• DOI:10.1112/BLMS/21.2.202
• Corpus ID: 126291954

REPRESENTATION THEORY OF SEMISIMPLE GROUPS: An Overview Based on Examples

@article{Plymen1989REPRESENTATIONTO,
  title={REPRESENTATION THEORY OF SEMISIMPLE GROUPS: An Overview Based on Examples},
  author={Roger Plymen},
  journal={Bulletin of The London Mathematical Society},
  year={1989},
  volume={21},
  pages={202-203}
}

• R. Plymen
• Published 1 March 1989
• Mathematics
• Bulletin of The London Mathematical Society

Page 55, proof of Lemma 3.13. This proof is incorrect as it stands because it involves an interchange of limits that has not been justified. A naive attempt to fix the proof might involve assuming that the given representation is continuous into the uniform operator topology, but this assumption is not necessarily valid. Instead, a different approach is needed, and such an approach may be found in the paper by M. Welleda Baldoni, “General representation theory of real reductive groups,” Proc. Sumposia in Pure Math. 61 (1997), 61–72. The relevant proof is the proof of Proposition 1 on page 63. The argument is straightforward enough, but it makes use of the fact that g → π(g)v is of class C1 if and only if for each w in the Hilbert space, g → hπ(g)v, wi is of class C1. She attributes this result to Grothendieck and points to J. B. Neto, Trans. Amer. Math. Soc. 111 (1964), 381–391, for “discussion and results.” Reading Proposition 1 and its proof, one sees that by approaching the development in a slightly different order, one gets around the temptation to use the uniform operator topology and obtains a valid proof of Proposition 3.14.