Let q be a power of a prime p, let G be a finite Chevalley group over F q and let U be a Sylow p-subgroup of G; we assume that p is not a very bad prime for G. We explain a procedure of reduction of irreducible complex characters of U , which leads to an algorithm whose goal is to obtain a parametrization of the irreducible characters of U along with a means to construct these characters as induced characters. A focus in this paper is determining the parametrization when G is of type F 4 , where we observe that the parametrization is "uniform" over good primes p > 3, but differs for the bad prime p = 3. We also explain how it has been applied for all groups of rank 4 or less. 1In fact for G of rank 4 or less, we obtain that each irreducible character of U can be obtained by inducing a linear character. In addition, we remark that our labelling of the irreducible characters is amenable to the action of a maximal torus and also the field automorphisms; thus it would be straightforward to determine these actions explicitly.The methods in this paper develop those used by Himstedt and the second and third authors in [HLM1] and [HLM2], and make significant further progress. A full parametrization of the irreducible characters of U for G of type D 4 and every prime p is given in [HLM1]. The so called single root minimal degree almost faithful irreducible characters are parameterized for every type and rank when p is not a very bad prime for G in [HLM2].The approach used in those papers and this paper is built on partitioning the irreducible characters of U in terms of the root subgroups that lie in their centre, but not in their kernel. Consequently, there are similarities to the theory of supercharacters, which were first studied for the case G is of type A by André, see for example [An]. This theory was fully developed by Diaconis and Isaacs in [DI]. Subsequently it was applied to the characters of U for G of types B, C and D by André and Neto in [AN].Another approach to the character theory of U is via the Kirillov orbit method, which is applicable for p greater than the Coxeter number of G. In [GMR2], Mosch, Röhrle and the first author explain an algorithm for parameterizing the coadjoint orbits of U, which was applied for G of rank at most 8, except E 8 ; through the Kirillov orbit method this leads to a parametrization of the irreducible characters of U. This was preceded by an algorithm to determine the conjugacy classes of U, see [GMR1].We note that a reduction procedure for algebra groups similar to ours was given by Evseev in [Ev] and builds on work of Isaacs in [Is2]. For G = SL n (q), this led to a parametrization of the irreducible characters of U for n ≤ 13. Also recently Pak and Soffer have determined the coadjoint orbits of U for G = SL n (q) and n ≤ 16, see [PS]. The situation for G not of type A turns out to be more complicated and we comment more on this below.There has been considerable other interest in the character theory and conjugacy classes of U. We refer the reader to [LM1] or the introductio...

Let G be a finite reductive group defined over Fq, 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 = PGL3(q) for an arbitrary prime p, and when G = SO5(q) for p odd.Let G be the fixed point subgroupḠ F of a reductive algebraic groupḠ under a Frobenius endomorphism F . A special role in the representation theory of finite reductive groups is played by Gelfand-Graev representations. These are certain representations induced from a linear character of a maximal unipotent subgroup of G. If the center of the ambient algebraic groupḠ is connected, then a Gelfand-Graev representation Γ is unique up to conjugacy and the irreducible components of Γ were described in [DL76]. In general, these representations were parametrized and decomposed in [DM91, Section 14] and [DLM92, Section 3].Denote by H the Hecke algebra (that is, the G-endomorphism algebra) of the module affording Γ. Since Γ is multiplicity free [Car85, Theorem 8.1.3], the algebra H is abelian. C. Curtis parametrized in [Cur93] the irreducible representations of H by pairs (T, θ) where T =T F for an F -stable maximal torusT ofḠ and where θ is an irreducible character of T . Each irreducible representation f T,θ of the algebra H is shown to have the following factorization [Cur93, Theorem 4.2], f T,θ =θ • f T , where f T : H →Q ℓ T is a homomorphism of algebras andθ is the linear extension of θ to the group algebraQ ℓ T . The homomorphism f T is independent of θ. Curtis' homomorphism f T was considered in [BK08] in the context of ℓ-modular Gelfand-Graev representations with ℓ = p.Here it is shown that if ℓ does not divide the order of the Weyl group of G, then the behavior of such endomorphism algebras is generic along all prime powers q.In order to obtain a constructive description of each f T , one needs to consider structure constants of H with respect to some basis. Building on work of Kawanaka [Kaw77] and Deodhar [Deo85], Curtis described in [Cur15] an algorithm for obtaining structure constants for H with respect to a standard basis parametrized by certain elements in N G (T ). In general, the determination of the structure

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