Characters of F₄(ℂ) at involutions

@misc{karmakar2026characters,
  title = {{Characters of $F_4(\mathbbC)$ at involutions}},
  author = {CHAYAN KARMAKAR},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.05116},
  eprint = {2610.05116},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.05116},
}

TY  - UNPB
TI  - Characters of F_4(ℂ) at involutions
AU  - CHAYAN KARMAKAR
PY  - 2026
DA  - 2026/10/04
PB  - arXiv
DO  - 10.48550/arxiv.2610.05116
UR  - https://arxiv.org/abs/2610.05116
AB  - The exceptional group F_4(ℂ) has two conjugacy classes of involutions: an involution t with centralizer Spin(9,ℂ), and the principal involution, with centralizer (SL(2,ℂ)×Sp(6,ℂ))/{±1}. We determine the values of all irreducible characters of F_4(ℂ) at t. For the irreducible representation with highest weight aω_1+bω_2+cω_3+dω_4 (in Bourbaki's labelling), we show that the character vanishes at t if and only if c and d are both odd. In all other cases its value is ± 2^−7 times the dimension of an explicit irreducible representation of Spin(9,ℂ). The proof is a limiting argument in the Weyl character formula, combined with a decomposition over the three cosets of W(B_4) in W(F_4) and the triality action of W(F_4)=W(D_4)rtimes S_3 on the weight lattice modulo the D_4-lattice. At the principal involution, Nadimpalli, Pattanayak and Prasad showed that the character value is ± 1/2 times the dimension of an irreducible representation of SL(2,ℂ)×Spin(7,ℂ). Comparing the two results, we find that the characters of F_4(ℂ) vanish at the two involutions for exactly the same highest weights. Together these results determine all irreducible characters of F_4(ℂ) at all involutions. We also explain how our formula at t relates to the generalized Weyl character formula of Gross, Kostant, Ramond and Sternberg.
ER  -