No, he does not: if the house was left to the two of you, he is an owner; an owner of property is not required to pay rent to live there. That does not mean that you and he cannot come to an agreement under which if he lives there but you do not, then he will rent, or maintenance costs, or taxes, utilities, etc. But such an agreement would have to be voluntary; you cannot make him pay rent. What you can do, if the two of you can't work things out, is to bring a lawsuit seeking a court order requiring that the house be sold and the proceeds split between you.