The Geometric Weil Representation
  In this paper we construct a geometric analogue of the Weil representation over a finite field. Our construction is principally invariant, not choosing any specific realization. This eliminates most of the unpleasant formulas that appear in the traditional (non-invariant) approaches, and puts in the forefront some delicate geometric phenomena which underlie this representation. 
Shamgar Gurevich and Ronny Hadani