Sanath Devalapurkar: Ruminations about Langlands duality with generalized coefficients

Ruminations

Let $\mathbb{k}$ be a $\mathbb{Q}$-commutative algebra, $G$ a connected reductive group over $\mathbb{C}$. Then there is derived geometric Satake equivalence $$\operatorname{Shv}_ {G(\mathbb{O})}(\mathrm{Gr}_ {G},\mathbb{k})\cong \operatorname{QCoh}(\check{\mathfrak{g}}^{\ast}_ {\mathbb{k}}[2]/\check{G}_ {\mathbb{k}})$$ and renormalized version $$\operatorname{Shv}^{\text{ren}}_ {G(\mathbb{O})}(\mathrm{Gr}_ {G},\mathbb{k})\cong \operatorname{QCoh}(\check({\mathfrak{g}}^{\ast}_ {\mathbb{k}})^{\wedge}_ {\mathcal{N}}[2]/\check{G}_ {\mathbb{k}})$$ and loop rotation version $$\operatorname{Shv}_ {G(\mathbb{O})\rtimes \mathbb{G}_ {m}^{\text{rot}}}(\mathrm{Gr}_ {G},\mathbb{k}) \cong \mathrm{U}_ {\hbar}(\check{\mathfrak{g}})\text{-Mod}$$ and renormalized version $$\operatorname{Shv}^{\text{ren}}_ {G(\mathbb{O})}(\mathrm{Gr}_ {G},\mathbb{k})\cong \operatorname{QCoh}(\check({\mathfrak{g}}^{\ast}_ {\mathbb{k}})^{\wedge}_ {\mathcal{N}}[2]/\check{G}_ {\mathbb{k}}) .$$

Now assume that $G$ is simply connected and simply laced, then $\check{G}\cong G/Z(G)$, and $\check{\mathfrak{g}}^{\ast}\cong \mathfrak{g}$. In this case, $G\twoheadrightarrow \check{G}$ acts on $\mathfrak{g}$.

Let $X$ be a space and then one can get $C^{\ast}(X,\mathbb{Z})$ and $H^{\ast}(X,\mathbb{Z})$ via a cohomology theory. There are also generalized cohomology theories.

\begin{align} \operatorname{Spc} &\rightarrow \operatorname{GrAb} \\ \ast & \mapsto \mathbb{Z}\text{ in every even weight} \\ X &\mapsto \operatorname{KU}^{\ast}(X) \end{align} is the complex $K$-theory.

Morally, spectra linearize spaces.

where the vertical arrow is $X\mapsto C^{\ast}(X,k):=\mathrm{Maps}_ {p}(\Sigma^{\infty}X,k)$

Just as one can form $\mathrm{Shv}(X,\mathbb{Z})$, one can also form $\operatorname{Shv}(X,k)$ for any commutative ring spectra $k$.

Deep insight à la Quillen, Morava,…

Let $k$ be a commutative ring spectra such that $\pi_ {\ast}C^{\ast}(BS^{1},k)\cong \pi_ {\ast}(k)[t]^{\wedge}$ by complex oriented.

The group law $\otimes: BS^{1}\times BS^{1}\rightarrow BS^{1}$ corresponds to $\pi_ {\ast}(k)[t]^{\wedge}\rightarrow \pi_ {2}(k)[t_ {1},t_ {2}]^{\wedge}$ sending $t=c_ {1}^{k}(\mathscr{L})\mapsto F(t_ {1},t_ {2})=c_ {1}^{k}(\mathscr{L}_ {1}\otimes \mathscr{L_ {2}})$.

Then $F$ is a 1-dimensional formal group law.

Now let $k$ be a $\mathbb{C}$-oriented commutative ring spectra and $G=T$ a torus.

Then $\operatorname{Shv}_ {T}(\operatorname{Gr}_ {T},k)\cong \oplus_ {\mathbb{X}_ {\ast}(T)}\operatorname{Shv}_ {T}(\text{pt},k)\cong \oplus_ {\mathbb{X}_ {\ast}(T)}\operatorname{Mod}(k_ {T})$, where $k_ {T}$ is either Borel $C^{\ast}(BT,k)$ or “genuine” $T$-equivariant version of $k$.

Take homotopy groups of $k_ {T}$: Given a spectra $A$, get a filtered complex $\tau_ {\geq\ast}A$ (or cohomology notation $\tau^{\leq -\ast}A$). Taking graded and shear, one gets $\pi_ {\ast}A$ ($\cong H_ {\ast}(A)$ since $A$ is a spectra).

So $k_ {T}$ is a 1 parameter degeneration (i.e. $\tau_ {\geq \ast} k_ {T}$) of $\pi_ {\ast}(k_ {T})$.

Then one get $\operatorname{Shv}_ {T}(\operatorname{Gr}_ {T},k)$ a 1-parameter degeneration of $\oplus_ {\mathbb{X}_ {\ast}(T)}\operatorname{Mod}^{\text{gr}}(\pi_ {\ast}(k_ {T})).$ If $\mathbb{H}=\operatorname{Spec}(k_ {S^{1}})$ (or $\operatorname{Spf}$), 1-dimensional (formal) group over $\pi_ {\ast}(k)$, then

$$\operatorname{Spec} \pi_ {\ast}(k_ {T}) = \operatorname{Hom}(\mathbb{X}^{\ast}(T), \mathbb{H}) =: T_ {\mathbb{H}},$$ and $$\operatorname{QCoh}(T_ {\mathbb{H}}\times B\check{T}_ {\pi_ {\ast}k})\cong \oplus_ {\mathbb{X}^{\ast}(T)}\operatorname{QCoh}^{\text{gr}}(T_ {\mathbb{H}}).$$

Summary:

We have a 1-parameter degeneration $\operatorname{Shv}_ {T(\mathbb{O})}(\mathrm{Gr}_ {T},k)$ to $\operatorname{QCoh}^{\text{gr}}(T_ {\mathbb{H}}\times B\check{T})$ from a $k$-linear category to graded $\pi_ {\ast}k$-linear category.

How to replace $T_ {\mathbb{H}}$ in general?

What is loop rotation in this generalized coefficients situation?

Classically, there eixsts an object $\check{R}\in \operatorname{Shv}_ {G(\mathbb{O})\rtimes \mathbb{G}_ {m}^{\text{rot}}}(\operatorname{Gr}_ {G},\mathbb{Q})$ and a $\delta$-sheaf at the base point, such that $$\operatorname{Ext}^{\bullet}_ {\operatorname{Shv}}(\delta,\check{R})\cong \mathrm{U}_ {\hbar}(\check{\mathfrak{g}}).$$

Expectation: THere eixsts $\check{R}_ {k}\in \operatorname{Shv}_ {G(\mathbb{O})\rtimes \mathbb{G}_ {m}^{\text{rot}}}(\operatorname{Gr}_ {G},k)$ and $\delta$, with $$\pi_ {\ast}R\operatorname{Hom}_ {\operatorname{Shv}}(\delta,\check{R}_ {k})=:\mathscr{A}$$ and we expect $\mathscr{A}$ to sit inside an algebra.

For $\operatorname{SL}_ {2}$, $$\mathrm{U}_ {\mathbb{H}}(\operatorname{SL}_ {2}) = \frac{\pi_ {\ast}k[t]^{\wedge}\langle e,f,h \rangle}{fh=(h+_ {F}t)f, eh=(h-_ {F}t)e, [e,f]=h(t-_ {F}h)-\overline{h}(h+_ {F}t)},$$ where $\pi_ {\ast}k[t]^{\wedge} = \pi_ {\ast}C^{\ast}(BS^{1},k)$, and $\overline{h}$=inverse of $h$ in $F$.