On a counting formula of Djoković for elements of finite order in compact Lie groups

Destrempes, F.; Pianzola, A.

doi:10.1090/S0002-9939-1994-1185259-5

On a counting formula of Djoković for elements of finite order in compact Lie groups
HTML articles powered by AMS MathViewer

by F. Destrempes and A. Pianzola

Proc. Amer. Math. Soc. 121 (1994), 943-950

DOI: https://doi.org/10.1090/S0002-9939-1994-1185259-5

PDF | Request permission

Abstract:

Given a compact connected simple Lie group $\mathfrak {G}$ and a positive integer N relatively prime to the order of the Weyl group we give a counting formula for the number of conjugacy classes of elements x of order N in $\mathfrak {G}$ with the property that the N-cyclotonic field when viewed as a Galois extension of the field of characters of x has Galois group containing a fixed chosen cyclic group $\mathcal {G}$. The case $\mathcal {G} = \{ 1\}$ recovers a formula, due to Djoković, which counts the number of conjugacy classes of elements of order dividing N in $\mathfrak {G}$.

References

R. W. Carter, Conjugacy classes in the Weyl group, Compositio Math. 25 (1972), 1–59. MR 318337
F. Destrempes and A. Pianzola, Elements of compact connected simple Lie groups with prime power order and given field of characters, Geom. Dedicata 45 (1993), no. 2, 225–235. MR 1202101, DOI 10.1007/BF01264522
Dragomir Ž. Djoković, On conjugacy classes of elements of finite order in compact or complex semisimple Lie groups, Proc. Amer. Math. Soc. 80 (1980), no. 1, 181–184. MR 574532, DOI 10.1090/S0002-9939-1980-0574532-7
Dragomir Ž. Djoković, On conjugacy classes of elements of finite order in complex semisimple Lie groups, J. Pure Appl. Algebra 35 (1985), no. 1, 1–13. MR 772157, DOI 10.1016/0022-4049(85)90026-X
R. V. Moody and J. Patera, Characters of elements of finite order in Lie groups, SIAM J. Algebraic Discrete Methods 5 (1984), no. 3, 359–383. MR 752042, DOI 10.1137/0605037
A. Pianzola, On the arithmetic of the representation ring and elements of finite order in Lie groups, J. Algebra 108 (1987), no. 1, 1–33. MR 887189, DOI 10.1016/0021-8693(87)90119-0
A. Pianzola, On the regularity and rationality of certain elements of finite order in Lie groups, J. Reine Angew. Math. 377 (1987), 40–48. MR 887398, DOI 10.1515/crll.1987.377.40
A. Pianzola and A. Weiss, The rationality of elements of prime order in compact connected simple Lie groups, J. Algebra 144 (1991), no. 2, 510–521. MR 1140619, DOI 10.1016/0021-8693(91)90119-S
Louis Solomon, Invariants of finite reflection groups, Nagoya Math. J. 22 (1963), 57–64. MR 154929, DOI 10.1017/S0027763000011028
T. A. Springer, Regular elements of finite reflection groups, Invent. Math. 25 (1974), 159–198. MR 354894, DOI 10.1007/BF01390173