\documentclass[b1.tex]{subfiles} \externaldocument{b1} \begin{document} \noindent \noindent\fcolorbox{black}{subcolor!50}{\parbox{\textwidth}{ \emph{\ref{DC1}: A derived motivic Satake equivalence (Advisor: \Scholbach; Coadvisor: \Richarz).}}} %\ref{DC1} will extend the scope of the motivic Satake equivalence. \resu \thlabel{Satake.DM} %(Integral derived motivic Satake equivalence; (\ref{OBJ1}, \ref{OBJ3}) \ref{DC1} aims to fully compute the category of motivic sheaves on the Hecke stack $L^+G\setminus LG/ L^+G$ by proving an equivalence as in the bottom row: $$\xymatrix{ \Rep_{G^\vee}(\MTM(S)) \ar@{^{(}->}[d] \ar[r]^-{\refeq{MTM.Satake}}_-{\sim} & \MTM(L^+ G \backslash LG / L^+ G) \ar@{^{(}->}[d] \\ \Mod_{\DM(S)}(\Sym \left (\mathfrak g^\vee(-1)[-2] \right )) / G^\vee) \ar[r]^-{?}_-{\sim} & \DM(L^+ G \backslash LG / L^+ G)^{\locc}. }\eqlabel{wow}$$ The bottom left category is the category of motives $M \in \DM(S)$ endowed with an action of $G^\vee$ and a $G^\vee$-equivariant action map $\Sym (\mathfrak g^\dual(-1)[-2]) \otimes M \r M$. %Here we use that the dual group $G^\vee$ and hence its Lie algebra $\mathfrak g^\dual$ admits an extra $\Z$-grading coming from the above-mentioned adjoint action of $\Gm$. We therefore consider (a shift and Tate twist of) this as an object in $\DTM(S)$. The superscript $\locc$ at the bottom right refers to objects that are compact after forgetting the $L^+G$-equivariance. \xresu This computation will unify two currently unrelated derived Satake equivalences: the original one by Bezrukavnikov--Finkelberg \cite{BezrukavnikovFinkelberg:Equivariant} and Riche's version for parity sheaves \cite{Riche:Kostant}. The categories appearing in the bottom line in \refeq{wow} are larger than the top ones in two respects. First there is no condition on (stratified) Tate motives, which for applications means that there is no need for fine-grained stratification analyses. Second, they are triangulated categories. In the absence of an abelian category of mixed motives $\MM(k)$, any Satake-type statement including non-Tate motives must necessarily be a statement about triangulated (or stable \ii-) categories, as opposed to abelian categories. A key step towards that result is to perform a similar computation for derived stratified \emph{Tate} motives. \chal \rodella{Refer to any important challenges you may have identified in the chosen methodology and how you intend to overcome them. Please also list these challenges and proposed risk-mitigation measures in the implementation risk table 3.1g in the implementation section. THIS REQUEST IS NEW IN THE 2023 TEMPLATE, SO CITE EACH CHALLENGE IN RISK TABLE ACCORDING TO MITIGATION MEASURES} %(Integral derived Satake equivalence for Tate motives) \thlabel{Satake.DTM} Extend the motivic Satake equivalence on the abelian level~\refeq{MTM.Satake} %(top row) mentioned above to a derived statement involving $\DTM(L^+ G \setminus LG / L^+G)$. \xchal \meth \thlabel{deformation} Using \ref{MET1}--\ref{MET5}, \ref{DC1} will approach this by considering it as a special case of a derived Satake equivalence fibered over $\A^1$, which is amenable to hyperbolic localization arguments and enables reductions to the case where the group $G$ is either a torus or the group $\PGL_2$. The necessary geometric preliminaries for these reduction arguments are present in \cite{CassvandenHoveScholbach:Motivic}. \thref{Satake.DM} should follow from this key step by establishing the compatibility of stratifications and tensor product (of stable \ii-categories, due to Lurie; cf.~\ref{MET3}--\ref{MET4}). Concretely, given an algebraic group $G$ acting on a cellularly stratified scheme $X$ over a base scheme $S$ (with appropriate compatibilities), we expect that the canonical functor $\DTM(G \backslash X) \otimes_{\DTM(S)} \DM(S) \r \DM(G \backslash X)$ is an equivalence. The same principles should apply when $G \backslash X$ is replaced by the Hecke stack $L^+ G\backslash LG / L^+ G$. %Similarly, but technically more demandingly, one can expect such a statement to hold if the scheme $X$ is replaced by the Hecke stack $L^+ G\backslash LG / L^+ G$. \xmeth %As an ultimate goal, we may aim for a description of the entire category $\DM(L^+ G \setminus LG / L^+ G)$ (i.e., dropping any finiteness conditions) in terms of a combination of Ind-coherent sheaves with singular support on the nilpotent cone, as put forth by Arinkin--Gaitsgory \cite{ArinkinGaitsgory:Singular}. \noindent\fcolorbox{black}{subcolor!50}{\parbox{\textwidth}{ \emph{\ref{DC2}: Derived Tate motives on the affine Grassmannian (Advisor: \Scholbach; Coadvisor: \Riche)}. }} %The research of \ref{DC2} will extend the motivic Satake equivalence \refeq{MTM.Satake} in a different direction. \resu \thlabel{DTM.Gr} %(The derived category of mixed Tate motives on the affine Grassmannian; \ref{OBJ1}--\ref{OBJ3}) (\ref{OBJ1}--\ref{OBJ3}) \ref{DC2} seeks to compute the derived category $\DTM(\Gr_G)$ of mixed Tate motives on the affine Grassmannian (with respect to the stratification by orbits of an Iwahori subgroup $I$) in terms of equivariant coherent sheaves on the Springer resolution $\widetilde{\mathcal{N}}$.% := \widehat G \x_{\widehat B} \widehat {\mathfrak n}$. \xresu The category $\DTM(\Gr_G)$ here is related to the category $\DTM(L^+ G \setminus LG / L^+ G)$ in \thref{Satake.DTM}, but they differ in two ways: the former has no equivariance condition and involves the stratification by $I$-orbits; the latter imposes $L^+G$-equivariance, and hence involves the (coarser) stratification by $L^+G$-orbits. %For comparison, the category $\DTM(L^+ G \setminus LG / L^+ G)$ in \thref{Satake.DTM}: the latter consists of motivic sheaves that are $L^+G$-equivariant, and thus necessarily Tate along $L^+G$-orbits. In contrast, the category $\DTM(\Gr_G)$ in \thref{DTM.Gr} involves no equivariance condition, and imposes the Tate condition with respect to a finer stratification. %For comparison, the category $\DTM(L^+ G \setminus LG / L^+ G)$ in \thref{Satake.DTM} consists of motivic sheaves on $\Gr_G$ that are equivariant with respect to the positive loop group. Such sheaves are necessarily Tate with respect to the stratification by $L^+G$-orbits. %In contrast, the category appearing in \thref{DTM.Gr} features no equivariance data, but the sheaves are stratified Tate with respect to the stratification by $I$-orbits, which is finer than the one by $L^+G$-orbits. Our approach to this result is inspired by work of Arkhipov--Bezrukavnikov--Ginzburg~\cite{ArkhipovBezrukavnikovGinzburg:Quantum}. In this approach, there are two substantial milestones to achieve. %There are two hurdles to overcome. \chal Compute the motive of $\Gr_G$ including its Hopf algebra structure. \xchal \meth \thlabel{convolution.method} To tackle this challenge, \ref{DC2} will draw upon Ginzburg's computation of the cohomology of $\Gr_G$ \cite{Ginzburg:Perverse} which expressed $\H^*(\Gr_G, \C)$ and crucially its Hopf algebra structure, in terms of the geometry of the nilpotent cone (inside $\mathfrak g^\vee$). % 1.7.2 there A computation of the motive of $\Gr_G$, without taking into account the Hopf algebra structure, is due to Bachmann \cite{Bachmann:Affine}. % Corollary 23 there \ref{DC2} shall employ \ref{MET1}--\ref{MET3} to combine this result with a statement of Yun--Zhu that also identifies the Hopf algebra structure on cohomology, at least away from certain bad primes \cite{YunZhu:Integral}. % Corollary 6.4 there These computations relate to the proof of \Achar--Rider \cite{Achar:ParitySheavesAffine2015} of the Mirkovi{\'c}--Vilonen conjecture on the torsion-freeness of stalks of costandard sheaves. \xmeth The second key step is to show that $\DTM(\Gr_G)$ is ``governed'' by an ordinary graded algebra, rather than by some homologically more complicated object such as a differential-graded or $A_\infty$-algebra. An $A_\infty$-algebra that is quasi-isomorphic to an ordinary graded algebra is said to be \emph{formal}. %A second key step in \cite{ArkhipovBezrukavnikovGinzburg:Quantum} is to procure a rich enough set of generators that capture the finer stratification by $I$-orbits. %In op.~cit., this is done by using the Wakimoto sheaves $(W_\lambda : \lambda \in \mathbf{X}^\vee)$, $\mathbf X^\vee$ being the cocharacter group of $G$, as introduced by Mirkovi{\'c}. These are certain convolution sheaves of costandard sheaves for $I$-orbits on $\Fl_G$. %As such, they act by convolution on $I$-equivariant sheaves on $\Gr_G$. %The techniques of \cite{RicharzScholbach:Intersection} readily imply that these can be regarded as Tate motives (with respect to the Iwahori stratification). \chal \thlabel{Wakimoto} %(Wakimoto sheaves) Show that the derived algebra $\bigoplus_{\lambda \in \mathbb{X}^\vee} \RHom_{\DM(\Gr_G)}(1, W_\lambda \star T)$ is formal. %, i.e., appropriately isomorphic to the sum of its cohomologies. %Prove that Wakimoto sheaves $W_\lambda$ are mixed Tate motives on $\Gr_G$ with respect to the Iwahori stratification. %Cf. \cite[Prop 8.2.4]{ArkhipovBezrukavnikovGinzburg:Quantum}. This uses that Iwahori strata are affine, Artin vanishing Here the $\mathbb{X}^\vee$ is the cocharacter group of $G$; the $W_\lambda$ are the Wakimoto sheaves (introduced by Mirkovi\'c); and $T$ is the object in $\MTM(\Gr_G)$ corresponding to the regular representation $\mathcal O_{G^\vee}$ under the Satake equivalence in \refeq{MTM.Satake}. %Compute the Ext-algebra %$$\bigoplus_{\substack{k \in \Z \\ \lambda \in \mathbf{X}^\vee}} \Hom_{\DM(\Gr_G)} (1, W_\lambda \star R (k)[2k]),$$ %$ = \Gamma (\tilde N, \mathcal O_{\tilde N(\lambda)})$$ \heysimon{Do these line bundles on $\tilde N$ have no higher (coherent) cohomology?} %Compute the LHS in terms of a version of hyperbolic localization, but for Iwahori strata (8.7.2) there. %One needs to refine the Satake equivalence by also taking into account the action of a Cartan subalgebra $\widehat {\mathfrak h }\subset \widehat {\mathfrak g}$ \xchal \meth This challenge will be worked on by \ref{DC2} using motivic weight structure considerations, which refine the formality arguments in \cite{ArkhipovBezrukavnikovGinzburg:Quantum} based on arguments of Frobenius actions in positive characteristic (\ref{MET1}, \ref{MET3}). \xmeth %\jakob{\url{https://mathscinet.ams.org/mathscinet/search/publdoc.html?loc=refcit&refcit=2422266&sort=Newest&vfpref=html&r=30&mx-pid=3314590} -- functoriality of ABG w.r.t. restriction} %\jakob{\cite{HoLi:Revisiting} claim they will do Bez equivalence in their mixed categories} \end{document}