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Let $G$ a complex reductive algebraic group, $X$ be a smooth compact complex curve. The moduli stack $Bun_G$ of principal $G$-bundles on $X$ is generally speaking not quasi-compact (so we do not have Verdier duality). However, Drinfeld has conjectured that a different functor, $Ps-Id_{Bun_G, !}$, gives us an equivalence between the category of D-modules on $Bun_G$ and its dual $$ Ps-Id_{Bun_G, !}:(D-mod(Bun_G))^\vee\rightarrow D-mod(Bun_G). $$ This conjecture has been proved by Gaitsgory. As a by-product of the proof, Gaitsgory has found that there is a canonical isomorphism of functors $D-mod(Bun_M)\rightarrow D-mod(Bun_G)$ $$ Eis_!^-\cdot Ps-Id_{Bun_M, !}\simeq Ps-Id_{Bun_G, !}\cdot (CT_*)^\vee, $$ where $M$ is the Levi quotient for a parabolic $P \subset G$, $Eis_!^-$ is the geometric Eisenstein series functor for the opposite parabolic $P^-$ and $CT_*^\vee$ is the dual of geometric constant term functor. He dubbed this isomorphism 'strange functional equation'.

My question is: have decategorifications of this functional equation been studied? Does it descend to an interesting relation on the level of Hochschild cohomologies, for example?

Let $G$ a complex reductive algebraic group, $X$ be a smooth compact complex curve. The moduli stack $Bun_G$ of principal $G$-bundles on $X$ is generally speaking not quasi-compact (so we do not have Verdier duality). However, Drinfeld has conjectured that a different functor, $Ps\text{-}Id_{Bun_G, !}$, gives us an equivalence between the category of D-modules on $Bun_G$ and its dual $$ Ps\text{-}Id_{Bun_G, !}\colon (D\text{-}mod(Bun_G))^\vee\rightarrow D\text{-}mod(Bun_G). $$ This conjecture has been proved by Gaitsgory. As a by-product of the proof, Gaitsgory has found that there is a canonical isomorphism of functors $D\text{-}mod(Bun_M)\rightarrow D\text{-}mod(Bun_G)$ $$ Eis_!^-\cdot Ps\text{-}Id_{Bun_M, !}\simeq Ps\text{-}Id_{Bun_G, !}\cdot (CT_*)^\vee, $$ where $M$ is the Levi quotient for a parabolic $P \subset G$, $Eis_!^-$ is the geometric Eisenstein series functor for the opposite parabolic $P^-$ and $CT_*^\vee$ is the dual of geometric constant term functor. He dubbed this isomorphism 'strange functional equation'.

My question is: have decategorifications of this functional equation been studied? Does it descend to an interesting relation on the level of Hochschild cohomologies, for example?

Let $G$ a complex reductive algebraic group, $X$ be a smooth compact complex curve. The moduli stack $Bun_G$ of principal $G$-bundles on $X$ is generally speaking not quasi-compact (so we do not have Verdier duality). However, Drinfeld has conjectured that a different functor, $Ps-Id_{Bun_G, !}$, gives us an equivalence between the category of D-modules on $Bun_G$ and its dual $$ Ps-Id_{Bun_G, !}:(D-mod(Bun_G))^\vee\rightarrow D-mod(Bun_G). $$ This conjecture has been proved by Gaitsgory. As a by-product of the proof, Gaitsgory has found that there is a canonical isomorphism of functors $D-mod(Bun_M)\rightarrow D-mod(Bun_G)$ $$ Eis_!^-\cdot Ps-Id_{Bun_M, !}\simeq Ps-Id_{Bun_G, !}\cdot (CT_*)^\vee, $$ where $M$ is the Levi quotient for a parabolic $P \subset G$, $Eis_!^-$ is the geometric Einstein series functor for the opposite parabolic $P^-$ and $CT_*^\vee$ is the dual of geometric constant term functor. He dubbed this isomorphism 'strange functional equation'.

My question is: have decategorifications of this functional equation been studied? Does it descend to an interesting relation on the level of Hochschild cohomologies, for example?

Let $G$ a complex reductive algebraic group, $X$ be a smooth compact complex curve. The moduli stack $Bun_G$ of principal $G$-bundles on $X$ is generally speaking not quasi-compact (so we do not have Verdier duality). However, Drinfeld has conjectured that a different functor, $Ps-Id_{Bun_G, !}$, gives us an equivalence between the category of D-modules on $Bun_G$ and its dual $$ Ps-Id_{Bun_G, !}:(D-mod(Bun_G))^\vee\rightarrow D-mod(Bun_G). $$ This conjecture has been proved by Gaitsgory. As a by-product of the proof, Gaitsgory has found that there is a canonical isomorphism of functors $D-mod(Bun_M)\rightarrow D-mod(Bun_G)$ $$ Eis_!^-\cdot Ps-Id_{Bun_M, !}\simeq Ps-Id_{Bun_G, !}\cdot (CT_*)^\vee, $$ where $M$ is the Levi quotient for a parabolic $P \subset G$, $Eis_!^-$ is the geometric Eisenstein series functor for the opposite parabolic $P^-$ and $CT_*^\vee$ is the dual of geometric constant term functor. He dubbed this isomorphism 'strange functional equation'.

My question is: have decategorifications of this functional equation been studied? Does it descend to an interesting relation on the level of Hochschild cohomologies, for example?

Let $G$ a complex reductive algebraic group, $X$ be a smooth compact complex curve. The moduli stack $Bun_G$ of principal $G$-bundles on $X$ is generally speaking not quasi-compact (so we do not have Verdier duality). However, Drinfeld have conjectured that a different functor, $Ps-Id_{Bun_G, !}$, gives us an equivalence between the category of D-modules on $Bun_G$ and its dual $$ Ps-Id_{Bun_G, !}:(D-mod(Bun_G))^\vee\rightarrow D-mod(Bun_G). $$ This conjecture has been proved by Gaitsgory. As a by-product of the proof, Gaitsgory has found that there is a canonical isomorphism of functors $D-mod(Bun_M)\rightarrow D-mod(Bun_G)$ $$ Eis_!^-\cdot Ps-Id_{Bun_M, !}\simeq Ps-Id_{Bun_G, !}\cdot (CT_*)^\vee, $$ where $M$ is the Levi quotient for a parabolic $P \subset G$, $Eis_!^-$ is the geometric Einstein series functor for the opposite parabolic $P^-$ and $CT_*^\vee$ is the dual of geometric constant term functor. He dubbed this isomorphism 'strange functional equation'.

My question is: have decategorifications of this functional equation been studied? Does it descend to an interesting relation on the level of Hochschild cohomologies, for example?

Let $G$ a complex reductive algebraic group, $X$ be a smooth compact complex curve. The moduli stack $Bun_G$ of principal $G$-bundles on $X$ is generally speaking not quasi-compact (so we do not have Verdier duality). However, Drinfeld has conjectured that a different functor, $Ps-Id_{Bun_G, !}$, gives us an equivalence between the category of D-modules on $Bun_G$ and its dual $$ Ps-Id_{Bun_G, !}:(D-mod(Bun_G))^\vee\rightarrow D-mod(Bun_G). $$ This conjecture has been proved by Gaitsgory. As a by-product of the proof, Gaitsgory has found that there is a canonical isomorphism of functors $D-mod(Bun_M)\rightarrow D-mod(Bun_G)$ $$ Eis_!^-\cdot Ps-Id_{Bun_M, !}\simeq Ps-Id_{Bun_G, !}\cdot (CT_*)^\vee, $$ where $M$ is the Levi quotient for a parabolic $P \subset G$, $Eis_!^-$ is the geometric Einstein series functor for the opposite parabolic $P^-$ and $CT_*^\vee$ is the dual of geometric constant term functor. He dubbed this isomorphism 'strange functional equation'.

My question is: have decategorifications of this functional equation been studied? Does it descend to an interesting relation on the level of Hochschild cohomologies, for example?