By Allen Tannenbaum
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Extra info for Invariance and System Theory: Algebraic and Geometric Aspects
Hk,K ) with the algebra F ⊗O Hk,O (resp. 42 Representations of general linear groups K ⊗O Hk,O ). We will usually write simply Hk in place of Hk,F over the modular field. We wish to construct a right action of Hk,R on the module M k (σ)R , as a very special case of the theory of Howlett and Lehrer [HL1 ]. Gn Write NR = M (σ)R · · · M (σ)R (k times) and MR = M k (σ)R = RG NR k (d ) for short. We identify MR with RGn ⊗RP NR , where P is the standard parabolic subgroup of Gn with Levi factor G(dk ) and unipotent radical Y(dk ) .
By adjointness, HomH (α(V ), α(W )) ∼ = HomC (β ◦ α(V ), W ). By the lemma, there is a natural homomorphism β ◦ α(V ) → V which is surjective as OP V = V , and has kernel Z contained in OP (β ◦ α(V )). Moreover, any homomorphism from β ◦ α(V ) to W must act as zero on Z since OP (W ) = 0, hence factors through the quotient V of β ◦ α(V ). Thus, HomC (β ◦ α(V ), W ) ∼ = HomC (V, W ). 1d. Theorem. The restrictions of the functors α and AP ◦ β induce mutually inverse equivalences of categories between mod(H) and the full subcategory M of mod(C) consisting of all V ∈ mod(C) such that OP (V ) = 0, OP (V ) = V .
Lb ), set b a 2 (lj ) − m(λ) = j=1 ili . 15)]: χλ (1) = q m(λ) (q n − 1)(q n−1 − 1) . . 3) where h runs over the hook lengths of λ. 1). 16)]. 5), we have for each λ = (λ1 , . . , λa ) κ(s) the irreducible unipotent character χλ1 . . χλa (outer product) of CG (s). Note we are abusing notation here (and later) by allowing q to vary to give the definition of the unipotent character χλi ∈ X(GLki (q di )), i = 1, . . , a. Define the sign εs by εs = (−1)n+k1 +···+ka . χλ1 . . 5) Then, is precisely the irreducible character of G parametrized by the pair (s, λ).
Invariance and System Theory: Algebraic and Geometric Aspects by Allen Tannenbaum