Zeta function of the affine Grassmanian and Weil conjecture

Let $$G$$ be a split connective reductive group over $$k=\mathbb F_q$$, then the affine Grassmannian $$X=Gr_G$$ is representable by an ind-projective strict ind-scheme over $$k$$. (That is, there exists an inductive system of projective schemes over $$\mathbb F_q$$, and the maps are closed inclusions.)

Is it possible to define zeta function for $$Gr_G$$? And what about the analog of Weil conjecture for $$Gr_G$$ with relations to the cohomology of $$Gr_G$$?

The naive example: $$Gr_{GL_n}=\text{colim}_{i} \ Gr^{(i)}_{GL_n}$$, one computes $$\# Gr^{(i)}_{GL_n}(\Bbb F_q)= \sum_{d=0}^{\infty} a_{d,i} q^d$$ is a polynomial of $$q$$. The limit of $$\frac{a_{d,i}}{2i+1} (i \rightarrow +\infty)$$ exists, and we call the limit $$a_d$$ and form the formal series $$\sum_{d=0}^{\infty}a_dq^d \in \Bbb Z[[q]]$$ . Then one define the zeta function as an element in $$\Bbb Q[[q,t]]$$ by the usual definition using exponential.

• It's better to work with a fixed connected component of $Gr_{G}$, so that the power series converges without the division by $2i+1$. Then the Weil conjectures for the colimit follow immediately from the Weil conjectures for the individual spaces, which are singular, but I belive their cohomology groups are pure so the Weil conjectures hold for compactly supported cohomology. – Will Sawin Nov 15 '18 at 2:47
• What's the distinction between compactly supported and usual cohomology here? My guess is that usual cohomology would just be defined as an inverse limit of cohomology on each piece, but presumably compactly supported cohomology should only pick up classes which are supported on finitely many pieces. If you know any references on etale cohomology of these spaces, I'd love to see them. – dorebell Nov 15 '18 at 3:10
• @dorebell The difference was just me not being careful enough, Because the individual closed strata are compact, the two theories are equal on those spaces. I agree that the natural way to extend them to the full space are inverse limit vs. direct limit over strata, but because the cohomology groups of the closed strata stabilize in each degree, it won't matter which sort of limit we take. – Will Sawin Nov 15 '18 at 3:20
• @dorobell My understanding is that even when doing the most sophisticated geometric Langlands stuff, people define categories of sheaves on affine Grassmanians as suitable limits of categories of sheaves on strata, and cohomology theories as appropriate limits of cohomology on strata. I know I've seen people do this, I'm not 100% confident they do this 100% of the time. Basically it's philosophically what you should get so there's no reason to define a theory for ind-schemes and check that it gives you the answer you want when you can just define it that way. – Will Sawin Nov 15 '18 at 3:22
• Ah, that makes sense. Do you know why the cohomology stabilizes in each degree? – dorebell Nov 15 '18 at 3:25