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Leading coefficients of character values of Hecke algebras. (English) Zbl 0657.20037

Representations of finite groups, Proc. Conf., Arcata/Calif. 1986, Pt. 2, Proc. Symp. Pure Math. 47, 235-262 (1987).
[For the entire collection see Zbl 0627.00009.]
Let W be a Weyl group and let H be the corresponding Hecke algebra over \(C(q^{1/2})\). Let E be an irreducible H-module; the trace of the standard basis element \(T_ w\in H\) in E is a polynomial in \(q^{1/2}\) of form \[ Tr(T_ w,E)=(-1)^{l(w)}c_{w,E} q^{(\ell (w)-a_ E)/2}+higher\quad powers\quad of\quad q^{1/2} \] where \(c_{w,E}\) are integers; the integer \(a_ E\) is uniquely defined by the condition that \(c_{w,E}\neq 0\) for some \(w\in W\). The integers \(c_{w,E}\) are called the leading coefficients of the character values \(Tr(T_ w,E)\). In [Characters of reductive groups over a finite field (Ann. Math. Stud. 107, 1984; Zbl 0556.20033)], the \(c_{w,E}\) were computed in certain cases, and were used in an essential way in the proof of the main theorem. Here we shall instead use the main theorem of [loc. cit.] to determine the \(c_{w,E}\) in all cases. (For example, if W is of classical type, then \(c_{w,E}\) is 0 or \(\pm 1\); if W is of type \(E_ 8\), then \(| c_{w,E}| \leq 8.)\) We show that \(c_{w,E}=0\), unless w and \(w^{-1}\) are in the same “left cell” of W. If \(\Gamma\) is a left cell of W, we show that \((\pm c_{w,E})\) for \(w\in \Gamma \cap \Gamma^{-1}\) can be regarded as the “character table” of a certain algebra \(J_{\Gamma \cap \Gamma^{-1}}\) (defined over Z) with respect to a canonical basis.
The main observation in this paper is that to each left cell \(\Gamma\) of W one can associate naturally two finite groups \({\mathcal G}\supset {\mathcal H}\) and that \((\pm c_{w,E})\) for w in \(\Gamma \cap \Gamma^{-1}\) can be interpreted in terms of an equivariant K-theory ring \(K_{{\mathcal G}}({\mathcal G}/{\mathcal H}\times {\mathcal G}/{\mathcal H})\). We also interpret in K- theoretic terms the nonabelian Fourier transform of [loc. cit.].

MSC:

20G05 Representation theory for linear algebraic groups
20G40 Linear algebraic groups over finite fields
18F25 Algebraic \(K\)-theory and \(L\)-theory (category-theoretic aspects)

Online Encyclopedia of Integer Sequences:

Number of fusion rings of multiplicity one and rank n.