Remote Access Transactions of the Moscow Mathematical Society

Transactions of the Moscow Mathematical Society

ISSN 1547-738X(online) ISSN 0077-1554(print)

   
 
 

 

On some modules of covariants for a reflection group


Authors: C. De Concini and P. Papi
Original publication: Trudy Moskovskogo Matematicheskogo Obshchestva, tom 78 (2017), vypusk 2.
Journal: Trans. Moscow Math. Soc. 2017, 257-273
DOI: https://doi.org/10.1090/mosc/272
Published electronically: December 1, 2017
Full-text PDF

Abstract | References | Additional Information

Abstract: Let $ \mathfrak{g}$ be a simple Lie algebra with Cartan subalgebra $ \mathfrak{h}$ and Weyl group $ W$. We build up a graded isomorphism $ \smash {\bigl (\bigwedge \mathfrak{h}\otimes \mathcal H\otimes \mathfrak{h}\bi... ... )^W}\to \bigl (\bigwedge \mathfrak{g}\otimes \mathfrak{g}\big )^{\mathfrak{g}}$ of $ \bigl (\bigwedge \mathfrak{g}\big )^{\mathfrak{g}}\cong S(\mathfrak{h})^W$-modules, where $ \mathcal H$ is the space of $ W$-harmonics. In this way we prove an enhanced form of a conjecture of Reeder for the adjoint representation.


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

  • [Bro95] Abraham Broer, The sum of generalized exponents and Chevalley's restriction theorem for modules of covariants, Indag. Math. (N.S.) 6 (1995), no. 4, 385-396. MR 1365182
  • [CKM17] Rocco Chirivì, Shrawan Kumar, and Andrea Maffei, Components of $ V(\rho)\otimes V (\rho)$, Transform. Groups 22 (2017), no. 3, 645-650. MR 3682832
  • [DCPP15] Corrado De Concini, Paolo Papi, and Claudio Procesi, The adjoint representation inside the exterior algebra of a simple Lie algebra, Adv. Math. 280 (2015), 21-46. MR 3350211
  • [DCMFPP14] Corrado De Concini, Pierluigi Möseneder Frajria, Paolo Papi, and Claudio Procesi, On special covariants in the exterior algebra of a simple Lie algebra, Atti Accad. Naz. Lincei Rend. Lincei Mat. Appl. 25 (2014), no. 3, 331-344. MR 3256213
  • [DdJO94] C. F. Dunkl, M. F. E. de Jeu, and E. M. Opdam, Singular polynomials for finite reflection groups, Trans. Amer. Math. Soc. 346 (1994), no. 1, 237-256. MR 1273532
  • [Hum90] James E. Humphreys, Reflection groups and Coxeter groups, Cambridge Studies in Advanced Mathematics, vol. 29, Cambridge University Press, Cambridge, 1990. MR 1066460
  • [KP90] A. A. Kirillov and I. M. Pak, Covariants of the symmetric group and its analogues in A. Weil algebras, Funktsional. Anal. i Prilozhen. 24 (1990), no. 3, 9-13, 96 (Russian); English transl., Funct. Anal. Appl. 24 (1990), no. 3, 172-176 (1991). MR 1082026
  • [Kos65] Bertram Kostant, Eigenvalues of the Laplacian and commutative Lie subalgebras, Topology 3 (1965), no. suppl. 2, 147-159 (German). MR 0167567
  • [Kos97] Bertram Kostant, Clifford algebra analogue of the Hopf-Koszul-Samelson theorem, the $ \rho$-decomposition $ C(\mathfrak{g})={\rm End}\, V_\rho\otimes C(P)$, and the $ \mathfrak{g}$-module structure of $ \bigwedge\mathfrak{g}$, Adv. Math. 125 (1997), no. 2, 275-350. MR 1434113
  • [MM65] Marvin Marcus and Henryk Minc, Generalized matrix functions, Trans. Amer. Math. Soc. 116 (1965), 316-329. MR 0194445
  • [Mol92] V. F. Molchanov, Poincaré series of representations of finite groups that are generated by reflections, Funktsional. Anal. i Prilozhen. 26 (1992), no. 2, 82-85 (Russian); English transl., Funct. Anal. Appl. 26 (1992), no. 2, 143-145. MR 1173093
  • [Pan12] Dmitri I. Panyushev, Invariant theory of little adjoint modules, J. Lie Theory 22 (2012), no. 3, 803-816. MR 3012155
  • [Ree97] Mark Reeder, Exterior powers of the adjoint representation, Canad. J. Math. 49 (1997), no. 1, 133-159. MR 1437204
  • [RS] V. Reiner and A. Shepler, Invariant derivations and differential forms for reflection groups. arxiv:1612.01031
  • [Sol63] Louis Solomon, Invariants of finite reflection groups, Nagoya Math. J. 22 (1963), 57-64. MR 0154929
  • [Ste87] John R. Stembridge, First layer formulas for characters of $ {\rm SL}(n,{\bf C})$, Trans. Amer. Math. Soc. 299 (1987), no. 1, 319-350. MR 869415
  • [Ste05] John R. Stembridge, Graded multiplicities in the Macdonald kernel. I, IMRP Int. Math. Res. Pap. 4 (2005), 183-236. MR 2199453


Additional Information

C. De Concini
Affiliation: Dipartimento di Matematica, Sapienza Università di Roma, Italy
Email: deconcin@mat.uniroma1.it

P. Papi
Affiliation: Dipartimento di Matematica, Sapienza Università di Roma, Italy
Email: papi@mat.uniroma1.it

DOI: https://doi.org/10.1090/mosc/272
Keywords: Exterior algebra, covariants, small representation, Dunkl operators
Published electronically: December 1, 2017
Dedicated: To Ernest Vinberg on the occasion of his 80th birthday
Article copyright: © Copyright 2017 C. De Concini, P.Papi

American Mathematical Society