By Edwin Hewitt, Kenneth A. Ross
This e-book is a continuation of vol. I (Grundlehren vol. a hundred and fifteen, additionally to be had in softcover), and encompasses a special remedy of a few vital components of harmonic research on compact and in the community compact abelian teams. From the stories: "This paintings goals at giving a monographic presentation of summary harmonic research, way more entire and accomplished than any publication already present at the subject...in reference to each challenge taken care of the publication deals a many-sided outlook and leads as much as newest advancements. Carefull cognizance is usually given to the historical past of the topic, and there's an intensive bibliography...the reviewer believes that for a few years to return this can stay the classical presentation of summary harmonic analysis." Publicationes Mathematicae
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Additional info for Abstract harmonic analysis. Structure and analysis for compact groups
So we have nk − 1 = q(nk−1 − 1), from which it follows that nk = 1 + (n0 − 1)qk . So it is enough to prove the result for the case k = 0, that is, for a hyperbolic line. In the symplectic case, each of the q + 1 projective points on a line is isotropic. Consider the unitary case. We can take the form to be B((x1 , y1 ), (x2 , y2 )) = x1 y2 + y1 x2 , 42 where x = xσ = xr , r2 = q. So the isotropic points satisfy xy + yx = 0, that is, Tr(xy) = 0. How many pairs (x, y) satisfy this? If y = 0, then x is arbitrary.
Most are straightforward but n = 16 and n = 81 require some effort. 4 A technical result The result in this section will be needed at one point in our discussion of the unitary groups. It is a method of recognising the groups PSp(4, F) geometrically. Consider the polar space associated with PSp(4, F). Its points are all the points of the projective space PG(3, F), and its lines are the flat lines (those on which the symplectic form vanishes). We call them F-lines for brevity. Note that the F-lines through a point p of the porojective space form the plane pencil consisting of all the lines through p in the plane p⊥ , while dually the F-lines in a plane Π are all those lines of Π containing the point Π⊥ .
Moreover, any pair of distinct points spans either a flat subspace or a hyperbolic plane. Again, Witt’s Lemma shows that the group is transitive on the pairs of each type. ) Now a non-trivial equivalence relation preserved by G would have to consist of the diagonal and one other orbit. So to finish the proof, we must show: (a) if B(x, y) = 0, then there exists z such that B(x, z), B(y, z) = 0; (b) if B(x, y) = 0, then there exists z such that B(x, z) = B(y, z) = 0. This is a simple exercise. 2 Prove (a) and (b) above.