Math. Z. 184, 203-233 (1983) Mathematische Zeitschrift 9 Springer-Verlag 1983 Character Correspondences in Finite General Linear, Unitary and Symmetric Groups Gerhard O. Michler 1 and Jorn B. Olsson 2. 1 Universit~it-Gesamthochschule Essen, Fachbereich Mathematik, Universit~itsstraBe 3, D-4300 Essen, Federal Republic of Germany 2 Universit~it Dortmund, Abteilung Mathematik, D-4600 Dortmund 50, Federal Republic of Germany Introduction Let r>0 be a prime integer, and let G be a finite general linear group GL(n, q), a unitary group U(n, q) with q = qg, or a symmetric group S(n) of degree n. The partition of the irreducible (ordinary) characters of S(n) into r-blocks of S(n) was given by Brauer's and Robinson's solution of Nakayama's conjecture, see [7], p. 245. In their fundamental paper [5] Fong and Srinivasan have recently classified the r-blocks of GL(n, q) and U(n, q) for all primes r>2 with (r, q)= 1. Using these classifications of the r-blocks of G we show in this article that there is a natural one-to-one correspondence ~b between the irreducible charac- ters of height zero of an r-block B of G with defect group R and the irreducible characters of height zero of the Brauer correspondent b of B in N=NG(R ) (Theorem (4.10)). In particular, ko(B)=ko(b), where ko(B ) denotes the number of all irreducible characters )~ of B with height htz=0. Therefore Alperin's conjec- ture on the numbers of irreducible characters of height zero is verified for all general linear, unitary and symmetric groups. If G=S(n), then Theorem(4.10) also holds for r = 2. In order to establish the character correspondence ~, three other correspon- dences are studied, the product of which is ~b. In Sect. 1 we construct for every r-block B of G with defect group R a subgroup G of G with a Sylow r-sub- group /~-~R such that there is a natural height preserving one-to-one corres- pondence T between the set Irr(B) of all irreducible characters of B and the set Irr(/~0) of all irreducible characters of the principal r-block/~o of d (Reduction Theorems (1.9) and (1.10)). The map ~ respects the geometric conjugacy classes of characters. If /~0 denotes the principal r-block of N=Nd(/~), and if b is the Brauer correspondent of B in N=N6(R), then by Theorems (3.8) and (3.10) the block ideals/~o and b are Morita equivalent, have the same decomposition numbers, and there is a natural height preserving one-to-one correspondence a between Irr(bo) and Irr(b). * The 2"a author was supported in part by Deutsche Forschungsgemeinschaft