Dr. Katz DEq Homework Solutions 42

Dr. Katz DEq Homework Solutions 42 - a central function f...

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42 N.M. Katz: Because G is finite, any representation of G on a finite dimensional K-space comes by extension of scalars from a representation of G in a free (9-module of finite rank (take the lattice generated by the G-translates of any lattice), and hence its character takes values in (9. It follows that the functions ei take values in (9; i.e., lie in (gEG]. Because the Z~ are absolutely irreducible, we have (3.2.2.4) deg(xi)l ~ G, hence p4Vdeg(gi). Hence the values of the e~ at the identity element 1G G are units in (9, because deg(zi) (deg(gi)) 2 (3.2.2.5) ei(1)= trace (g~(1))= ~:G :~G Thus iffG (9 [G] is central, f= ~, ai ei and (ai ei)(1) (f ei)(1) G(9. (3.2.2.6) ai ei(1) - ei(1) Thus the e~ give an (9-base of the center of (9 [G]. Now let f~Gk[G] be the indecomposable central idempotent cor- responding to an irreducible representation X- We can certainly lift f~ to
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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 I-G] 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 K-space. Q.E.D....
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