Let be a modular form for a congruence subgroup of .
For , let with :
In particular, for a prime, with;
where if is not divisible by .
The Hecke operators leave the space of modular forms and cusp forms invariant and turn out to be self-adjoint for a scalar product called the Petersson scalar product. In particular they have real eigenvalues. Hecke operators also satisfy multiplicative properties that are best summarized by the formal identity:
That equation in particular implies that whenever .
The set of all Hecke operators is usually denoted and is called the Hecke algebra.
0.1 Group algebra example
Definition 0.1 Let be a locally compact totally disconnected group; then the Hecke algebra of the group is defined as the convolution algebra of locally constant complex-valued functions on with compact support.
|Date of creation||2013-03-22 14:08:13|
|Last modified on||2013-03-22 14:08:13|
|Last modified by||olivierfouquetx (2421)|
|Defines||Hecke algebra of the group G|