First of all, what’s the Republican calendar? The Republican calendar was used during the French Republic (think 1790s -1810s). Most early civil records in Luxembourg use it.
What’s the Gregorian calendar? It’s the calendar we use today. You’ll need to be able to convert between the two to find records of an ancestor who was born under the Republican calendar but died under the Gregorian.
The national archives of Luxembourg has a tool to help. Enter the Republican date; it will give you the Gregorian.