Abstract
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 with respect to a standard basis of the endomorphism algebra of a Gelfand-Graev representation of G when G = PGL3(q) for an arbitrary prime p, and when G = SO5(q) for p odd.
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