Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)



Linear maps preserving invariants

Author: Gerald W. Schwarz
Journal: Proc. Amer. Math. Soc. 136 (2008), 4197-4200
MSC (2000): Primary 20G20, 22E46, 22E60
Published electronically: July 23, 2008
MathSciNet review: 2431032
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: Let $ G\subset\mathrm{GL}(V)$ be a complex reductive group. Let $ G'$ denote $ \{\varphi\in\mathrm{GL}(V)\mid p\circ\varphi=p$ for all $ p\in\mathbb{C}[V]^G\}$. We show that, ``in general'', $ G'=G$. In case $ G$ is the adjoint group of a simple Lie algebra $ \mathfrak{g}$, we show that $ G'$ is an order 2 extension of $ G$. We also calculate $ G'$ for all representations of $ \mathrm{SL}_2$.

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

  • [Dix79] J. Dixmier, Champs de vecteurs adjoints sur les groupes et algèbres de Lie semi-simples, J. Reine Angew. Math. 309 (1979), 183–190 (French). MR 542047, 10.1515/crll.1979.309.183
  • [Hum72] James E. Humphreys, Introduction to Lie algebras and representation theory, Graduate Texts in Mathematics, vol. 9, Springer-Verlag, New York-Berlin, 1978. Second printing, revised. MR 499562
  • [Jac62] Nathan Jacobson, Lie algebras, Interscience Tracts in Pure and Applied Mathematics, No. 10, Interscience Publishers (a division of John Wiley & Sons), New York-London, 1962. MR 0143793
  • [Lun73] Domingo Luna, Slices étales, Sur les groupes algébriques, Soc. Math. France, Paris, 1973, pp. 81–105. Bull. Soc. Math. France, Paris, Mémoire 33 (French). MR 0342523
  • [Rai72] Mustapha Raïs, Distributions homogènes sur des espaces de matrices, Société Mathématique de France, 1972 (French). Thèse Sci. Math., Paris, 1970; Bull. Soc. Math. France Mém., No. 30; Supplément au Bull. Soc. Math. France, Tome 100, no. 2. MR 0507255
  • [Rai07] -, Notes sur la notion d'invariant caractéristique, abs/0707.0782v1.
  • [Sch95] Gerald W. Schwarz, Lifting differential operators from orbit spaces, Ann. Sci. École Norm. Sup. (4) 28 (1995), no. 3, 253–305. MR 1326669
  • [Sol05] S. Solomon, Irreducible linear group-subgroup pairs with the same invariants, J. Lie Theory 15 (2005), no. 1, 105–123. MR 2115231
  • [Sol06] S. Solomon, Orthogonal linear group-subgroup pairs with the same invariants, J. Algebra 299 (2006), no. 2, 623–647. MR 2228331, 10.1016/j.jalgebra.2005.11.008

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2000): 20G20, 22E46, 22E60

Retrieve articles in all journals with MSC (2000): 20G20, 22E46, 22E60

Additional Information

Gerald W. Schwarz
Affiliation: Department of Mathematics, Brandeis University, Waltham, Massachusetts 02454-9110

Keywords: Invariant polynomials
Received by editor(s): November 14, 2007
Published electronically: July 23, 2008
Additional Notes: This work was partially supported by NSA Grant H98230-06-1-0023
Communicated by: Gail R. Letzter
Article copyright: © Copyright 2008 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.