"Indo-Persian culture" refers to those Persian aspects that have been integrated into or absorbed into the cultures of the Indian Subcontinent (hence the prefix "Indo"), and in particular, into North India, and modern-day Pakistan.
Persian influence was first introduced to the South Asia by Muslim rulers of Turkic and Afghan origin, especially with the Delhi Sultanate from the 13th century, and in the 16th to 19th century by the Mughal Empire. In general, from its earliest days, aspects of the culture and language were brought to the subcontinent by various Persianized Central Asian Turkic and Afghan rulers and conquerors, amongst them the most notable being Mahmud of Ghazni in the 11th century AD.
Persian was the official language of the Delhi Sultanate, the Mughal Empire, and their successor states, as well as the cultured language of poetry and literature. Many of the Sultans and nobility in the Sultanate period were Persianised Turks from Central Asia who spoke Turkic languages as their mother tongues. The Mughals were also culturally Persianized Central Asians (of Turco-Mongol origin), but spoke Chagatai Turkic as their first language at the beginning, before eventually adopting Persian. Persian became the preferred language of the Muslim elite of north India. Muzaffar Alam, a noted scholar of Mughal and Indo-Persian history, suggests that Persian became the lingua franca of the empire under Akbar for various political and social factors due to its non-sectarian and fluid nature. The influence of these languages led to a vernacular that is the ancestor of today's Urdu.
Indo-Persian culture Wikipedia
Indo-Persian culture has helped produce certain composite traditions within the South Asia that survive to this day, of which the Urdu language and literature is notable. The legacy of Indo-Persianate culture moreover can also be seen in much of the Mughal architecture within Lahore, Delhi and Agra, latterly of which the Taj Mahal is world-renowned. Indian classical music also owes much, including some ragas and instruments, to the Persian culture. In many ways, the absorption and assimilation of Persian or Persianate culture within India may be compared to the gradual (if sometimes problematic) absorption of English, British or Western culture generally of which the English language is perhaps the most notable and controversial within both India and Pakistan today. The influence of Persian language moreover may be seen in the considerable proportion of loan words absorbed into the vernaculars of the north and north-west of the South Asia including Punjabi, Gujarati, Urdu-Hindi, Kashmiri and Pashto.
With the presence of Muslim culture in the region in the Ghaznavid period, Lahore and Uch were established as centers of Persian literature. Abu-al-Faraj Runi and Masud Sa'd Salman (d. 1121) were the two earliest major Indo-Persian poets based in Lahore. The earliest of the "great" Indo-Persian poets was Amir Khusrow (d. 1325) of Delhi, who has since attained iconic status within the Urdu speakers of South Asia as, among other things, the "father" of Urdu literature.
Indo-Persian culture and to varying degrees also Turkic culture flourished side-by-side during the period of the Delhi Sultanate (1206–1526). The invasion of Babur in 1526, the end of the Delhi Sultanate, and the establishment of what would become the Mughal Empire would usher the golden age of Indo-Persian culture with particular reference to the art and architecture of the Mughal era.
The Mughal Era to the British Raj: Persian persisted as the language of the Mughal regime up to and including the year 1707 which marked the death of the Emperor Aurangzeb, generally considered the last of the "Great Mughals". Thereafter, with the decline of the Mughal empire, the 1739 invasion of Delhi by Nader Shah and the gradual growth initially of the Hindu Marathas and later the European power within the South Asia, Persian or Persian culture commenced a period of decline although it nevertheless enjoyed patronage and may even have flourished within the many regional empires or kingdoms of South Asia including that of the Sikh Maharaja Ranjit Singh (r. 1799–1837).
Persian as a language of governance and education was abolished in 1839 by the British and the last Mughal emperor Bahadhur Shah Zafar, even if his rule was purely symbolic or ceremonial, was overthrown in 1857 by the British.
Further, C.E. Bosworth wrote about the Central Asian's Persian (Tajiks→Ghurids) influence on India: "...The sultans were generous patrons of the Persian literary traditions of Khorasan, and latterly fulfilled a valuable role as transmitters of this heritage to the newly conquered lands of northern India, laying the foundations for the essentially Persian culture which was to prevail in Muslim India until the 19th century..."
Given that the Mughals had historically symbolized Indo-Persian culture to one degree or another, the overthrow of Bahadhur Shah Zafar and the institution of the British Raj in 1858 may be considered as marking the end of the Indo-Persian era, even if, after 1857, Persian would still retain an audience and even produce commendable literature such as the philosophical poetry of Muhammad Iqbal (d. 1938).