Bruhat cells which contain only regular elements

Let (G) over tilde be a connected reductive algebraic group defined over an algebraically closed field (F) over bar. Let (T) over tilde be a maximal torus of (G) over tilde and let (B) over tilde be a Borel subgroup of (G) over tilde containing (T) over tilde. Let W denote the Weyl group of (G) over tilde and let (w) over dot denote an element in the preimage of the natural map from N-(G) over tilde ((T) over tilde) to W. We investigate what conditions are necessary and sufficient for the double cosets (B) over tilde B(w) over dot (B) over tilde (called Bruhat cells) to contain only regular elements. An application is provided on how this information about Bruhat cells can be used to determine character values for certain Hecke algebras.