In Shakespeare's play "King Lear," the character who refuses to marry Cordelia without a dowry is the Duke of Burgundy. He is interested in marrying Cordelia, but when Lear disowns her and she is left without a dowry, the Duke decides not to pursue the marriage. This moment highlights the themes of wealth, power, and the value placed on material possessions in relationships during the time.

If you have any more questions about "King Lear" or its characters, feel free to ask!