I can not understand Remark 12.8.8 in the preprint "SINGULAR SUPPORT OF COHERENT SHEAVES AND THE GEOMETRIC LANGLANDS CONJECTURE". I am somewhat embarrased by the degree of my confusion, hopefully someone knowledgeable could help me.

Authors claim that for any (connected) algebraic stack $Y$ and any compact object $M$ of the DG category of D-modules on $Y$ the map $$ H_{dR}(Y)\otimes M\rightarrow M \qquad (*) $$ vanishes on a sufficiently high power of the augmentation ideal of $H_{dR}(Y)$.

The first difficulty is that $H_{dR}$, I believe, was only defined for the stack $pt/G$ where $G$ is a connected reductive group over an algebraically closed field of characteristic 0. So I am not completely sure what is that supposed to mean for other stacks.

The second difficulty is what would the map (*) be.

The third difficulty is why do we talk about augmentation ideal. Is it true that the image of the map $H_{dR}(pt/G)\rightarrow H_{dR}(Bun_G)$ lies in the augmentation ideal?

The fourth difficulty is how do we actually show that the above claim is true?