What influence did the Aryan invasion have in the ancient civilization of India?

First of all, it wasn't an Aryan Invasion. It was a Migration. The Aryans moved north from their homeland in South India around 40,000 BCE, according to DNA research. Some of them left Pakistan and settled in Iran and started the Persian Empire and Zoroastrianism. We know this because of glossochronology, which is a technique used by linguists to study languages and how they change over time. We know that Sanskrit was a descendant of the original Aryan language because of how similar certain words are across India, Iran, and even ancient Greece.
If you look at Zoroastrianism, you will see a lot of similarities with Hinduism, such as worship of fire. Similar structures of Gods and their holy books are called Avesta, whereas Hindus call them Vedas. Similarly, Hindus have Kathas, they have Gathas. During their migration north through India, the Aryans composed the Vedas.
Some Aryans left India gradually. Some remained in Pakistan and northwest India and built the Indus Valley Civilization.