// get the currently selected features in the map
var selectedFeatures = ArcGIS.Desktop.Mapping.MapView.Active.Map.GetSelection();
// get the first layer and its corresponding selected feature OIDs
var firstSelectionSet = selectedFeatures.ToDictionary().First();
// create an instance of the inspector class
var inspector = new ArcGIS.Desktop.Editing.Attributes.Inspector();
// load the selected features into the inspector using a list of object IDs
await inspector.LoadAsync(firstSelectionSet.Key, firstSelectionSet.Value);