By Edwin Hewitt, Kenneth A. Ross
This ebook is a continuation of vol. I (Grundlehren vol. a hundred and fifteen, additionally to be had in softcover), and features a specific therapy of a few vital components of harmonic research on compact and in the neighborhood compact abelian teams. From the reports: ''This paintings goals at giving a monographic presentation of summary harmonic research, way more whole and accomplished than any e-book already present at the subject...in reference to each challenge handled the publication bargains a many-sided outlook and leads as much as newest advancements. Carefull cognizance can be given to the historical past of the topic, and there's an intensive bibliography...the reviewer believes that for a few years to return it will stay the classical presentation of summary harmonic analysis.'' Publicationes Mathematicae
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Additional info for Abstract harmonic analysis, v.2. Structure and analysis for compact groups
PRAEGER, J. SAXL f + e for some Fqm/2 -basis e, f of V2 , then s has Jordan form J2m on V , and satisﬁes (v, s(v)) = 0 for some v ∈ V (for example take v = λf (λ ∈ Fqm/2 ), where Tr(λ2 [e, f ]) = 0). 1, some conjugate of s ﬁxes a point of Ω, a contradiction. Hence U , and indeed every S-invariant m-subspace of V , is totally singular. In particular S < LU = Pm , a parabolic subgroup of L. Write Pm = QR, where Q is the unipotent radical and R ∼ = GLm (q) is a Levi subgroup. Then Q is elementary abelian, and has the structure of an Fq R-module, with composition factors Vm , ∧2 Vm , if L = Sp2m (q), and just ∧2 Vm if L = P Ω+ 2m (q) (where Vm denotes a natural module for R).
1 in the case where G has socle L = P Ω− 2m (q) (m ≥ 4). Suppose G = AB, A ∩ B = 1 and A max G. 5) L = Ω10 (2). 1 q m−1 (q m + 1). 1). 4, from which we check that the only possibility − − 4 is m = 4, q = 2 and B Ω− m/2 (q ) = Ω2 (16). 2 that every involution class in G is represented in N1 , so no regular subgroup occurs. 2). If A = P1 then |B| = |G : A| = (q m + 1)(q m−1 − 1)/(q + 1) and we have a factorization N (SUm (q)) = B(A ∩ N (SUm (q)) ≤ BP1 . For m ≥ 5 odd, the only factorization of Um (q) (or an automorphism group) with P1 as a factor is U9 (2) = J3 P1 .
1: SYMPLECTIC AND ORTHOGONAL GROUPS 43 V1 ⊗V2 , a tensor product of 2-dimensional spaces over Fqm/2 . Let Ti ﬁx a symplectic form ( , )i on Vi , and let ei , fi be a basis of Vi with (ei , fi )i = 1. Then T1 × T2 ﬁxes the symmetric form [ , ] on V1 ⊗ V2 which is the product of ( , )1 and ( , )2 , and we may assume the Fq -form ( , ) on V preserved by L to be TrkK [ , ], where K = Fqm/2 , k = Fq . We may take ti to send ei → ei , fi → αi ei + fi for i = 1, 2 and some αi ∈ Fqm/2 . Then t1 t2 ﬁxes the vector v = α1 e1 ⊗ f2 − α2 f1 ⊗ e2 .
Abstract harmonic analysis, v.2. Structure and analysis for compact groups by Edwin Hewitt, Kenneth A. Ross