## Finite-dimensional representations of minimal nilpotent W-algebras and zigzag algebras

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- by Alexey Petukhov
- Represent. Theory
**22**(2018), 223-245 - DOI: https://doi.org/10.1090/ert/516
- Published electronically: November 13, 2018
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## Abstract:

Let $\frak g$ be a simple finite-dimensional Lie algebra over an algebraically closed field $\mathbb F$ of characteristic 0. We denote by $\mathrm {U}(\frak g)$ the universal enveloping algebra of $\frak g$. To any nilpotent element $e\in \frak g$ one can attach an associative (and noncommutative as a general rule) algebra $\mathrm {U}(\frak g, e)$ which is in a proper sense a “tensor factor” of $\mathrm {U}(\frak g)$. In this article we consider the case in which $e$ belongs to the minimal nonzero nilpotent orbit of $\frak g$. Under these assumptions $\mathrm {U}(\frak g, e)$ was described explicitly in terms of generators and relations. One can expect that the representation theory of $\mathrm {U}(\frak g, e)$ would be very similar to the representation theory of $\mathrm {U}(\frak g)$. For example one can guess that the category of finite-dimensional $\mathrm {U}(\frak g, e)$-modules is semisimple.

The goal of this article is to show that this is the case if $\frak g$ is not simply-laced. We also show that, if $\frak g$ is simply-laced and is not of type $A_n$, then the regular block of finite-dimensional $\operatorname {U}(\frak g, e)$-modules is equivalent to the category of finite-dimensional modules of a zigzag algebra.

## References

- Hyman Bass,
*Algebraic $K$-theory*, W. A. Benjamin, Inc., New York-Amsterdam, 1968. MR**0249491** - Dan Barbasch and David Vogan,
*Primitive ideals and orbital integrals in complex classical groups*, Math. Ann.**259**(1982), no. 2, 153–199. MR**656661**, DOI 10.1007/BF01457308 - Dan Barbasch and David Vogan,
*Primitive ideals and orbital integrals in complex exceptional groups*, J. Algebra**80**(1983), no. 2, 350–382. MR**691809**, DOI 10.1016/0021-8693(83)90006-6 - 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** - A. Borel and J. De Siebenthal,
*Les sous-groupes fermés de rang maximum des groupes de Lie clos*, Comment. Math. Helv.**23**(1949), 200–221 (French). MR**32659**, DOI 10.1007/BF02565599 - W. Borho and J.-L. Brylinski,
*Differential operators on homogeneous spaces. III. Characteristic varieties of Harish-Chandra modules and of primitive ideals*, Invent. Math.**80**(1985), no. 1, 1–68. MR**784528**, DOI 10.1007/BF01388547 - Walter Borho and Jens Carsten Jantzen,
*Über primitive Ideale in der Einhüllenden einer halbeinfachen Lie-Algebra*, Invent. Math.**39**(1977), no. 1, 1–53 (German, with English summary). MR**453826**, DOI 10.1007/BF01695950 - N. Bourbaki,
*Éléments de mathématique. Fasc. XXXIV. Groupes et algèbres de Lie. Chapitre IV: Groupes de Coxeter et systèmes de Tits. Chapitre V: Groupes engendrés par des réflexions. Chapitre VI: systèmes de racines*, Actualités Scientifiques et Industrielles [Current Scientific and Industrial Topics], No. 1337, Hermann, Paris, 1968 (French). MR**0240238** - Jonathan Brown,
*Representation theory of rectangular finite $W$-algebras*, J. Algebra**340**(2011), 114–150. MR**2813566**, DOI 10.1016/j.jalgebra.2011.05.014 - Jonathan S. Brown and Simon M. Goodwin,
*Finite dimensional irreducible representations of finite $W$-algebras associated to even multiplicity nilpotent orbits in classical Lie algebras*, Math. Z.**273**(2013), no. 1-2, 123–160. MR**3010154**, DOI 10.1007/s00209-012-0998-8 - Jonathan Brundan and Alexander Kleshchev,
*Schur-Weyl duality for higher levels*, Selecta Math. (N.S.)**14**(2008), no. 1, 1–57. MR**2480709**, DOI 10.1007/s00029-008-0059-7 - David H. Collingwood and William M. McGovern,
*Nilpotent orbits in semisimple Lie algebras*, Van Nostrand Reinhold Mathematics Series, Van Nostrand Reinhold Co., New York, 1993. MR**1251060** - Christopher Dodd,
*Injectivity of the cycle map for finite-dimensional $W$-algebras*, Int. Math. Res. Not. IMRN**19**(2014), 5398–5436. MR**3267375**, DOI 10.1093/imrn/rnt106 - J. Matthew Douglass,
*Cells and the reflection representation of Weyl groups and Hecke algebras*, Trans. Amer. Math. Soc.**318**(1990), no. 1, 373–399. MR**1035211**, DOI 10.1090/S0002-9947-1990-1035211-6 - Dimitar Grantcharov and Vera Serganova,
*Cuspidal representations of ${\mathfrak {sl}}(n+1)$*, Adv. Math.**224**(2010), no. 4, 1517–1547. MR**2646303**, DOI 10.1016/j.aim.2009.12.024 - E. B. Dynkin,
*Semisimple subalgebras of semisimple Lie algebras*, Mat. Sbornik N.S.**30(72)**(1952), 349–462 (3 plates) (Russian). MR**0047629** - Ruth Stella Huerfano and Mikhail Khovanov,
*A category for the adjoint representation*, J. Algebra**246**(2001), no. 2, 514–542. MR**1872113**, DOI 10.1006/jabr.2001.8962 - James E. Humphreys,
*Representations of semisimple Lie algebras in the BGG category $\scr {O}$*, Graduate Studies in Mathematics, vol. 94, American Mathematical Society, Providence, RI, 2008. MR**2428237**, DOI 10.1090/gsm/094 - A. W. Knapp and Gregg J. Zuckerman,
*Classification of irreducible tempered representations of semisimple groups. II*, Ann. of Math. (2)**116**(1982), no. 3, 457–501. MR**678478**, DOI 10.2307/2007019 - Anthony Joseph,
*On the associated variety of a primitive ideal*, J. Algebra**93**(1985), no. 2, 509–523. MR**786766**, DOI 10.1016/0021-8693(85)90172-3 - Anthony Joseph,
*Orbital varietes of the minimal orbit*, Ann. Sci. École Norm. Sup. (4)**31**(1998), no. 1, 17–45 (English, with English and French summaries). MR**1604290**, DOI 10.1016/S0012-9593(98)80017-7 - Bertram Kostant,
*On Whittaker vectors and representation theory*, Invent. Math.**48**(1978), no. 2, 101–184. MR**507800**, DOI 10.1007/BF01390249 - Ivan Losev and Victor Ostrik,
*Classification of finite-dimensional irreducible modules over $W$-algebras*, Compos. Math.**150**(2014), no. 6, 1024–1076. MR**3223881**, DOI 10.1112/S0010437X13007604 - Ivan Losev,
*Finite-dimensional representations of $W$-algebras*, Duke Math. J.**159**(2011), no. 1, 99–143. MR**2817650**, DOI 10.1215/00127094-1384800 - Ivan Losev,
*Quantized symplectic actions and $W$-algebras*, J. Amer. Math. Soc.**23**(2010), no. 1, 35–59. MR**2552248**, DOI 10.1090/S0894-0347-09-00648-1 - Thomas Emile Lynch,
*GENERALIZED WHITTAKER VECTORS AND REPRESENTATION THEORY*, ProQuest LLC, Ann Arbor, MI, 1979. Thesis (Ph.D.)–Massachusetts Institute of Technology. MR**2940885** - Barry Mitchell,
*Theory of categories*, Pure and Applied Mathematics, Vol. XVII, Academic Press, New York-London, 1965. MR**0202787** - Alexey Petukhov,
*On the Gelfand-Kirillov conjecture for the W-algebras attached to the minimal nilpotent orbits*, J. Algebra**470**(2017), 289–299. MR**3565435**, DOI 10.1016/j.jalgebra.2016.08.021 - Alexander Premet,
*Special transverse slices and their enveloping algebras*, Adv. Math.**170**(2002), no. 1, 1–55. With an appendix by Serge Skryabin. MR**1929302**, DOI 10.1006/aima.2001.2063 - Alexander Premet,
*Enveloping algebras of Slodowy slices and the Joseph ideal*, J. Eur. Math. Soc. (JEMS)**9**(2007), no. 3, 487–543. MR**2314105**, DOI 10.4171/JEMS/86 - Alexander Premet and Lewis Topley,
*Derived subalgebras of centralisers and finite $W$-algebras*, Compos. Math.**150**(2014), no. 9, 1485–1548. MR**3260140**, DOI 10.1112/S0010437X13007823 - David A. Vogan Jr.,
*A generalized $\tau$-invariant for the primitive spectrum of a semisimple Lie algebra*, Math. Ann.**242**(1979), no. 3, 209–224. MR**545215**, DOI 10.1007/BF01420727

## Bibliographic Information

**Alexey Petukhov**- Affiliation: The University of Manchester, Oxford Road M13 9PL, Manchester, United Kingdom; and Institute for Information Transmission Problems, Bolshoy Karetniy 19-1, Moscow 127994, Russia
- MR Author ID: 828039
- Email: alex-{}-2@yandex.ru
- Received by editor(s): June 3, 2017
- Received by editor(s) in revised form: July 3, 2018
- Published electronically: November 13, 2018
- Additional Notes: The main result of the research presented in Sections 7, 8 was carried out at the IITP at the expense of Russian Science Foundation (project No. 14-50-00150). Other sections of this work were carried out as a part of a project on W-algebras supported by Leverhulme Trust Grant RPG-2013-293.
- © Copyright 2018 American Mathematical Society
- Journal: Represent. Theory
**22**(2018), 223-245 - MSC (2010): Primary 22E47, 16D50, 16D90
- DOI: https://doi.org/10.1090/ert/516
- MathSciNet review: 3875450