PROCEEDINGS of the AMERICAN MATHEMATICAL SOCIETY Volume 121, Number 3, July 1994 ON A COUNTING FORMULAOF DJOKOVIC FOR ELEMENTS OF FINITE ORDER IN COMPACT LIE GROUPS F. DESTREMPESAND A. PIANZOLA (Communicated by Roe W. Goodman) 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 0 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 9. The case 'S = {1} recovers a formula, due to Djokovic, which counts the number of conjugacy classes of elements of order dividing N in «5. 0. Introduction In this paper we derive polynomial formulas for the number of conjugacy classes of elements of order TV coprime with the order of the Weyl group W of a compact connected simple Lie group whose character values generate an extension of Q contained in certain number fields. This result generalizes that of [DP] ([PW]) where the case N = pK (N = p, respectively) was treated. It also generalizes Djokovic [Djkl]. Let 0 be a (real) compact connected simple Lie group. As in [DP] and [PW] we define a lattice Y corresponding to a fixed maximal torus X of (5 via the following short exact sequence (i) o^r^t CTp2"'(,),s^i where t = it', with t' the Lie algebra of X, and / = %/-T • The Weyl group W of © acts on the lattice Y and hence on V = C ®z Y. Given an element w £ W, the multiplicity of the dth root of unity C¿ := e2l"ld as an eigenvalue of w in V is denoted by (2) fd(w). Received by the editors June 1, 1992 and, in revised form, October 5, 1992. 1991 Mathematics Subject Classification. Primary 22E15, 17B20. The authors acknowledge the support of the CRM in Montréal and of NSERC Canada. Work of the first author was supported in part by the Natural Sciences and Engineering Research Council of Canada and by the funds FCAR du Québec. Research of the second author supported in part by the Natural Sciences and Engineering Re- search Council of Canada. © 1994American MathematicalSociety 0002-9939/94 $1.00+ $.25 per page 943 License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use