The 55-year-old man was pronounced dead at the scene.
Police have confirmed that the man fell from the third floor of Buchanan Galleries.
Police and ambulance staff - including a trauma team - were called to the incident at around 9.40am and shoppers were turned away from the main entrance, on Buchanan Street.
Inside, police tape cordoned off the area below the escalators on the ground floor.
Police and ambulance staff could be seen entering and exiting the building, and a stretcher was seen being carried into the building by paramedics.
A guest at the Premier Inn in Greenock, Inverclyde, discovered 23-year-old Nicola McDonough in a hallway. Her mother Margaret, 52, was found in a nearby room at around 7am today.
Mrs McDonough died from her injuries at Inverclyde Royal Hospital. Her daughter is still being treated and her condition is described as critical.
Police are trying to piece together the last movements of the women, who are from Paisley, Renfrewshire.
Their red Suzuki Swift car has been taken from the car park of the hotel, in James Watt Way.
Brian Harkins should have been enjoying a double celebration on Sunday – Mother’s Day and his 46th birthday.
Instead, his two older sisters made a heartfelt appeal for information after their brother was shot just over a week ago.
His elderly mother Ellen, 79, has been left inconsolable following Mr Harkins death last Wednesday.
He was shot dead at his home in Rutherglen, South Lanarkshire, around 8am. Police today revealed the killer struck at the side door of Mr Harkins home in Westmuir Place.
More than 30 police officers stopped early morning commuters exactly one week on from the murder of Brian Harkins, in Rutherglen, South Lanarkshire.
Police revealed they had narrowed down the key period when the gunman struck, and are focusing on a time frame of between 7am and 9am last Wednesday.
Mr Harkins is understood to have been shot around 8am just yards from the busy main road. The roadblock was set up in Westmuir Place, close to the 45-year-old's home.