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To Bertram Huppert on his 60th birthdayRepresentation theory of finite groups has to a large extent been character theory, i.e. the study of traces. The present paper stands in that tradition but instead of looking at traces of group elements in a given representation, it focuses on traces of endomorphisms of that representation. The group G comes in by the backdoor: first, multiplication with a conjugacy class sum is an endomorphism and second, much more important, for every subgroup H of G one has a map T% from the //-endomorphisms into the G-endomorphisms; the trivial observation that tr Tu(a) = \G : H\ tr a is exploited extensively.The starting point of the investigation was the following problem. Let M be an /?G-lattice affording an irreducible character %; let D be a vertex and 5 a £>-source of M, so 5 is an indecomposable /?D-lattice with vertex D. Which (if any) additional properties of S can be deduced from the fact that % is irreducible? Such additional properties are highly desirable, since the class of all finitely generated indecomposable RD-lattices is far too large to be useful.A partial answer to this question is given in this paper: S will be 'virtually irreducible' (defined in 1.2 below). A systematic study of virtually irreducible lattices shows that this class of lattices is closed under standard constructions like the Heller operator, taking duals, the Green correspondence, and taking sources. Under additional assumptions, it is also reasonably well-behaved under induction and restriction from and to subgroups. These results are developed in the first three paragraphs. There were some tempting sidetracks which I could not resist. In 1.12, a short proof is given for Ito's theorem that the degree of an irreducible character divides the index of a normal abelian subgroup. In 2.11, it is shown that an irreducible character which vanishes on all elements of order p will vanish on all elements with an order divisible by p.In § 4, Clifford Theory is used to study the vertices of virtually irreducible lattices. The main result is 4.15 which states that the defect group D of a block is abelian if and only if all virtually irreducible lattices in that block have D as a vertex. This characterization of blocks with abelian defect group is loosely related to Brauer's height 0 conjecture. However, it is not true in general that all virtually irreducible lattices in a block with abelian defect group are of height 0; a counter-example is given in 4.20.Finally, in § 5, a version of Frobenius reciprocity is given for virtually irreducible lattices (5.3), and in 5.8, it is shown that (rk U) p ^(rkM) p , if U | M ® X for some X and M is virtually irreducible.

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not only laid the foundations of modular representation theory of finite groups, he also raised a number of questions and made conjectures (see [1], [2] for instance) which since then have attracted the interest of many people working in the field and continue to guide the research efforts to a good extent. One of these is known as the "Height zero conjecture". It may be stated as follows: CONJECTURE. Let B be a p-block of the finite group G. All irreducible ordinary characters of G belonging to B are of height 0 if and only if a defect group of B is abelian.The conjecture is known to be true in special cases: Reynolds [14] treated the case of a normal defect group. Fong [6] proved the "if "-part for p-solvable groups and the "only if "-part for the principal block of a p-solvable group. Very recently, the proof for p-solvable groups has been completed by papers of Wolf and of Gluck and Wolf jointly [15], [8], [9].The present paper deals with the "if"-part of the conjecture. We show that this part holds true provided it holds for all quasi-simple groups, i.e. for the covering groups of non-abelian simple groups and their factor groups.Since the finite simple groups are classified, there is good reason to assume that in due time, the conjecture will be verified for, or a counterexample will show up among, these groups. In fact, this should be just a byproduct of a general effort to study the representation theory of simple groups. Relevant results are contained in papers of Fong-Srinivasan [7] and Michler-Olsson [13].We should mention that, ironically, this way of proving the "if "-part of Brauer's conjecture runs contrary to Brauer's intentions. He seems to have hoped to somehow structure the theory of simple groups using