## Projective modules in the category ${\scr O}_ S$: Loewy series

HTML articles powered by AMS MathViewer

- by Ronald S. Irving PDF
- Trans. Amer. Math. Soc.
**291**(1985), 733-754 Request permission

## Abstract:

Let $\mathfrak {g}$ be a complex, semisimple Lie algebra with a parabolic subalgebra ${\mathfrak {p}_S}$. The Loewy lengths and Loewy series of generalized Verma modules and of their projective covers in ${\mathcal {O}_S}$ are studied with primary emphasis on the case in which ${\mathfrak {p}_S}$ is a Borel subalgebra and ${\mathcal {O}_S}$ is the category $\mathcal {O}$. An examination of the change in Loewy length of modules under translation leads to the calculation of Loewy length for Verma modules and for self-dual projectives in $\mathcal {O}$, assuming the Kazhdan-Lusztig conjecture (in an equivalent formulation due to Vogan). In turn, it is shown that the Loewy length results imply Vogan’s statement, and lead to the determination of Loewy length for the self-dual projectives and certain generalized Verma modules in ${\mathcal {O}_S}$. Under the stronger assumption of Jantzen’s conjecture, the radical and socle series are computed for self-dual projectives in $\mathcal {O}$. An analogous result is formulated for self-dual projectives in ${\mathcal {O}_S}$ and proved in certain cases.## References

- Emil Artin, Cecil J. Nesbitt, and Robert M. Thrall,
*Rings with Minimum Condition*, University of Michigan Publications in Mathematics, no. 1, University of Michigan Press, Ann Arbor, Mich., 1944. MR**0010543** - Dan Barbasch,
*Filtrations on Verma modules*, Ann. Sci. École Norm. Sup. (4)**16**(1983), no. 3, 489–494 (1984). MR**740080**, DOI 10.24033/asens.1457 - Alexandre Beĭlinson and Joseph Bernstein,
*Localisation de $g$-modules*, C. R. Acad. Sci. Paris Sér. I Math.**292**(1981), no. 1, 15–18 (French, with English summary). MR**610137** - I. N. Bernšteĭn, I. M. Gel′fand, and S. I. Gel′fand,
*Structure of representations that are generated by vectors of highest weight*, Funckcional. Anal. i Priložen.**5**(1971), no. 1, 1–9 (Russian). MR**0291204**, DOI 10.1007/BF01075841
—, - Brian D. Boe and David H. Collingwood,
*A comparison theory for the structure of induced representations*, J. Algebra**94**(1985), no. 2, 511–545. MR**792968**, DOI 10.1016/0021-8693(85)90197-8 - J.-L. Brylinski and M. Kashiwara,
*Kazhdan-Lusztig conjecture and holonomic systems*, Invent. Math.**64**(1981), no. 3, 387–410. MR**632980**, DOI 10.1007/BF01389272 - Vinay V. Deodhar and James Lepowsky,
*On multiplicity in the Jordan-Hölder series of Verma modules*, J. Algebra**49**(1977), no. 2, 512–524. MR**463253**, DOI 10.1016/0021-8693(77)90255-1 - Thomas J. Enright and Brad Shelton,
*Decompositions in categories of highest weight modules*, J. Algebra**100**(1986), no. 2, 380–402. MR**840583**, DOI 10.1016/0021-8693(86)90083-9 - O. Gabber and A. Joseph,
*Towards the Kazhdan-Lusztig conjecture*, Ann. Sci. École Norm. Sup. (4)**14**(1981), no. 3, 261–302. MR**644519**, DOI 10.24033/asens.1406
R. S. Irving, - Ronald S. Irving,
*Projective modules in the category ${\scr O}_S$: self-duality*, Trans. Amer. Math. Soc.**291**(1985), no. 2, 701–732. MR**800259**, DOI 10.1090/S0002-9947-1985-0800259-9 - Jens Carsten Jantzen,
*Moduln mit einem höchsten Gewicht*, Lecture Notes in Mathematics, vol. 750, Springer, Berlin, 1979 (German). MR**552943**, DOI 10.1007/BFb0069521 - Jens Carsten Jantzen,
*Einhüllende Algebren halbeinfacher Lie-Algebren*, Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], vol. 3, Springer-Verlag, Berlin, 1983 (German). MR**721170**, DOI 10.1007/978-3-642-68955-0 - David Kazhdan and George Lusztig,
*Representations of Coxeter groups and Hecke algebras*, Invent. Math.**53**(1979), no. 2, 165–184. MR**560412**, DOI 10.1007/BF01390031 - David Kazhdan and George Lusztig,
*Schubert varieties and Poincaré duality*, Geometry of the Laplace operator (Proc. Sympos. Pure Math., Univ. Hawaii, Honolulu, Hawaii, 1979) Proc. Sympos. Pure Math., XXXVI, Amer. Math. Soc., Providence, R.I., 1980, pp. 185–203. MR**573434** - Alvany Rocha-Caridi,
*Splitting criteria for ${\mathfrak {g}}$-modules induced from a parabolic and the Berňsteĭn-Gel′fand-Gel′fand resolution of a finite-dimensional, irreducible ${\mathfrak {g}}$-module*, Trans. Amer. Math. Soc.**262**(1980), no. 2, 335–366. MR**586721**, DOI 10.1090/S0002-9947-1980-0586721-0 - Richard P. Stanley,
*Weyl groups, the hard Lefschetz theorem, and the Sperner property*, SIAM J. Algebraic Discrete Methods**1**(1980), no. 2, 168–184. MR**578321**, DOI 10.1137/0601021 - David A. Vogan Jr.,
*Irreducible characters of semisimple Lie groups. II. The Kazhdan-Lusztig conjectures*, Duke Math. J.**46**(1979), no. 4, 805–859. MR**552528**

*A category of*$g$

*-modules*, Functional Anal. Appl.

**10**(1976), 87-92.

*Projective modules in the category*$\mathcal {O}$, typescript, 1982.

## Additional Information

- © Copyright 1985 American Mathematical Society
- Journal: Trans. Amer. Math. Soc.
**291**(1985), 733-754 - MSC: Primary 17B10; Secondary 22E47
- DOI: https://doi.org/10.1090/S0002-9947-1985-0800260-5
- MathSciNet review: 800260