groups. The philosophy of cusp forms highlights the kinship of representation theoretic aspects of these types of groups with general linear groups of local fields such as Qp and of the ring of adeles, see . Outlook—Representations of compact groups The theory of representations of compact groups may be, to some degree, extended to locally compact groups. The representation theory unfolds in this context great importance for harmonic analysis and the study of automorphic forms. For proofs, further information and for a more detailed insight which is beyond the scope of this chapter please consult [4] and [5]. Definition and properties A topological group is a group together with a topology with respect to which the group composition and the inversion are continuous. Such a group is called compact, if any cover of which is open in the topology, has a finite subcover. Closed subgroups of a compact group are compact again. Let be a compact group and let be a finite-dimensional –vector space. A linear representation of to is a continuous group homomorphism i.e. is a continuous function in the two variables and A linear representation of into a Banach space is defined to be a continuous group homomorphism of into the set of all bijective bounded linear operators on with a continuous inverse. Since we can do without the last requirement. In the following, we will consider in particular representations of compact groups in Hilbert spaces. Just as with finite groups, we can define the group algebra and the convolution algebra. However, the group algebra provides no helpful information in the case of infinite groups, because the continuity condition gets lost during the construction. Instead the convolution algebra takes its place. Most properties of representations of finite groups can be transferred with appropriate changes to compact groups. For this we need a counterpart to the summation over a finite group: Existence and uniqueness of the Haar measure On a compact group there exists exactly one measure such that: It is a left-translation-invariant measure The whole group has unit measure: Such a left-translation-invariant, normed measure is called Haar measure of the group Since is compact, it is possible to show that this measure is also right-translation-invariant, i.e. it also applies By the scaling above the Haar measure on a finite group is given by for all All the definitions to representations of finite groups that are mentioned in the