Cuspidal representation

In number theory, cuspidal representations are certain representations of algebraic groups that occur discretely in spaces. The term cuspidal is derived, at a certain distance, from the cusp forms of classical modular form theory. In the contemporary formulation of automorphic representations, representations take the place of holomorphic functions; these representations may be of adelic algebraic groups.

When the group is the general linear group , the cuspidal representations are directly related to cusp forms and Maass forms. For the case of cusp forms, each Hecke eigenform (newform) corresponds to a cuspidal representation.

Formulation

Let G be a reductive algebraic group over a number field K and let A denote the adeles of K. The group G(K) embeds diagonally in the group G(A) by sending g in G(K) to the tuple (gp)p in G(A) with g=gp for all (finite and infinite) primes p. Let Z denote the centre of G and let ω be a continuous unitary character from Z(K) \ Z(A)× to C×. Fix a Haar measure on G(A) and let L20(G(K) \ G(A), ω) denote the Hilbert space of measurable complex-valued functions, f, on G(A) satisfying

  1. fg) = f(g) for all γ ∈ G(K)
  2. f(gz) = f(g)ω(z) for all zZ(A)
  3. for all unipotent radicals, U, of all proper parabolic subgroups of G(A).

The vector space L20(G(K) \ G(A), ω) is called the space of cusp forms with central character ω on G(A). A function appearing in such a space is called a cuspidal function.

A cuspidal function generates a unitary representation of the group G(A) on the complex Hilbert space generated by the right translates of f. Here the action of gG(A) on is given by

.

The space of cusp forms with central character ω decomposes into a direct sum of Hilbert spaces

where the sum is over irreducible subrepresentations of L20(G(K) \ G(A), ω) and the mπ are positive integers (i.e. each irreducible subrepresentation occurs with finite multiplicity). A cuspidal representation of G(A) is such a subrepresentation (π, Vπ) for some ω.

The groups for which the multiplicities mπ all equal one are said to have the multiplicity-one property.

gollark: This makes sense as a display thing, but how do you run it *backward*?
gollark: I tried playing a 10Hz sine wave just now and I can't hear it.
gollark: The position of the pen clearly can't be being directly mapped to voltage on a speaker or something, because the frequency would be waaaaay too low to hear.
gollark: What property of the waveforms it's generating varies as you change X/Y?
gollark: I'm aware it's converting it into waveforms somehow. That's just very vague.

See also

References

  • James W. Cogdell, Henry Hyeongsin Kim, Maruti Ram Murty. Lectures on Automorphic L-functions (2004), Chapter 5.
This article is issued from Wikipedia. The text is licensed under Creative Commons - Attribution - Sharealike. Additional terms may apply for the media files.