BUG: read_parquet does not respect index for arrow dtype backend
- [x] closes #51717 (Replace xxxx with the GitHub issue number)
- [x] [Tests added and passed](https://pandas.pydata.org/pandas-docs/dev/development/contributing_codebase.html#writing-tests) if fixing a bug or adding a new feature
- [x] All [code checks passed](https://pandas.pydata.org/pandas-docs/dev/development/contributing_codebase.html#pre-commit).
- [x] Added [type annotations](https://pandas.pydata.org/pandas-docs/dev/development/contributing_codebase.html#type-hints) to new arguments/methods/functions.
- [ ] Added an entry in the latest `doc/source/whatsnew/vX.X.X.rst` file if fixing a bug or adding a new feature.

Hey, thanks for tackling this so quickly, I appreciate it. One additional request is that RangeIndex can have a name, which would be inside `params`.

If it helps,  there's an implementation here already: https://github.com/apache/arrow/blob/45918a90a6ca1cf3fd67c256a7d6a240249e555a/python/pyarrow/pandas_compat.py#L943 and Wes introduced the dict-based index description [here](https://github.com/pandas-dev/pandas/issues/25672). It feels like an under-defined format so could be hard to write tests for this. thx, adjusted Was this supposed to be addressed in https://github.com/pandas-dev/pandas/pull/51766? Not directly, but I added type_mapper support for the index in pyarrow a couple of days ago. We will get this out of the box with pyarrow 12.0, so I wouldn't do anything on our side.