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Let $G$ be a reductive group. If one can associate to $G$ a Shimura datum $(G,X)$, then the étale cohomology of the associated Shimura variety $\operatorname{Sh}(G,X)$ is a strong tool for the Langlands correspondence because it carries an action of $G(\mathbb{A})\times\operatorname{Gal}(\mathbb{Q}^{\operatorname{al}},\mathbb{Q})$.

(It is of course more complicated, one should first show that these Shimura varieties admit a model over a number field but this known in general using the theory of special points.)

I have recently heard that some group don't admit a Shimura datum, so my questions are

0- Is this really true? What is the list of known groups which don't have a Shimura datum (any reference)?

1- Is there a criterion to know if $G$ admits a Shimura datum?

2- Is there a criterion to know if $G$ doesn't admit a Shimura datum?

3- If $G$ doesn't have any Shimura datum, then with what to replace Shimura variety? Can Flag varieties be helpful for example?

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    $\begingroup$ 0 - yes, some examples are $GL_n$ for $n>2$ or $GL_{2,K}$ for $K$ a number field which is not totally real. 3 - if you find a good answer to this question, you may be able to help propel Langlands program quite considerably. AFAIK the best idea we have is to relate the $G$ and its locally symmetric spaces to some other $G'$ which do have Shimura varieties, like unitary groups. $\endgroup$
    – Wojowu
    Commented Nov 27 at 19:38
  • $\begingroup$ @Wojowu Thank you, do you know where I can find an explanation of why those groups don't admit Shimura datum? $\endgroup$
    – Loading
    Commented Nov 27 at 21:05
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    $\begingroup$ A Shimura variety can exist only when the group admits a miniscule cocharacter and a compact inner form. This is explained in Deligne's Corvallis notes. The restriction is quite severe, but, because of the necessity of both polarizations and Griffith's transversality, is the best we can hope to do in the context of algebraic moduli spaces of motives. To get 'Shimura varieties' for other groups (and motivic weights) one needs to find a better category of spaces than schemes and algebraic stacks, or even complex analytics spaces. $\endgroup$
    – Keerthi Madapusi
    Commented Nov 27 at 22:50
  • $\begingroup$ Thank you for the reference @KeerthiMadapusi $\endgroup$
    – Loading
    Commented Nov 28 at 19:18
  • $\begingroup$ Is the picture in your last sentence elaborate in more details somewhere ? $\endgroup$
    – Loading
    Commented Nov 28 at 19:20

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