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Session 31. Representation Theory, Transformation Groups, and Applications
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Parameters for defining characteristic representations and counting semisimple classes
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Frank Lübeck, RWTH Aachen, Germany
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The talk is based on the joint work with Olivier Brunat (Paris)
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In this talk we consider connected reductive algebraic groups \(G\) over an
algebraic closure \(\bar{\mathbb F}_p\) of a finite prime field with
\(p\) elements. We assume that \(G\) is defined over a finite subfield
\(\mathbb{F}_q\) via a Frobenius morphism \(F: G \to G\), and we are interested
in the corresponding finite group of Lie type \(G(q) := G^F\), the fixed
points of \(F\) in \(G\).
In the special case that \(G\) is of simply-connected type, Steinberg showed
that the irreducible representations of \(G(q)\) over \(\bar{\mathbb F}_p\) are
obtained as restrictions of \(q^l\) (where \(l\) is the rank of \(G\)) highest
weight representations of the algebraic group \(G\). As a consequence we get
that \(G(q)\) has \(q^l\) semisimple conjugacy classes.
We consider general groups \(G\) and Frobenius morphisms \(F\) given in terms
of a root datum with Frobenius action on it. Starting from these data we
describe a parameterization of the irreducible representations of \(G(q)\)
over \(\bar{\mathbb F}_p\) (in general they are not all restrictions from the
algebraic group). As an application we compute from this parameterization
for all simple algebraic groups \(G\) as above and all Frobenius morphisms on
\(G\) the number of semisimple classes of the corresponding finite group
\(G(q)\).
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References-
O. Brunat, F. Lübeck, On defining characteristic representations of finite reductive groups,
Journal of Algebra, Volume 395 (2013), 121-141.
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