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Algebraic Solutions of Differential Equations
43
(3.2.3)
Corollary.
Hypotheses and notations as in
(3.2.2),
the indecom
posable central idempotents in K
[G],
which all lie in (P
have reductions
modulo t~ in k
[G]
which remain indecomposable.
Proof.
The number of e i is equal to the number off~, both being the
number of conjugacy classes in G. Hence
every e i
lifts an ft.
Q.E.D.
In terms of representations, this gives:
(3.2.3 bis)
Hypotheses as in
(3.2.2bis),
the reduction
mod p
of
any Kirreducible representation of G in a free Cmodule of finite rank is
irreducible (and every irreducible representation of G in a finitedimen
sional kspace arises this way).
(3.2.3.1)
(3.2.2his),
every irreducible repre
sentation of G in a finitedimensional kspace is absolutely irreducible.
This follows from (3.3.0bis), applied to arbitrary finite exten
sions K' of K, and arbitrary extensions of the valuation of K to K', by
which we can realize arbitrary finite extensions k' of k as residue field.
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This note was uploaded on 12/21/2011 for the course MAP 4341 taught by Professor Normankatz during the Fall '11 term at UNF.
 Fall '11
 NormanKatz

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