The adverbial form of 'deep'; indicates a great degree, depth, or intensity (e.g., deeply grateful, think deeply).
ご協力いただき、深く感謝します。
I deeply appreciate your cooperation.
問題の原因についてもっと深く考える必要がある。
We need to think more deeply about the cause of the problem.
彼は深く眠っているので、起こさないでください。
He's sleeping deeply, so please don't wake him.