16-dimensional compact projective planes with a collineation by Salzmann H. PDF

By Salzmann H.

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8). 8)). Now if 8cr(8) is semisimple, with eigenvalues A, 1, A - 1 , then N8 is the stable class in H with eigenvalues A, A - 1 . If A ^ —1 then N18 is the class in Hi with eigenvalues A, 1, A - 1 . 47 II. Orbital integrals 48 Denote by ZG(5a) the group of x in G with 5a = Int(x)(5a), by ZH(J) the centralizer of 7 in H, and by Z^iji) the centralizer of 71 in H\. For / G C™(G), /o G C™(H), /1 G C ~ ( # i ) , define the or&itaZ integrate $(5a,fdg) = f(lnt(x)(5a))^, at JG/ZaiSrr) r (-ji, fidhi) = fi(lnt(x)(ji)) dx — , (i = 0,1), where we put f(gcr) = f(g)- These depend on choices of Haar measures, denoted dg or dx or dhi depending on the context.

Define F(5-f K |det(l-Ad((g)dg (n()(h) = [ f(g)ct>(a(h)g)dg = f JG = f f JG 1 f{a{h- )nak){S T])(a)(p(k)S-1(a)dndadk.

DEFINITION. This sa is called the absolutely semisimple part of ka and u is the topologically unipotent part of ka. PROOF. , then u\ = u% for a = 0ia2- Since (a, q) = 1, there are integers a^, (3N with a^a + (3NqN — 1. Then u2u^ = u«»a+e»oNu-a»°-0»"N = u02NqNu^NqN -+1 as N —> oo. For the existence, recall that the prime-to-p part of the number n of elements in GL(n,F q ) is c = f] {q1 — 1). Let {{ka)q *} be a convergent i=i subsequence in the sequence {{ka) qm ; qm = 1 (mod c«)} in (K, a). Denote the limit by sa,s £ K.

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16-dimensional compact projective planes with a collineation group of dimension >= 35 by Salzmann H.


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