Regular characters of $GL_n(O)$ and Weil representations over finite fields

In this paper, we will point out a gap in the proof of a theorem of G.Hill (J. Algebra, 174 (1995), 610-635) and will give new arguments to give a remedy in the non-dyadic case modulo a conjecture on the triviality of certain Schur multiplier associated with a symplectic space over finite field. The new argument uses the Schr\"odinger representation of the Heisenberg group associated with a symplectic space over a finite field, and a simple application of Weil representation. This argument is applicable to the regular characters in general which include the cuspidal cases as well as the regular split cases.