The Principal Representations of Reductive Algebraic Groups with Frobenius Maps

Abstract

We introduce the principal representation category of reductive groups with Frobenius maps and show that this category is a highest weight category when the ground field is complex field C. We also study certain kind of bound quiver algebras whose representations are related to the principal representation category.

Alvis-Curtis Duality for Representations of Reductive Groups with Frobenius Maps

“…The costandard module ∇(θ) J = M(θ, J ′ ) = kG ⊗ kP J ′ k θ , where J ′ = I(θ)\J for J ⊂ I(θ) was studied in [7]. By [7, Proposition 2.1], all the composition factors of…”

“…In paper [7], we introduce the principal representation category O(G). It is the full subcategory of kG-Mod such that any object M in O(G) is of finite length and its composition factors are E(θ) J for some θ ∈ T and J ⊂ I(θ).…”

“…Then we get the following corollary by Theorem 4.2. When the ground field k is the complex field C or an algebraically closed field of characteristic 0, the paper [7] shows that O(G) is highest weight category with decomposition…”

“…which is an isomorphism of k-algebras. Each A θ is a bound quiver algebra (see [7]), then we denote the representation of A θ by (V J , f α ). So we have a functor D θ : Rep A θ −→ Rep A θ w 0 , (V J , f α ) −→ (V * D θ (J) , f * α ) which is an equivalence.…”

“…All the composition factors of M(θ) are given when char k = char Fq . Based on these results, we introduced and studied the principal representation category O(G) in the paper [7]. The purpose of this paper is to generalize Alvis-Curtis duality of finite reductive group to the principal representation category O(G).…”