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

Authors:
F. Destrempes and A. Pianzola

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

MSC:
Primary 22E40; Secondary 22C05

MathSciNet review:
1185259

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Abstract: Given a compact connected simple Lie group 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 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 . The case recovers a formula, due to Djoković, which counts the number of conjugacy classes of elements of order dividing *N* in .

**[Ctr]**R. W. Carter,*Conjugacy classes in the Weyl group*, Compositio Math.**25**(1972), 1–59. MR**0318337****[DP]**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**, 10.1007/BF01264522**[Djk1]**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**, 10.1090/S0002-9939-1980-0574532-7**[Djk2]**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**, 10.1016/0022-4049(85)90026-X**[MPt]**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**, 10.1137/0605037**[Pzl1]**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**, 10.1016/0021-8693(87)90119-0**[Pzl2]**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**, 10.1515/crll.1987.377.40**[PW]**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**, 10.1016/0021-8693(91)90119-S**[Slm]**Louis Solomon,*Invariants of finite reflection groups*, Nagoya Math. J.**22**(1963), 57–64. MR**0154929****[Spg]**T. A. Springer,*Regular elements of finite reflection groups*, Invent. Math.**25**(1974), 159–198. MR**0354894**

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

Article copyright:
© Copyright 1994
American Mathematical Society