The 59-year-old, the party’s former CEO, was previously arrested by Police Scotland on April 5 last year in the same probe and was taken into custody again this morning.
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Bosses at the corporation are said to be eyeing up the 39-year-old to take over presenting the Radio 4 show when veteran journalist Kearney, 66, stands down from her role after the general election.
The soprano singer has become embroiled in controversy after footage emerged showing her leading a choir in the protest chant at a Sing For Palestine event in Caerphilly, south Wales.
Hundreds of people have marched on the Russian embassy in London after the death of Vladimir Putin’s opponent Alexei Navalny.
Lord Walney, the former Labour MP who is now the political violence tsar, recommended a law change in a review that comes after a serious of mass protests against Israeli military action.