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](http://www.math.harvard.edu/~gaitsgde/GL/Strange.pdf) 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?



Decategorification of Gaitsgory's strange functional equation?
This is essentially the question studied in <https://arxiv.org/abs/1503.04705v4>.


A (possibly wrong) comment: Unlike $\operatorname{Eis}\_\*$, which decategorifes (under Drinfeld-Wang's conventions) to the function-theoretic Eisenstein series, there is no obvious decategorification of $
\operatorname{Eis}\_!.$ The strange functional equation, rather than giving some classical identity for Eisenstein series, tells us you how to decategorify $\operatorname{Eis}\_!.$


The decategorification is via the fonction-faisceaux correspondence (see e.g., [fonctions-faisceaux correspondence](https://mathoverflow.net/questions/5693/fonctions-faisceaux-correspondence) ). Your Eisenstein $\*$ and $!$ functors becomes maps from the space of functions on $\operatorname{Bun}\_T(\mathbb{F}\_p)$ to the space of functions on $\operatorname{Bun}\_G(\mathbb{F}\_p)$.


In the usual fonction-faisceaux correspondence, $!$-pushforward corresponds to "integration", i.e., summing over points in fibers. The Drinfeld-Wang conventions are actually such that instead $\*$-pushforward plays this role, and it is $!$-pushforward that admits no classical analogue. This in turn leads to $\operatorname{Eis}\_\*$ being the categorical counterpart of classical Eisenstein series, while $\operatorname{Eis}\_!$ a priori doesn't necessarily have a function-theoretic analogue.


