Deligne-Lusztig restriction of a Gelfand-Graev module
  Using Deodhar's decomposition of a double Schubert cell, we study the regular representations of finite groups of Lie type arising in the cohomology of Deligne-Lusztig varieties associated to tori. We deduce that the Deligne-Lusztig restriction of a Gelfand-Graev module is a shifted Gelfand-Graev module. 
Olivier Dudas (LM-Besan\c{c}on)