Remote Access Representation Theory
Green Open Access

Representation Theory

ISSN 1088-4165



The Hom-spaces between projective functors

Author: Erik Backelin
Journal: Represent. Theory 5 (2001), 267-283
MSC (2000): Primary 17B10, 18G15, 17B20
Published electronically: September 10, 2001
MathSciNet review: 1857082
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: The category of projective functors on a block of the category $\mathcal O(\mathfrak g)$ of Bernstein, Gelfand and Gelfand, over a complex semisimple Lie algebra $\mathfrak g$, embeds to a corresponding block of the category $\mathcal O(\mathfrak g \times \mathfrak g)$. In this paper we give a nice description of the object $V$ in $\mathcal O(\mathfrak g \times \mathfrak g)$ corresponding to the identity functor; we show that $V$ is isomorphic to the module of invariants, under the diagonal action of the center $\mathcal Z$ of the universal enveloping algebra of $\mathfrak g$, in the so-called anti-dominant projective.

As an application we use Soergel's theory about modules over the coinvariant algebra $C$, of the Weyl group, to describe the space of homomorphisms of two projective functors $T$ and $T'$. We show that there exists a natural $C$-bimodule structure on $\operatorname{Hom}_{\{\operatorname{Functors}\}}(T, T')$ such that this space becomes free as a left (and right) $C$-module and that evaluation induces a canonical isomorphism $k \otimes_C \operatorname{Hom}_{\{\operatorname{Functors}\}} (T, T') \cong \operatorname{Hom}_{\mathcal O(\mathfrak g)}(T(M_e), T'(M_e))$, where $M_e$ denotes the dominant Verma module in the block and $k$ is the complex numbers.

References [Enhancements On Off] (What's this?)

  • [Bac] E. Backelin, Koszul duality for singular and parabolic category $\ensuremath{\mathcal{O}} $, Representation theory ( 3, (1999) 1-31. MR 2001c:17034
  • [Bass] H. Bass, Algebraic K-theory, Benjamin, 1968. MR 40:2736
  • [BB] A. Beilinson and J. Bernstein, Localisation de $\ensuremath{\mathfrak{g}} $-modules, C. R. Acad. Sc. Paris, 292 (Série I) (1981) 15-18. MR 82k:14015
  • [BeilGin] A. Beilinson and V. Ginzburg, Wall-crossing functors and $\mathcal{D}$-modules, Representation theory ( 3, (1999) 1-31. MR 2000d:17007
  • [BGS] A. Beilinson, V. Ginzburg and W. Soergel, Koszul duality patterns in representation theory, J. Amer. Math. Soc. 9, (1996) 473-527. MR 96k:17010
  • [B] J. Bernstein, Trace in categories, in: Proceedings of the Dixmier Colloquium, Birkhäuser, Progress in Math. 92, Boston, 1990, 417-423. MR 92d:17010
  • [BG] J. Bernstein and S.I. Gelfand, Tensor products of finite- and infinite-dimensional representations of semisimple Lie algebras, Compositio Math. 41 (1981), 245-285. MR 82c:17003
  • [BGG] J. Bernstein, I.M. Gelfand and S.I. Gelfand, On a category of $\mathfrak{g}$-modules, Functional Anal. Appl. 10 (1976), 87-92.
  • [D] P. Deligne, Catégories tannakiennes, The Grothendieck Festschrift, Vol II, Progr. Math. 87, Birkhäuser Boston (1990), 111-195. MR 92d:14002
  • [Jan] J. C. Jantzen, Einhüllende Algebren halbeinfacher Lie-Algebren, Springer-Verlag (1983). MR 86c:17011
  • [KL] D. Kazhdan and G. Lusztig, Representations of Coxeter groups and Hecke Algebras, Inventiones Math. 53, (1979) 165-184. MR 81j:20066
  • [BM] B. Müller, Projective Funktoren auf der parabolischen Kategorie $\ensuremath{\mathcal{O}} $, Diplomarbeit (1999), Mathematisches Fakultät, Albert-Ludwigs-Universität Freiburg i. Breisgau, unpublished.
  • [S] W. Soergel, Kategorie $\mathcal{O}$, Perverse Garben und Moduln über der Koinvarianten Algebra zur Weylgruppe, J. Amer. Math. Soc. 3 (1990), 421-444. MR 91e:17007
  • [S2] W. Soergel, Harish-Chandra bimodules, J. Reine. Angew. Math. 429 (1992), 49-74. MR 94b:17011

Similar Articles

Retrieve articles in Representation Theory of the American Mathematical Society with MSC (2000): 17B10, 18G15, 17B20

Retrieve articles in all journals with MSC (2000): 17B10, 18G15, 17B20

Additional Information

Erik Backelin
Affiliation: Sorselevägen 17, 16267 Vällingby, Stockholm, Sweden

Received by editor(s): May 16, 2000
Received by editor(s) in revised form: May 2, 2001
Published electronically: September 10, 2001
Article copyright: © Copyright 2001 American Mathematical Society

American Mathematical Society