On refined Bruhat decompositions and endomorphism algebras of Gelfand-Graev representations

Let $G$ be a finite reductive group defined over $\mathbb{F}_q$, with $q$ a power of a prime $p$. Motivated by a problem recently posed by C. Curtis, we first develop an algorithm to express each element of $G$ into a canonical form in terms of a refinement of a Bruhat decomposition, and we then use the output of the algorithm to explicitly determine the structure constants of the endomorphism algebra of a Gelfand-Graev representation of $G$ when $G=\mathrm{PGL}_3(q)$ for an arbitrary prime $p$, and when $G=\mathrm{SO}_5(q)$ for $p$ odd.