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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

MSC:
17B35 Universal enveloping (super)algebras
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[1] Curtis, C.W; Reiner, I, Representation theory of finite groups and associative algebras, (1962), Interscience London · Zbl 0131.25601
[2] Green, J.A, The modular representation algebra of a finite group, Illinois J. math., 6, 607-619, (1962) · Zbl 0131.26401
[3] Hamermesh, M, Group theory, (1964), Addison-Wesley London · Zbl 0151.34101
[4] Joseph, A, W-module structure in the primitive spectrum of the enveloping algebra of a semisimple Lie algebra, (), 116-135
[5] Joseph, A, Goldie rank in the enveloping algebra of a semisimple Lie algebra, III, J. algebra, 73, 295-326, (1981) · Zbl 0482.17002
[6] Joseph, A, On the variety of a highest weight module, J. algebra, 88, 238-278, (1984) · Zbl 0539.17006
[7] Joseph, A, On the classification of primitive ideals in the enveloping algebra of a semisimple Lie algebra, (), 30-76
[8] Joseph, A, On the cyclicity of vectors associated with Duflo involutions, (), 144-188
[9] \scA. Joseph, Multiplicity of the adjoint representatipn in simple quotients of the enveloping algebra of a simple Lie algebra, Trans. Amer. Math. Soc., in press. · Zbl 0676.17009
[10] Joseph, A; Stafford, J.T, Modules of I finite vectors over semi-simple Lie algebras, (), 361-384 · Zbl 0543.17004
[11] Lusztig, G, On a theorem of benson and Curtis, J. algebra, 71, 490-498, (1981) · Zbl 0465.20042
[12] Lusztig, G; Lusztig, G, A class of irreducible representations of a Weyl group I, II, (), 219-226 · Zbl 0511.20034
[13] Lusztig, G, Characters of reductive groups over a finite field, Ann. of math. stud., 107, (1984) · Zbl 0556.20033
[14] Lusztig, G, Sur LES cellules gauches des groupes de Weyl, C. R. acad. sci., Paris, 302, 5-8, (1986) · Zbl 0615.20020
[15] Lusztig, G, Cells in affine Weyl groups, I, II, Adv. stud. pure math., 6, 255-287, (1985), and a preprint, 1985
[16] Vogan, D.A, The orbit method and primitive ideals for semisimple Lie algebras, (), 281-316
[17] Joseph, A, Second commutant theorems in enveloping algebras, Amer. J. math., 99, 1167-1192, (1977) · Zbl 0378.17006
[18] Passman, D, The algebraic structure of group rings, (1977), Wiley New York
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