Langlands functoriality conjecture

In mathematics, the Langlands program is a web of far-reaching and influential conjectures that relate Galois groups in algebraic number theory to automorphic forms and representation theory of algebraic groups over local fields and adeles. It was proposed by Robert Langlands (1967, 1970).

In a very broad context, the program built on existing ideas: the philosophy of cusp forms formulated a few years earlier by Harish-Chandra and Gelfand (1963), the work and approach of Harish-Chandra on semisimple Lie groups, and in technical terms the trace formula of Selberg and others.

What initially was very new in Langlands' work, besides technical depth, was the proposed direct connection to number theory, together with the rich organisational structure hypothesised (so-called functoriality).

For example, in the work of Harish-Chandra one finds the principle that what can be done for one semisimple (or reductive) Lie group, should be done for all. Therefore, once the role of some low-dimensional Lie groups such as GL(2) in the theory of modular forms had been recognised, and with hindsight GL(1) in class field theory, the way was open at least to speculation about GL(n) for general n > 2.

The cusp form idea came out of the cusps on modular curves but also had a meaning visible in spectral theory as 'discrete spectrum', contrasted with the 'continuous spectrum' from Eisenstein series. It becomes much more technical for bigger Lie groups, because the parabolic subgroups are more numerous.

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