The fine structure of translation functors

Günzl, Karen

doi:10.1090/S1088-4165-99-00056-4

The fine structure of translation functors
HTML articles powered by AMS MathViewer

by Karen Günzl

Represent. Theory 3 (1999), 223-249

DOI: https://doi.org/10.1090/S1088-4165-99-00056-4

Published electronically: August 16, 1999

PDF | Request permission

Abstract:

Let $E$ be a simple finite-dimensional representation of a semisimple Lie algebra with extremal weight $\nu$ and let $0 \neq e \in E_{\nu }$. Let $M(\tau )$ be the Verma module with highest weight $\tau$ and $0 \neq v_{\tau } \in M(\tau )_{\tau }$. We investigate the projection of $e \otimes v_{\tau } \in E \otimes M(\tau )$ on the central character $\chi (\tau +\nu )$. This is a rational function in $\tau$ and we calculate its poles and zeros. We then apply this result in order to compare translation functors.

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
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
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, Structure of representations that are generated by vectors of highest weight, Funkcional. Anal. i Priložen. 5 (1971), no. 1, 1–9 (Russian). MR 0291204, DOI 10.1007/BF01075841
—, Category of $\mathfrak {g}$-modules. Funct. Anal. App. 10 (1976), 87–92.
Jacques Dixmier, Algèbres enveloppantes, Cahiers Scientifiques, Fasc. XXXVII, Gauthier-Villars Éditeur, Paris-Brussels-Montreal, Que., 1974 (French). MR 0498737
Pavel Etingof and Konstantin Styrkas, Algebraic integrability of Schrödinger operators and representations of Lie algebras, Compositio Math. 98 (1995), no. 1, 91–112. MR 1353287
Etingof, P., Varchenko, A.: Exchange Dynamical Quantum Groups. On the xxx-preprint server under math.QA/9801135.
Peter John Hilton and Urs Stammbach, A course in homological algebra, Graduate Texts in Mathematics, Vol. 4, Springer-Verlag, New York-Berlin, 1971. MR 0346025, DOI 10.1007/978-1-4684-9936-0
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
Masaki Kashiwara, The universal Verma module and the $b$-function, Algebraic groups and related topics (Kyoto/Nagoya, 1983) Adv. Stud. Pure Math., vol. 6, North-Holland, Amsterdam, 1985, pp. 67–81. MR 803330, DOI 10.2969/aspm/00610067
Bertram Kostant, On the tensor product of a finite and an infinite dimensional representation, J. Functional Analysis 20 (1975), no. 4, 257–285. MR 0414796, DOI 10.1016/0022-1236(75)90035-x
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