---
tag: EMZTGD63V3
kind: concept
name: Petersson inner product
msc: [11F11, 11F25]
sources:
  - url: https://en.wikipedia.org/wiki/Petersson_inner_product
same_as:
  wikidata: Q2079742
  wikipedia: Petersson inner product
---

## Summary

The Petersson inner product is an [[inner product|0NB0AWGJBW]] on the space $S_k$ of [[cusp forms|A50Y012RBP]] of weight $k$.
It integrates $f \overline{g}\, y^k$ against the hyperbolic area over a fundamental domain of the [[modular group|VEKV1EM77B]].
The [[Hecke operators|S4SV3QZ8B1]] are self-adjoint for it, so $S_k$ has a basis of common eigenforms.
The [[Eisenstein series|EYREVY31BN]] are orthogonal to the cusp forms in a suitable sense.

## Explanations

### L1 human

To speak of angles and lengths in a space of functions, one needs an inner product.
For cusp forms the natural one multiplies $f$ by the conjugate of $g$, adds the weight factor $y^k$, and integrates over one copy of the quotient of the upper half-plane.
The weight factor makes the integrand invariant, so the choice of copy does not matter.
With this inner product, the Hecke operators behave like symmetric matrices, and so they can be diagonalized together.