the city and the state are investigating a employer, can they both charge the company with discrimination, if only one which should take the case
Divorce / Separation Lawyer
Yes, and both can proceed simultaneously.
Employment / Labor Attorney
I admit that it is a very confusing area of the law. Many agencies can begin an EEOC-type investigation, but usually one will defer to the other. In terms of an ultimate claim, it is typically up to the claimant to decide (assuming all deadlines are met, and depending on the nature of the specific claims involved) whether the claimant wants to proceed in federal court, before the Illinois Human Rights Commission, or the Chicago Commission on Human Relations. Different relief is available from different agencies, so it's important to review all of your options.
If you have questions on the best way to proceed, you should contact an experienced employment attorney. Many of us, myself included, will provide a free consultation.
This answer is provided for informational purposes only and should not be relied on as legal advice. You should be aware that no attorney-client relationship is established through this answer and none will be established without a personal consultation and the signing of an engagement agreement.