git mv

git mv improper_Case improve_case2
git mv improve_case2 improve_case
git commit -m ""

考慮設定

git config core.ignorecase false

core.ignoreCase

If true, this option enables various workarounds to enable Git to work better on filesystems that are not case sensitive, like FAT. For example, if a directory listing finds "makefile" when Git expects "Makefile", Git will assume it is really the same file, and continue to remember it as "Makefile".
The default is false, except git-clone[1] or git-init[1] will probe and set core.ignoreCase true if appropriate when the repository is created.