}(B) = \wedge^{\ast}(\check{\mathfrak{g}}^{\ast}[-1])$. Note $B\rightarrow \pi_ {0}(B)=\mathbb{F}_ {p}$.

Then $R\operatorname{Hom}_ {B}(\mathbb{F}_ {p},\mathbb{F}_ {p})^{h\check{G}}$ deforms $R\operatorname{Hom}_ {\pi_ {\ast}B} (\mathbb{F}_ {p},\mathbb{F}_ {p})^{h\check{G}}\cong \operatorname{Sym}(\check{\mathfrak{g}}[-2])^{h\check{G}}.$

This is similar to $\operatorname{Sym}(\mathfrak{g}^{\ast}[-2])^{hG}$ but linear dual is replaced with Langlands dual here.

Example 6. For $(\operatorname{Spin}_ {11})^{\vee} = \mathbb{P}\operatorname{Sp}_ {10}$, there is no clear way to relate Hodge cohomology and “Satake approximation”.

Remark 7.

  1. For simply laced, Hodge cohomology is the same as “Satake approximation”.
  2. If $\widehat{\mathfrak{g}^{\vee}_ {0}}\cong G_ {e}^{\vee}$ a $G^{\vee}$-equivariant isomorphism, the deformation above is trivial.

Example 8. For $(\operatorname{Spin}_ {12})^{\vee}\cong \mathbb{P}\operatorname{SO}_ {12}$, $\mathfrak{g}^{\vee}\cong \mathfrak{g}^{\ast}$ as $\mathbb{P}\operatorname{SO}_ {12}12$-representations.

Theorem 9 (Kubrak-Prikhodko). Let $n=3,4,5 \operatorname{mod} 8$ and $n\geq 11$. Then $$\text{dim}H_ {\text{sing}}^{k}(B\operatorname{SPin}_ {n},\mathbb{F}_ {2})< \text{dim}H_ {\text{dR}}^{k}(B\operatorname{Spin}_ {n},\mathbb{F}_ {2})$$ for $k>\!>0$.