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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 \(\{L(M(\lambda),L(\tau\lambda)):\) \(\tau\in {\mathcal C}\cap \Sigma\}.\)
See also the following review.
Reviewer: Yu.N.Mal’tsev

17B35 Universal enveloping (super)algebras
Full Text: DOI
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