Translation and shuffling of projectively presentable modules and a categorification of a parabolic Hecke module

Mazorchuk, Volodymyr; Stroppel, Catharina

doi:10.1090/S0002-9947-04-03650-5

Translation and shuffling of projectively presentable modules and a categorification of a parabolic Hecke module
HTML articles powered by AMS MathViewer

by Volodymyr Mazorchuk and Catharina Stroppel PDF

Trans. Amer. Math. Soc. 357 (2005), 2939-2973 Request permission

Abstract:

We investigate certain singular categories of Harish-Chandra bimodules realized as the category of $\mathfrak {p}$-presentable modules in the principal block of the Bernstein-Gelfand-Gelfand category $\mathcal {O}$. This category is equivalent to the module category of a properly stratified algebra. We describe the socles and endomorphism rings of standard objects in this category. Further, we consider translation and shuffling functors and their action on the standard modules. Finally, we study a graded version of this category; in particular, we give a graded version of the properly stratified structure, and use graded versions of translation functors to categorify a parabolic Hecke module.

References

H. H. Andersen, J. C. Jantzen, and W. Soergel, Representations of quantum groups at a $p$th root of unity and of semisimple groups in characteristic $p$: independence of $p$, Astérisque 220 (1994), 321 (English, with English and French summaries). MR 1272539
H. H. Andersen and N. Lauritzen, Twisted Verma modules, Studies in memory of Issai Schur (Chevaleret/Rehovot, 2000) Progr. Math., vol. 210, Birkhäuser Boston, Boston, MA, 2003, pp. 1–26. MR 1985191
Maurice Auslander, Representation theory of Artin algebras. I, II, Comm. Algebra 1 (1974), 177–268; ibid. 1 (1974), 269–310. MR 349747, DOI 10.1080/00927877408548230
Erik Backelin, The Hom-spaces between projective functors, Represent. Theory 5 (2001), 267–283. MR 1857082, DOI 10.1090/S1088-4165-01-00099-1
Joseph Bernstein, Trace in categories, Operator algebras, unitary representations, enveloping algebras, and invariant theory (Paris, 1989) Progr. Math., vol. 92, Birkhäuser Boston, Boston, MA, 1990, pp. 417–423. MR 1103598
Joseph Bernstein, Igor Frenkel, and Mikhail Khovanov, A categorification of the Temperley-Lieb algebra and Schur quotients of $U(\mathfrak {sl}_2)$ via projective and Zuckerman functors, Selecta Math. (N.S.) 5 (1999), no. 2, 199–241. MR 1714141, DOI 10.1007/s000290050047
J. N. Bernstein and S. I. Gel′fand, Tensor products of finite- and infinite-dimensional representations of semisimple Lie algebras, Compositio Math. 41 (1980), no. 2, 245–285. MR 581584
I. N. Bernšteĭn, I. M. Gel′fand, and S. I. Gel′fand, A certain category of ${\mathfrak {g}}$-modules, Funkcional. Anal. i Priložen. 10 (1976), no. 2, 1–8 (Russian). MR 0407097
Alexander Beilinson, Victor Ginzburg, and Wolfgang Soergel, Koszul duality patterns in representation theory, J. Amer. Math. Soc. 9 (1996), no. 2, 473–527. MR 1322847, DOI 10.1090/S0894-0347-96-00192-0
Nicolas Bourbaki, Lie groups and Lie algebras. Chapters 4–6, Elements of Mathematics (Berlin), Springer-Verlag, Berlin, 2002. Translated from the 1968 French original by Andrew Pressley. MR 1890629, DOI 10.1007/978-3-540-89394-3
E. Cline, B. Parshall, and L. Scott, Finite-dimensional algebras and highest weight categories, J. Reine Angew. Math. 391 (1988), 85–99. MR 961165
David H. Collingwood and Ronald S. Irving, A decomposition theorem for certain self-dual modules in the category ${\scr O}$, Duke Math. J. 58 (1989), no. 1, 89–102. MR 1016415, DOI 10.1215/S0012-7094-89-05806-7
Vinay V. Deodhar, On some geometric aspects of Bruhat orderings. II. The parabolic analogue of Kazhdan-Lusztig polynomials, J. Algebra 111 (1987), no. 2, 483–506. MR 916182, DOI 10.1016/0021-8693(87)90232-8
Jacques Dixmier, Enveloping algebras, Graduate Studies in Mathematics, vol. 11, American Mathematical Society, Providence, RI, 1996. Revised reprint of the 1977 translation. MR 1393197, DOI 10.1090/gsm/011
Vlastimil Dlab, Properly stratified algebras, C. R. Acad. Sci. Paris Sér. I Math. 331 (2000), no. 3, 191–196 (English, with English and French summaries). MR 1781825, DOI 10.1016/S0764-4442(00)01612-8
Yu. A. Drozd, S. A. Ovsienko, and V. M. Futornyĭ, The Harish-Chandra $S$-homomorphism and ${\mathfrak {G}}$-modules generated by a semiprimitive element, Ukrain. Mat. Zh. 42 (1990), no. 8, 1031–1037 (Russian, with Ukrainian summary); English transl., Ukrainian Math. J. 42 (1990), no. 8, 919–924 (1991). MR 1078818, DOI 10.1007/BF01099221
Thomas J. Enright, On the fundamental series of a real semisimple Lie algebra: their irreducibility, resolutions and multiplicity formulae, Ann. of Math. (2) 110 (1979), no. 1, 1–82. MR 541329, DOI 10.2307/1971244
V. Futorny, S. König, and V. Mazorchuk, Categories of induced modules and standardly stratified algebras, Algebr. Represent. Theory 5 (2002), no. 3, 259–276. MR 1921761, DOI 10.1023/A:1016579318115
Vyacheslav Futorny, Steffen König, and Volodymyr Mazorchuk, $\scr S$-subcategories in $\scr O$, Manuscripta Math. 102 (2000), no. 4, 487–503. MR 1785327, DOI 10.1007/s002290070038
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
James E. Humphreys, Reflection groups and Coxeter groups, Cambridge Studies in Advanced Mathematics, vol. 29, Cambridge University Press, Cambridge, 1990. MR 1066460, DOI 10.1017/CBO9780511623646
Ronald Irving, Shuffled Verma modules and principal series modules over complex semisimple Lie algebras, J. London Math. Soc. (2) 48 (1993), no. 2, 263–277. MR 1231714, DOI 10.1112/jlms/s2-48.2.263
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, 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
O. Khomenko, Categories with projective functors, PhD thesis. Universität Freiburg (Germany), 2004.
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
Steffen König and Volodymyr Mazorchuk, Enright’s completions and injectively copresented modules, Trans. Amer. Math. Soc. 354 (2002), no. 7, 2725–2743. MR 1895200, DOI 10.1090/S0002-9947-02-02958-6
S. König and V. Mazorchuk, An equivalence of two categories of $\textrm {sl}(n,\Bbb C)$-modules, Algebr. Represent. Theory 5 (2002), no. 3, 319–329. MR 1921764, DOI 10.1023/A:1016531419024
Volodymyr Mazorchuk, Twisted and shuffled filtrations on tilting modules, C. R. Math. Acad. Sci. Soc. R. Can. 25 (2003), no. 1, 26–32 (English, with English and French summaries). MR 1962132
Claus Michael Ringel, The category of modules with good filtrations over a quasi-hereditary algebra has almost split sequences, Math. Z. 208 (1991), no. 2, 209–223. MR 1128706, DOI 10.1007/BF02571521
Jeremy Rickard, Translation functors and equivalences of derived categories for blocks of algebraic groups, Finite-dimensional algebras and related topics (Ottawa, ON, 1992) NATO Adv. Sci. Inst. Ser. C: Math. Phys. Sci., vol. 424, Kluwer Acad. Publ., Dordrecht, 1994, pp. 255–264. MR 1308990
Wolfgang Soergel, Kazhdan-Lusztig polynomials and a combinatoric[s] for tilting modules, Represent. Theory 1 (1997), 83–114. MR 1444322, DOI 10.1090/S1088-4165-97-00021-6
Wolfgang Soergel, Kategorie $\scr O$, perverse Garben und Moduln über den Koinvarianten zur Weylgruppe, J. Amer. Math. Soc. 3 (1990), no. 2, 421–445 (German, with English summary). MR 1029692, DOI 10.1090/S0894-0347-1990-1029692-5
Wolfgang Soergel, Équivalences de certaines catégories de ${\mathfrak {g}}$-modules, C. R. Acad. Sci. Paris Sér. I Math. 303 (1986), no. 15, 725–728 (French, with English summary). MR 872544
C. Stroppel, A generalization of Joseph’s functor and shuffled Harish-Chandra bimodules. in preparation.
C. Stroppel, Categorification of the Temperley-Lieb category, Tangles and cobordisms via projective functors, Preprint Aarhus University (2003).
Catharina Stroppel, Category ${\scr O}$: gradings and translation functors, J. Algebra 268 (2003), no. 1, 301–326. MR 2005290, DOI 10.1016/S0021-8693(03)00308-9