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We study the prime-to-$p$ Hecke action on the projective limit of the sets ofconnected components of Shimura varieties with fixed parahoric or Bruhat--Titslevel at $p$. In particular, we construct infinitely many Shimura varieties forCM unitary groups in odd variables for which the considering actions are nottransitive. We prove this result by giving negative examples on the question ofBruhat--Colliot-Th\'el\`ene--Sansuc--Tits or its variant, which is related tothe weak approximation on tori over $\mathbb{Q}$.

On the connected components of Shimura varieties for CM unitary groups in odd variables