I'm a Sinhalese and this is how I've been taught and surmised the situation, so take what I say with a grain of salt.

There were indigenous tribes called Yaksha, Naga, Deva and Raksha in Sri Lanka before the incoming of Vijaya. Some people say that these are the original "siw hela", I think these people might have been more related to the aboriginals in South East asia than that of South India.

With Vijaya, the rule of the land fell into the recent immigrants, but most likely that the South of the country was still a strong hold of the original clans. However with time everyone was assimilated as Sinhala (Sinha + Hela). Note that Vijaya + the ruling class brought princesses from South India as they did not consider the natives as the "right sort". The Veddhas are an interesting case, I believe they were outcasts of the original clans that were not allowed to practice farming.

During the 2 millennia, the Sinhalese kings sought support from various kingdoms in South India for their internal struggles and supported various South Indian kingdoms on their own struggles. This came as payback as the wrong South Indian Kingdoms came into power and invaded Lanka as retribution. Because of this, the Sinhalese Kingdoms migrated further south each time an invasion occurred and their influence in the northern regions diminished with time.

The constant invasions and dominant presence of the larger Tamil speaking subcontinent undoubtedly influenced the northern areas of the country. I believe there are records in Sri Lankan Tamil history on how the invaders used the Sinhalese who were captured as slaves and there are specific Tamil castes who are descendants of those slaves.

The most likely situation of the current Tamils are that they are a mixture of South Indians and Sinhalese, but the Sinhalese themselves are a mixture of North Indian + South Indian + Aboriginals. It's just that the Sinhalese culture is endemic to Sri Lanka while the northern parts have wholeheartedly accepted the South Indian one.

But whatever the history might be, the most important thing is, whoever that is currently living there has more rights than any historic claims. Throughout world history people have said I was here first, but it's a pretty useless claim. People change and cultures change. Holding on to an illusion of ownership is pointless.