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A sum rule for scale factors in the Goldie rank polynomials. (English) Zbl 0699.17014
The author by applying the results of a book of G. Lusztig [Characters of reductive groups over a finite field. Ann. Math. Stud. 107 (1984; Zbl 0556.20033] to that of his own paper [Lect. Notes Math. 1243, 144-188 (1987; Zbl 0621.17006)] proves that (i) if $$\tau_ 1$$, $$\tau_ 2$$ are involutions in a Lusztig cell then Soc $$L(L(\tau_ 1\lambda),L(\tau_ 2\lambda))$$ is determined by multiplying appropriate irreducible characters of A and identifying coefficients. In particular, if $$A=S_ 5$$ then it can happen that $$L(L(\tau\lambda),L(\tau\lambda)$$) is not multiplicity free; (ii) if $$\tau$$ is an involution in a Lusztig cell then $$z_{\tau}$$ is the degree of an irreducible character of A; (iii) $$z_{\omega}$$ divides $$| A|$$ for every $$\omega$$ of a Lusztig cell; (iv) if $$A_{{\mathcal D}{\mathcal C}}$$ is commutative then for each left cell $${\mathcal C}$$ of $${\mathcal D}{\mathcal C}$$ the modules $$\{Soc\;L(L(\omega\lambda),L(\omega\lambda)):$$ $$\omega\in {\mathcal C}\}$$ generate the same subgroup of the Grothendieck group as the modules $$\{L(M(\lambda),L(\tau\lambda)):$$ $$\tau\in {\mathcal C}\cap \Sigma\}.$$