The golden egg / Donna Leon.

Enhanced descriptions from Syndetics:

In The Golden Egg , as the first leaves of autumn begin to fall, Vice Questore Patta asks Brunetti to look into a minor shop-keeping violation committed by the mayor's future daughter-in-law. Brunetti has no interest in helping his boss amass political favors, but he has little choice but to comply. Then Brunetti's wife, Paola, comes to him with a request of her own. The mentally handicapped man who worked at their dry cleaner has just died of a sleeping pill overdose, and Paola loathes the idea that he lived and died without anyone noticing him, or helping him.

Brunetti begins to investigate the death and is surprised when he finds nothing on the man: no birth certificate, no passport, no driver's license, no credit cards. As far as the Italian government is concerned, he never existed. Stranger still, the dead man's mother refuses to speak to the police, and assures Brunetti that her son's identification papers were stolen in a burglary. As secrets unravel, Brunetti suspects that the Lembos, an aristocratic family, might be somehow connected to the death. But why would anyone want this sweet, simple-minded man dead?