Hecke algebra

Let f be a modular formMathworldPlanetmath for Γ a congruence subgroup of SL2().

f(z)=n=0anqn (1)

where q=e2iπτ.

For m, let Tmf(z)=n=0bnqn with :

bn=d|gcd(m,n)dk-1amn/d2 (2)

In particular, for p a prime, Tpf(z)=n=0bnqn with;

bn=apn+pk-1an/p (3)

where an/p=0 if n is not divisible by p.

The operator Tn is a linear operator on the space of modular forms called a Hecke operatorMathworldPlanetmath.

The Hecke operators leave the space of modular forms and cusp formsMathworldPlanetmathPlanetmath invariant and turn out to be self-adjointPlanetmathPlanetmath for a scalar productMathworldPlanetmath called the Petersson scalar product. In particular they have real eigenvaluesMathworldPlanetmathPlanetmathPlanetmathPlanetmath. Hecke operators also satisfy multiplicative properties that are best summarized by the formal identityPlanetmathPlanetmathPlanetmath:

n=1Tnn-s=𝑝(1-Tpp-s+pk-1-2s)-1 (4)

That equation in particular implies that TmTn=TnTm whenever gcd(n,m)=1.

The set of all Hecke operators is usually denoted 𝕋 and is called the Hecke algebra.

0.1 Group algebra example

Definition 0.1 Let Glcd be a locally compact totally disconnected group; then the Hecke algebra H(Glcd) of the group Glcd is defined as the convolution algebra of locally constant complex-valued functions on Glcd with compact support.

Such (G) algebrasPlanetmathPlanetmath play an important role in the theory of decomposition of group representationsMathworldPlanetmathPlanetmath into tensor productsPlanetmathPlanetmathPlanetmath.

Title Hecke algebra
Canonical name HeckeAlgebra
Date of creation 2013-03-22 14:08:13
Last modified on 2013-03-22 14:08:13
Owner olivierfouquetx (2421)
Last modified by olivierfouquetx (2421)
Numerical id 13
Author olivierfouquetx (2421)
Entry type Definition
Classification msc 11F11
Classification msc 20C08
Related topic ModularForms
Related topic AlgebraicNumberTheory
Defines Hecke operator
Defines Hecke algebra H(G) of the group G