Duke Mathematical Journal

Finite-dimensional representations of $W$-algebras

Ivan Losev

Abstract

$W$-algebras of finite type are certain finitely generated associative algebras closely related to the universal enveloping algebras of semisimple Lie algebras. In this article, we prove a conjecture of Premet that gives an almost complete classification of finite-dimensional irreducible modules for $W$-algebras. A key ingredient in our proof is a relationship between Harish-Chandra bimodules and bimodules over $W$-algebras that is also of independent interest.

Article information

Source
Duke Math. J., Volume 159, Number 1 (2011), 99-143.

Dates
First available in Project Euclid: 11 July 2011

https://projecteuclid.org/euclid.dmj/1310416364

Digital Object Identifier
doi:10.1215/00127094-1384800

Mathematical Reviews number (MathSciNet)
MR2817650

Zentralblatt MATH identifier
1235.17007

Citation

Losev, Ivan. Finite-dimensional representations of $W$ -algebras. Duke Math. J. 159 (2011), no. 1, 99--143. doi:10.1215/00127094-1384800. https://projecteuclid.org/euclid.dmj/1310416364

References

• R. Bezrukavnikov, M. Finkelberg, and V. Ostrik, On tensor categories attached to cells in affine Weyl groups, III, Israel J. Math. 170 (2009), 207–234.
• —, Character D-modules via Drinfeld center of Harish-Chandra bimodules, preprint.
• M. Bordemann and S. Waldmann, A Fedosov star product of the Wick type for Kähler manifolds, Lett. Math. Phys. 41 (1997), 243–253.
• W. Borho and H. Kraft, Über die Gelfand-Kirillov-Dimension, Math. Ann. 220 (1976), 1–24.
• J. Brundan, S. M. Goodwin, and A. Kleshchev, Highest weight theory for finite $W$-algebras, Int. Math. Res. Not. IMRN 2008, no. 15, art. ID rnn051.
• J. Brundan and A. Kleshchev, Shifted Yangians and finite $W$-algebras, Adv. Math. 200 (2006), 136–195.
• —, Representations of shifted Yangians and finite W-algebras, Mem. Amer. Math. Soc. 196 (2008), no. 918.
• —, Schur-Weyl duality for higher levels, Selecta Math. (N.S.) 14 (2008), 1–57.
• C. Dodd and K. Kremnizer, A localization theorem for finite W-algebras, preprint.
• D. Eisenbud, Commutative Algebra with a View toward Algebraic Geometry, Grad. Texts in Math. 150, Springer, New York, 1995.
• P. Etingof and V. Ginzburg, Symplectic reflection algebras, Calogero-Moser space, and deformed Harish-Chandra homomorphism, Invent. Math. 147 (2002), 243–348.
• B. Fedosov, A simple geometrical construction of deformation quantization, J. Differential Geom. 40 (1994), 213–238.
• —, Deformation Quantization and Index Theory, Mathematical Topics 9, Akademie, Berlin, 1996.
• —, Non-abelian reduction in deformation quantization, Lett. Math. Phys. 43 (1998), 137–154.
• W. L. Gan and V. Ginzburg, Quantization of Slodowy slices, Int. Math. Res. Not. IMRN 2002, no. 5, 243–255.
• V. Ginzburg, On primitive ideals, Selecta Math. (N.S.) 9 (2003), 379–407.
• —, Harish-Chandra bimodules for quantized Slodowy slices, Represent. Theory 13 (2009), 236–271.
• F. D. Grosshans, Algebraic Homogeneous Spaces and Invariant Theory, Lecture Notes in Math. 1673, Springer, Berlin, 1997.
• J. C. Jantzen, Einhüllende Algebren halbeinfacher Lie-Algebren, Ergeb. Math. Grenzgeb. (3), Springer, Berlin, 1983.
• A. Joseph, On the associated variety of a primitive ideal, J. Algebra 93 (1985), 509–523.
• B. Kostant, On Whittaker vectors and representation theory, Invent. Math. 48 (1978), 101–184.
• I. Losev, Quantized symplectic actions and $W$-algebras, J. Amer. Math. Soc. 23 (2010), 34–59.
• —, Completions of symplectic reflection algebras, preprint.
• G. Lusztig, Characters of Reductive Groups over a Finite Field, Ann. of Math. Stud. 107, Princeton Univ. Press, Princeton, 1984.
• W. M. Mcgovern, Completely prime maximal ideals and quantization, Mem. Amer. Math. Soc. 108 (1994), no. 519.
• A. Premet, Special transverse slices and their enveloping algebras, Adv. Math. 170 (2002), 1–55.
• —, Enveloping algebras of Slodowy slices and the Joseph ideal, J. Eur. Math. Soc. (JEMS) 9 (2007), 487–543.
• —, Primitive ideals, non-restricted representations and finite $W$-algebras, Mosc. Math. J. 7 (2007), 743–762, 768.
• —, Commutative quotients of finite W-algebras, Adv. Math. 225 (2010), 269–306.
• è. B. Vinberg and V. L. Popov, “Invariant theory” (in Russian) in Algebraicheskaya geometriya, 4, VINITI, Moscow, 1989, 137– 314; English translation in Algebraic Geometry, 4, Encyclopaedia Math. Sci. 55, Springer, Berlin, 1994, 123–284.
• D. A. Vogan, “Associated varieties and unipotent representations” in Harmonic Analysis on Reductive Groups (Brunswick, Maine, 1989), Progr. Math. 101, Birkhäuser, Boston, 1991, 315–388.