Vanishing linear periods of cuspidal automorphic sheaves for \(\mathrm{GL}_{m+n}\) (Q6554692)

Let \(X\) stand for a connected smooth projective curve over an algebraically closed field with genus greater than one. Let \(E\) stand for an irreducible local system of rank \(m + n\) over \(X\). We denote by \(\mathrm{Per}_E\) the linear period of \(\mathrm{Aut}_E\), the automorphic \(E\)-Hecke eigensheaf on the stack classifying rank \(m + n\) vector bundles on \(X\).\N\NIn the paper under the review, the author shows that if \(m \neq n\) and \(m, n \geq 1\), then all cohomology sheaves of \(\mathrm{Per}_E\) vanish.