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In his investigations of the
“natural domain of validity” for all Siegel Formulae over number fields, Igusa was lead
to a certain class of representations, which however, make sense over any field,
not just number fields. Calling these representations absolutely admissible,
Igusa analyzed their arithmetic nature in “Geometry of absolutely admissible
representations” [4], to find the ring of invariants, the stabilizers of various points etc.
The objective of [2] and of this paper is to show that for function fields, the
absolutely admissible representations arise from the same arithmetic questions
concerning the Siegel Formula, as was the case for number fields. In [2] we obtained
a list of the composition factors of the representations that arise in this
manner. In the present paper we show that these representations are in
fact completely reducible, whence for the characteristic of the function field
sufficiently large (a bound given explicitly for each group) the arithmetic of
invariants discussed in [4] hold for function fields, exactly as for number
fields.
The method of proof is cohomological, using the structure theory of semi-simple
groups to find a sufficient condition, which will guarantee that the extensions split. A
case by case examination shows that this condition is satisfied in every case save for
SL2 and E6, where further arguments are needed.
The author wishes to thank John Sullivan for helpful conversations concerning the
cohomology.
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