Decategorification of Gaitsgory's strange functional equation?

Question

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?

Answer 1

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 ). 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.