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Is it true that each finite group of Lie type has a non-principal real $2$-block? Here the principal block is the one containing the trivial character and a block is real if it contains the complex conjugates of its characters.

Note: G. O. Michler and W. Willems proved, a long time ago, that each such group has a $p$-block of defect zero, for every prime $p$. However I do not know if they thought about the reality of such blocks.

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    $\begingroup$ Have you checked which 2-block contains the Steinberg character (associated the natural characteristic of the Lie type group). This is a rational-valued character (and answers your question for Lie type groups of characteristic 2, though in odd characteristic I'm not sure how to proceeed). $\endgroup$
    – Geoff Robinson
    Commented May 25, 2020 at 20:41
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    $\begingroup$ Hi Geoff, for a Lie type group defined in odd characteristic p, the degree of St is a power of p, hence is odd. As St is real valued, it therefore belongs to the principal 2-block. So no joy there. $\endgroup$
    – John Murray
    Commented May 26, 2020 at 11:23
  • $\begingroup$ Srinivasan and Vinroot, in this paper, prove that a character is real-valued iff its label is the 'obvious' form, i.e., (s,chi) where s and its inverse are conjugate, and chi is real-valued. So the question now becomes whether all such characters lie in the principal 2-block. I haven't gone further than this though, although it should be approachable for classical groups, and also for exceptional groups. $\endgroup$
    – David A. Craven
    Commented Jun 4, 2020 at 13:27
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    $\begingroup$ Hi David, as Radha Kessar has pointed out, for $n>1$ and $q$ an odd prime power, the $2$-blocks of $GL(n,q)$ are parametrized by the conjugacy classes of odd-order semisimple elements. Moreover a block is real if and only if the characteristic polynomial of any corresponding semisimple element is self-dual. Using this idea, it is easy to show that, apart from $GL(2,3)$ and $GL(3,3)$, all groups $GL(n,q)$ have a real non-principal $2$-block. $\endgroup$
    – John Murray
    Commented Jun 5, 2020 at 15:17

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