I respectfully request you to look into this matter urgently.

respectfully 是形容词 respectful（恭敬的、有礼貌的）的副词形式，由 respect（尊敬）加后缀 -ful（充满...的）构成形容词，再加 -ly 变成副词，表示"恭敬地、有礼貌地"。