SEOUL, South Korea — The heir apparent to the Samsung empire, Jay Y. Lee, was trying to push through a corporate merger seen as critical to his plans to succeed his father as chairman.

For months, key shareholders fought the move. Then, suddenly, the standoff broke as South Korea’s government-controlled pension fund, which held the shares to cast the deciding vote, endorsed Mr. Lee’s deal.

A week later, President Park Geun-hye invited Mr. Lee to her office and asked for Samsung’s help with a campaign to promote South Korean culture and sports. Within months, Samsung had donated $17.4 million to two foundations controlled by the president’s confidante, Choi Soon-sil, and $6.2 million for the training of Korean equestrians, including Ms. Choi’s daughter.

Those donations — and whether they were part of a quid pro quo — are now at the heart of the impeachment case against Ms. Park. The nation’s full Constitutional Court will begin formal hearings on Tuesday into the case, the biggest influence-peddling scandal in South Korea’s history.