On Refined Bruhat Decompositions and Endomorphism Algebras of Gelfand-Graev Representations

Abstract

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

On endomorphism algebras of Gelfand–Graev representations II

“…Our theorem improves the equality ℤ[ [12,Theorem 2.3] whenever the adjoint group of 𝐺 is simple of type other than 𝐹 4 or 𝐺 2 (in these two excluded types, the bad primes and the primes dividing the order of the Weyl group coincide). Moreover, via the ℤ-model 𝖤 𝐺 of Λ𝖤 𝐺 from [12, section 1.5], if we denote by 𝑀 is the product of all bad primes for 𝐺, then the above theorem implies that ℤ[ 1 𝑝𝑀 ]𝖤 𝐺 = ℤ[ 1 𝑝𝑀 ]𝖪 𝐺 * .…”

“…Indeed, for GL 𝑛 , in the course of constructing this correspondence in joint work with Moss [14], Helm proved in [8,Theorem 10.1] the equality Λ𝖤 GL 𝑛 = Λ𝖡 GL ∨ 𝑛 for Λ being the ring of Witt vectors of 𝔽 𝓁 with 𝓁 ≠ 𝑝. In our current context (𝐺 a connected reductive group over 𝔽 𝑞 ), when 𝐺 * has simply connected derived subgroup, it is known that 𝖡 𝐺 ∨ = 𝖪 𝐺 * (see [12,Theorem 3.13]), so that our main theorem yields the equalities…”

“…), the map Cur 𝐺 𝑆 is the unique ring homomorphism making the following diagram commutative (see [12,Lemma 1.6]):…”

“…The first author thanks Professor Jean-François Dat, his Ph.D. thesis advisor, for his constant support and enlightening opinions on this work. The second author thanks Robert Kurinczuk for bringing the work [12] of the first author to his attention. The authors thank Jay Taylor for providing a helpful reference.…”

“…Moreover, via the ℤ-model 𝖤 𝐺 of Λ𝖤 𝐺 from [12, section 1.5], if we denote by 𝑀 is the product of all bad primes for 𝐺, then the above theorem implies that ℤ[ 1 𝑝𝑀 ]𝖤 𝐺 = ℤ[ 1 𝑝𝑀 ]𝖪 𝐺 * . Indeed, this amounts to showing that the identification ℤ[ 1 𝑝𝑀 ]𝖤 𝐺 = ℤ[ 1 𝑝𝑀 ]𝖪 𝐺 * in the above theorem is equivariant under the action of the Galois group Gal(ℚ∕ℚ) on the coefficients, and the proof of this equivariance is the same as that of [12,Corollary 2.4].…”