Abstract
If p is an arbitrary parabolic subsuperalgebra of g = gl(m + nɛ), p(m), a character formula for the generic finite-dimensional irreducible g-module, such that p is the stabilizer of its lowest weight space, is announced. Furthermore, an estimate for the character of any finite-dimensional irreducible g-module in terms of its highest weight with respect to a distinguished Borel subsuperalgebra is presented (inequality (4)) and a sufficient condition for this to be an equality is found. In this way, two generalizations of the Kac character formula for typical modules are obtained: a formula concerning an arbitrary Borel subsuperalgebra ((1)) and a more effective formula ((3)) for the special case of a distinguished Borel subsuperalgebra. The complete proofs will appear in [14].
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References
BalantekinA. and BarsB., Representations of supergroups, J. Math. Phys. 22, 1810–1818 (1981).
BalantekinA. and BarsB., Dimension and character formulas for Lie supergroups, J. Math. Phys. 23, 1239–1247 (1982).
BernsteinJ. N. and LeitesD. A., A character formula for irreducible finite-dimensional modules over the Lie superalgebras of series H and sl, C.R. Acad. Sci. Bulg. 33, 1049–1051 (1980).
DemazureM., Une démonstration algébrique d'un théorème de Bott, Inv. Math. 5, 349–356 (1968).
DemazureM., A very simple proof of Bott's theorem, Inv. Math. 33, 271–272 (1976).
Hughes, J. W. and King, R. C., A conjecture of character for sl(m/n)-modules, J. Phys. A 20, N16, L1047–1052.
KacV. G., Lie superalgebras, Adv. Math. 26, 8–96 (1977).
KacV. G., Characters of typical representations of classical Lie superalgebras, Comm. Alg. 5(8), 889–897 (1977).
KacV. G., Representations of classical Lie superalgebras, Lecture Notes in Math. 676, 597–626 (1978).
LeitesD. A., Character formulas for irreducible finite dimensional modules of simple Lie superalgebras, Funct. Anal. i ego Pril. 14, N2, 35–38 (1980) (in Russian).
LeitesD. A., Lie superalgebras, in Sovr. Probl. Mat. 25, VINITI, Moscow (1984), pp. 3–47 (in Russian).
PenkovI. B., Borel-Weil-Bott theory for classical Lie supergroups, Sovr. Probl. Mat. 32, VINITI, Moscow (1988), to appear (in Russian).
Penkov, I. B., Classical Lie supergroups and Lie superalgebras and their representations, Preprint of Institut Fourier, Grenoble (1988), to appear.
Penkov, I. B. and Serganova, V., Cohomology of G/P for classical Lie supergroups G=GL(m+nε), p(m) and characters of atypical G-modules, submitted to Ann. Fourier.
SergeevA. N., The tensor algebra of the standard representation as a module over the Lie superalgebras gl(n/m) and Q(n), Mat. Sbornik 123 (165), N3, 422–430 (1984) (in Russian).
Thierry-MiegJ., Irreducible representations of the basic classical Lie superalgebras SU(m/n)/U(1), OSP(m/2n), D(2/1, α), G(3), F(4), Lect. Notes in Phys. 201, 94–98 (1984).
Thierry-Mieg, J., Tables of irreducible representations of the basic classical Lie superalgebras, Preprint of Groupe d'astrophysique relativiste CNRS, Observatoire de Meudon, 1985.
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Partially supported by Contract 911/11.04.88 with the Bulgarian Ministry of Culture, Science, and Education.
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Penkov, I., Serganova, V. Character formulas for some classes of atypical gl(m+nε)- and p(m)-modules. Lett Math Phys 16, 251–261 (1988). https://doi.org/10.1007/BF00398962
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DOI: https://doi.org/10.1007/BF00398962