Unformatted text preview: a central function f with values in (9, which we may write (3.2.2.7) f =~ aiei, aiG(9. Because fl "f: =fl, we have (3.2.2.8) ~ a { ei =f2 =_f = ~ ai e, mod(p [G]) and multiplying both sides by e~, we get (3.2.2.9) a~ ei  ai ei mod p[G]. Evaluating at the identity element le G, we have (by (3.2.2.4)) (3.2.2.10) a~=ai modp, so that each coefficient ai is congruent to either 0 or 1 mod p. Thus we may lift f~ to a central idempotent (3.2.2.11) ~,eiei, ei=0 or 1 in (9 [G]. Because the e~ are a basis of the center of (9 [G], the indecompo sability of f~ as central idempotent in k IG] implies that e~ differs from zero for only one value of i, say i= 1. This shows that ei is the unique central idempotent in (9 [G] lifting f~. This shows that Z is the "reduction mod p" of a unique irreducible representation Xl of G in a finite dimensional Kspace. Q.E.D....
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 Fall '11
 NormanKatz
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