It is hard to say that the Duchess was flirtatious. Perhaps she was only friendly and in love with life. Either way, the Duke wanted her to only show affection to HIM. Take a look at the lines below and make your own judgment:
She had a heart--how shall I say?--too soon made glad, too easily impressed; she liked whate'er she looked on, and her lookes went everywhere. Sir, 'twas all one! My favor at her breast, the dropping of the daylight in the West, the bough of cherries some officious fool broke in the orchard for her, the white mule she rode with round the terrace--all and each would draw from her alike the approving speech, or blush, at least.
She seems very likely in love with nature, life, and small kindesses. What made him mad was the fact that she ranked his approval for her with everything else--sunsets, cherries that someone brought her, riding her mule. How dare she! (Smile)