By Biasi C., de Mattos D.
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Extra resources for A Borsuk-Ulam Theorem for compact Lie group actions
3 must coincide with 1 ~ Z 2 ~ ~r2(F ) ~ S ~ ( F ) when V=F. 2. Well's factor construction set for S-L2(F) as a group of operators. for this group. ,Xn) the character diY denote the as above. the limit = me~lim~pm~i(y2)diY (by Well [AT] it is an eighth root of unity). the invariant v(q,~) is well-defined. ,n, Y(Ti) this group y Given q(Xi'Xi)" F, realize such that the Fourier that such that Haar measure is known to exist F OF . V ~i=5 = T(giY) corresponding of on Recall is yield an a host of representations to quadratic = ~f(y)~(2xy)d (x) =f(-x).
Odd function a e F x. in Xo formula. By formula lal -1/2 = 1. (2oI$8) Therefore as d e s i r e d . , for example, For simplicity, that summation 8(9,g) 8(~,g) always defines is always it may often be the case that Suppose, follows from Poisson's suppose Finally rq(~)~(y,~) follows Then suppose that rq(~)~ also that rq([ -i0 - ~])~(X,t) But by the GF-invariance of 8(~) [ o invariant. e. ~ is also odd for all ~ e ~. F = Q. (2o~7) From formula = -~(X,t), -1 e(~, ~(-X,t) GF an is an it i e. o ]7) : - e(~,~).
18 and let By the above remarks, equal to ( | ~). v vcS'-S and For each up to a scalar of modulus isometric lim HS, space (or fixed by determined is a natural ~ ~ | ~v Hv S ~ S O , set of ~ v - r e p r e s e n t a t i o n S' ~ S, there Hs, , namely we can define the p r e - H i l b e r t Consequently (by completion) a Hilbert s p a c e H = ~ Hv on S which the ~A-representation V acts as follows: ~ = | ~v is any decomposable element of H, V 0 ~v = ~v with If for almost every v, then ~'(g)~ = | ~v'(gv)~v for g - (gv) in G~.
A Borsuk-Ulam Theorem for compact Lie group actions by Biasi C., de Mattos D.