Frobenius semisimplicity and convolution morphisms 
--Mark Andrea de Cataldo, September 16, 2016

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The decomposition theorem of Beilinson-Bernstein-Deligne-Gabber is a
great tool to study morphism of algebraic varieties. It is known to
hold when the ground field is algebraically closed. Even if the proof
involves algebraic geometry over finite fields, it is not known
whether it holds over a finite ground field. I will discuss the issues
involved over a finite field and discuss some results and conjectures
in this context developed in joint work with Li Li and Tom Haines. I
will then prove the conjectures in the case of convolution morphisms
associated with reductive groups. The key result is a geometric global
surjectivity result in (intersection) cohomology, which may be of
independent interest.