
TL;DR
The paper provides a geometric proof that certain representations of split reductive groups over non-archimedean fields are defined over algebraic closures of , with applications to inner forms of .
Contribution
It introduces a new geometric proof establishing that standard Iwahori-spherical representations are defined over , extending to inner forms of .
Findings
Standard Iwahori-spherical representations are defined over .
Unpublished Clozel theorem on square-integrable representations is reproved locally.
Standard representations of inner forms of are also shown to be defined over .
Abstract
Let be a connected reductive group defined and split over a non-archimedean local field . We give a new geometric proof of a special case of a recent theorem of Solleveld. Namely, we show that the class of standard Iwahori-spherical -representations, a notion a priori dependent on the coefficient field being the complex numbers, is actually defined over . An unpublished theorem of Clozel, proven with global techniques, says that the class of essentially square-integrable representations is also defined over . As an application of our main result, we give a local proof of this theorem for inner forms of , as well as showing that standard representations of these groups are defined over .
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