I recently set up a savings account in a bank.They invest my savings in other investments and share the profits with me.

What does the religion say about this?

This is halal and permissible. It is an investment and trade which is perfectly legitimate. This is actually the foundation of Islamic banking. The bank holds money for you and keeps it safe. It provides service in a way the money is reinvested in halal businesses; the profit is shared between the bank and the account holder. This has no doubt or grey areas as long as the money invested, the businesses invested in and profits are halal.

It is stated in Fatwa al-Lajnah al-Daa’imah, 13/310:

It is permissible to deal with banks and financial institutions that do not deal with interest. But if they deal with interest then it is not permissible to deal with them, and they are not Islamic banks.

And it also says in Fatwa al-Lajnah al-Daa’imah (13/365):

If the Islamic bank does not deal with interest, and rather it invests money according to Islamic principles, then it is permissible for you to deposit your money with them for investment purposes.

And Allah knows best.