So what: Republic Airways said that it is aware of "a complaint filed by Delta" but hasn't yet been served or received the full complaint. According to reports, Delta has sued over Republic Airways failing to supply a full schedule of flights, causing Delta to find replacement flights, rebook customers, and issue refunds.
This is likely fallout from the company's recent dispute with pilots, which left the airline understaffed.
Now what: It's not surprising that there's some disputes between Republic Airways and its partners over operational challenges the company has had recently. But a recent tentative agreement with pilots should bring some stability to the company and its operations. It was also reported that Delta's losses because of the disruption were only $1 million, so even the impact of a settlement would likely be small. I don't see this as a reason to run from Republic Airways today.