can I use the already installed cable splitter to split the signal from a digital antenna?

The splitter will work the same from an antenna input as it will from a cable service input.

However, are you sure about your choice of antenna? Understand that there are no "HD" or "digital" antennas. Words like that used in antenna advertising are absolute bunk; mere hype to make you think you need it for receiving digital channels. You don't. All TV antennas do exactly the same thing, both the old and the new, and whether the signal carries analog or digital modulation makes no difference whatsoever.

It is very easy for the unskilled to buy the wrong antenna, and there are a hoard of them on the market that are wrong for almost anyone. If you want help with that, update your question with your Zip code. If you'd rather try researching it yourself, the website below will help, but even that requires a high level of understanding about VHF and UHF transmission and propagation. Good luck.