As this post is being written, the jury is deliberating, and the prosecution has filed a motion for a mistrial with prejudice based in part on the cross-examination of the defendant on the witness stand about his post-arrest silence. Although the motion itself does not identify the constitutional right involved, much comment on the matter identifies it as a Fifth Amendment self-incrimination issue (or, more specifically, since this is a state case, a violation of the Fifth Amendment right to remain silent incorporated by the Fourteenth Amendment Due Process Clause).
I do not believe that there is a self-incrimination issue here. It is well established that a criminal defendant who has taken the witness stand can be impeached with evidence of post-arrest silence.
Fletcher v. Weir,
455 U.S. 603 (1982).
There is a clear constitutional violation, though, but it is a due process, not self-incrimination, violation. Rittenhouse turned himself in, and was accordingly warned of his constitutional rights, including the right to remain silent. It is well established that comments on a defendant's silence violate due process (not self-incrimination) if the silence occurs after the defendant was advised of a right to remain silent.
Doyle v. Ohio,
426 U.S. 610 (1976).
We know that Rittenhouse received this warning because of press reports and YouTube videos. I have not watched much of the trial, and don't know whether the warning, an essential element of the constitutional claim, is part of the formal trial record. If it was my case, I would make sure to identify this as a Doyle issue, and to put the warning on the record.