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A Murnaghan-Nakayama rule for values of unipotent characters in classical groups. (English) Zbl 1376.20015
Represent. Theory 20, 139-161 (2016); correction ibid. 21, 1-3 (2017).
Summary: We derive a Murnaghan-Nakayama type formula for the values of unipotent characters of finite classical groups on regular semisimple elements. This relies on Asai’s explicit decomposition of Lusztig restriction. We use our formula to show that most complex irreducible characters vanish on some $$\ell$$-singular element for certain primes $$\ell$$.
As an application we classify the simple endotrivial modules of the finite quasi-simple classical groups. As a further application we show that for finite simple classical groups and primes $$\ell \geq 3$$ the first Cartan invariant in the principal $$\ell$$-block is larger than 2 unless Sylow $$\ell$$-subgroups are cyclic.

##### MSC:
 20C33 Representations of finite groups of Lie type 20C20 Modular representations and characters
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##### References:
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