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A reflection group has a reflection representation in the natural sense. But what is the reflection representation for a Coxeter group if its simple roots cannot be regarded as reflections in the Euclidean space?

In this paper, the author says that this is Definition 1.5 in

W. Soergel, Kazhdan-Lusztig-Polynome und unzerlegbare Bimoduln uber Polynomringen, J. Inst. Math. Jussieu 6 (2007), no. 3, 501–525.

But I don't have this paper and I can't read in German. Would anyone please give some help?

sunkist
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  • Did you check Bourbaki, "Lie groups and Lie algebras" chapters 4-6 (English translation)? If I remember correctly, they define representations to $O(p,q)$ for general finitely generated Coxeter groups and explain what a pseudo-orthogonal reflection means in this context. If it is not there, it should be in Jim Humphreis book on Coxeter groups. – Moishe Kohan Mar 20 '14 at 10:38
  • @studiosus Thank you very much for the comment. Lie groups and Lie algebras, chapters 4-6 is due to be returned to the school library in a week. I will check it then. – sunkist Mar 20 '14 at 12:15
  • Try also Humphreys book "Reflection groups and Coxeter groups" too. http://books.google.com/books/about/Reflection_Groups_and_Coxeter_Groups.html?id=ODfjmOeNLMUC – Moishe Kohan Mar 20 '14 at 13:11
  • The definition in Soergel is: "Sei $(W, S)$ ein Coxetersystem. Bezeichne $T \subset W$ die Menge aller ,,Spiegelungen“, d.h. die Menge aller Elemente von $W$, die zu Elementen von $S$ konjugiert sind. Unter einer spiegelungstreuen Darstellung unseres Coxetersystems verstehen wir eine Darstellung $W \hookrightarrow GL(V)$ in einem endlichdimensionalen Vektorraum über einem Körper $k$ der Charakteristik $\operatorname{char} k \ne 2$ mit den folgenden Eigenschaften: – LSpice Oct 07 '15 at 20:10
  • (1) Unsere Darstellung ist treu. (2) Für $x ∈ W$ gilt $\dim(V/V^x) = 1$ $\Leftrightarrow$ $x ∈ T$. In Worten haben genau die Spiegelungen aus W eine Fixpunktmenge der Kodimension 1 in $V$." – LSpice Oct 07 '15 at 20:11
  • My very poor German translates this loosely as: "Let $(W, S)$ be a Coxeter system. Let $T$ be the set of 'reflections', that is, the elements conjugate to $S$. A reflection representation of such a system is a faithful representation in a finite-dimensional vector space over an odd-characteristic field for which the elements of $W$ whose fixed-point spaces in $V$ have codimension $1$ are precisely those in $T$." – LSpice Oct 07 '15 at 20:13

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