representation, 2-representation, ∞-representation group, ∞-group group algebra, algebraic group, Lie algebra vector space, n-vector space affine space, symplectic vector space action, ∞-action module, equivariant object bimodule, Morita equivalence induced representation, Frobenius reciprocity Hilbert space, Banach space, Fourier transform, functional analysis orbit, coadjoint orbit, Killing form unitary representation geometric quantization, coherent state socle, quiver module algebra, comodule algebra, Hopf action, measuring Quite generally, automorphic forms are suitably well-behaved functions on a quotient space where is typically a discrete group, hence suitable functions on which are invariant under the action of a discrete group. The precise definition has evolved a good bit through time. Henri Poincaré considered analytic functions invariant under a discrete infinite group of fractional linear transformations and called them Fuchsian functions (after his advisor Lazarus Fuchs). More generally, automorphic forms in the modern sense are suitable functions on a coset space , hence functions on groups which are invariant with respect to the action of the subgroup . The archetypical example here are modular forms regarded as functions on where is a congruence subgroup, and for some time the terms “modular form” and “automorphic form” were used essentially synonymously, see below. Based on the fact that a modular form is a section of some line bundle on the moduli stack of elliptic curves, Pierre Deligne defined an automorphic form to be a section of a line bundle on a Shimura variety. By pullback of functions the linear space of such functions hence constitutes a representation of and such representations are then called automorphic representations (e.g. Martin 13, p. 9) , specifically so if is the general linear group with coefficients in a ring of adeles of some global field and . This is the subject of the Langlands program. There one also considers...