Hecke algebra
The operator is a linear operator on the space of modular forms called a Hecke operator.
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:
(4) |
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.
Such algebras play an important role in the theory of decomposition of group representations into tensor products.
Title | Hecke algebra |
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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 of the group G |