An important branch of representation theory studies representations of reductive groups over finite fields, such as $GL(n,F_q), Sp(2n,F_q)$ etc.  A deep theory due mostly to Lusztig and Shoji provides a classification of irreducible representations and a formula for their characters in terms of certain algebro-geometric objects called character sheaves. In a joint work with M. Finkelberg and V. Ostrik we establish some new nice features of the geometric objects, motivated by an attempt to find a conceptual explanation for the beautiful but somewhat mysterious results of Lusztig and Shoji.