Abstract Harmonic Analysis, Pontryagin Duality, and the LCA Groups¶
Job: develop the LCA groups / Haar / dual group, the Pontryagin duality / Fourier on LCA, the adeles / Tate thesis / Poisson, the Peter–Weyl / non-LCA, and the \(p\)-adic Fourier — with the Pontryagin \(\widehat{\mathbb{R}}=\mathbb{R}\) / \(\widehat{\mathbb{T}}=\mathbb{Z}\) and the Tate thesis on the adele as the worked example.
Core¶
LCA groups and the Haar measure¶
The locally compact abelian (LCA) group \(G\) (the abelian / the locally compact, the \(\mathbb{R}^n\), \(\mathbb{T}^n\), \(\mathbb{Z}^n\), \(\mathbb{Q}_p\), the adele \(\mathbb{A}\)). The Haar measure (the left-invariant the Radon, the unique up to scalar, the measure theory / foundations). The dual group \(\hat G=\mathrm{Hom}(G,\mathbb{T})\) (the characters \(\chi:G\to\mathbb{T}\), the Pontryagin dual, the LCA).
| LCA / Haar / dual | Description |
|---|---|
| LCA group \(G\) | the abelian / locally compact (\(\mathbb{R}^n\), \(\mathbb{T}^n\), \(\mathbb{Z}^n\), \(\mathbb{Q}_p\), \(\mathbb{A}\)) |
| Haar measure | the left-invariant Radon (unique up to scalar, the measure theory) |
| Dual group \(\hat G\) | \(\mathrm{Hom}(G,\mathbb{T})\) (the characters \(\chi:G\to\mathbb{T}\), the Pontryagin dual) |
Pontryagin duality and the Fourier transform on LCA¶
Theorem (Pontryagin duality). \(\hat{\hat G}\cong G\) — the double dual of an LCA group is canonically isomorphic to \(G\).
The Fourier transform on LCA: \(\hat f(\chi)=\int_G f(g)\overline{\chi(g)}\,dg\) (the Fourier on the LCA, the Plancherel / the inversion on the LCA). The examples: \(\widehat{\mathbb{R}^n}=\mathbb{R}^n\) (\(\chi_\xi(x)=e^{2\pi ix\xi}\)), \(\widehat{\mathbb{T}^n}=\mathbb{Z}^n\) (\(\chi_n(e^{i\theta})=e^{in\theta}\), the Fourier series), \(\widehat{\mathbb{Z}^n}=\mathbb{T}^n\) (\(\chi_z(n)=z^n\), \(z\in\mathbb{T}\)), \(\widehat{\mathbb{Q}_p}=\mathbb{Q}_p\) (the self-dual \(p\)-adic), \(\widehat{\mathbb{A}}=\mathbb{A}/\mathbb{Q}\) (the self-dual adele).
| Pontryagin / Fourier on LCA | Description |
|---|---|
| Pontryagin duality | \(\hat{\hat G}\cong G\) (the double dual, the LCA the self-dual) |
| Fourier on LCA | \(\hat f(\chi)=\int_G f\overline{\chi}\,dg\) (the Plancherel / the inversion on the LCA) |
| Examples | \(\widehat{\mathbb{R}}=\mathbb{R}\), \(\widehat{\mathbb{T}}=\mathbb{Z}\), \(\widehat{\mathbb{Z}}=\mathbb{T}\), \(\widehat{\mathbb{Q}_p}=\mathbb{Q}_p\), \(\widehat{\mathbb{A}}=\mathbb{A}/\mathbb{Q}\) |
The adeles and the Tate thesis¶
The adele ring \(\mathbb{A}\) (the restricted product \(\mathbb{A}=\mathbb{R}\times\prod_p'\mathbb{Q}_p\) the adele, the algebraic number theory the global). The Tate thesis (the zeta / the \(L\)-function on the adele, the Fourier on \(\mathbb{A}\), the Poisson summation on the adele, the analytic number theory / L-functions the connection). The Poisson on the adele: \(\sum_{x\in\mathbb{Q}}f(x)=\sum_{\xi\in\mathbb{Q}}\hat f(\xi)\) (the Poisson, the self-dual \(\widehat{\mathbb{A}}=\mathbb{A}/\mathbb{Q}\)).
| Adele / Tate / Poisson | Description |
|---|---|
| Adele ring \(\mathbb{A}\) | \(\mathbb{R}\times\prod_p'\mathbb{Q}_p\) (the restricted product, the global) |
| Tate thesis | the zeta / \(L\)-function on the adele (the Fourier on \(\mathbb{A}\), the analytic NT / L-functions) |
| Poisson on adele | \(\sum_{\mathbb{Q}}f=\sum_{\mathbb{Q}}\hat f\) (the self-dual \(\widehat{\mathbb{A}}=\mathbb{A}/\mathbb{Q}\)) |
The abstract harmonic and the representation¶
The abstract harmonic analysis (the LCA the Fourier, the non-abelian the representation, the representation theory / foundations). The Peter–Weyl (the compact non-abelian: the \(L^2(G)=\bigoplus_{\pi\in\hat G}V_\pi\otimes V_\pi^*\) the decomposition into the irreducibles, the Peter–Weyl, the representation the Fourier the generalization). The non-LCA / the non-commutative (the non-abelian the representation, the representation theory, the NCG / operator algebras).
| Abstract / Peter–Weyl / non-LCA | Description |
|---|---|
| Abstract harmonic | the LCA the Fourier, the non-abelian the representation (representation theory) |
| Peter–Weyl | the compact non-abelian: \(L^2(G)=\bigoplus V_\pi\otimes V_\pi^*\) (the irreducibles, the Fourier the generalization) |
| Non-LCA / non-commutative | the non-abelian the representation, the NCG / operator algebras |
The \(p\)-adic Fourier and the local fields¶
The \(p\)-adic Fourier (the \(\hat f(\chi)=\int_{\mathbb{Q}_p}f(x)\overline{\chi(x)}\,dx\) the \(p\)-adic, the \(\mathbb{Q}_p\) the LCA, the characters the \(\chi_a(x)=e^{2\pi i\{ax\}_p}\) where \(\{ax\}_p\) is the fractional part). The Tate local (the \(p\)-adic the \(L\)-function / the zeta, the algebraic number theory, the prismatic cohomology the \(p\)-adic). The self-dual \(\widehat{\mathbb{Q}_p}=\mathbb{Q}_p\) (the \(p\)-adic the self-dual, the Haar the normalized).
| \(p\)-adic / Tate local / prismatic | Description |
|---|---|
| \(p\)-adic Fourier | \(\hat f(\chi)=\int_{\mathbb{Q}_p}f\overline{\chi}\,dx\) (the \(\chi_a(x)=e^{2\pi i\{ax\}_p}\), the self-dual) |
| Tate local | the \(p\)-adic \(L\) / zeta (the algebraic NT, the prismatic) |
| Self-dual | \(\widehat{\mathbb{Q}_p}=\mathbb{Q}_p\) (the \(p\)-adic the self-dual, the Haar the normalized) |
See it¶
Pontryagin duality \(\widehat{\mathbb{R}}=\mathbb{R}\), \(\widehat{\mathbb{T}}=\mathbb{Z}\), \(\widehat{\mathbb{Z}}=\mathbb{T}\) and the Tate thesis on the adele. The Pontryagin: \(\widehat{\mathbb{R}}=\mathbb{R}\) (the \(\chi_\xi(x)=e^{2\pi ix\xi}\) the characters, the \(\hat G=\mathbb{R}\)), \(\widehat{\mathbb{T}}=\mathbb{Z}\) (the \(\chi_n(e^{i\theta})=e^{in\theta}\) the Fourier series, the \(\hat G=\mathbb{Z}\)), \(\widehat{\mathbb{Z}}=\mathbb{T}\) (the \(\chi_z(n)=z^n\) the \(z\in\mathbb{T}\), the \(\hat G=\mathbb{T}\)), the double dual \(\hat{\hat G}=G\) (the Pontryagin). The Tate: the \(\mathbb{A}=\mathbb{R}\times\prod'\mathbb{Q}_p\) (the adele, the restricted product), the \(\hat f(\chi)=\int_\mathbb{A} f(x)\overline{\chi(x)}\,dx\) (the Fourier on the adele, the self-dual \(\widehat{\mathbb{A}}=\mathbb{A}/\mathbb{Q}\)), the Poisson on the adele (the \(\sum_{x\in\mathbb{Q}}f(x)=\sum_{\xi\in\mathbb{Q}}\hat f(\xi)\) the Poisson, the analytic number theory the zeta). The key point: the Pontryagin (the LCA the self-dual), the Tate (the adele the Fourier / the Poisson, the zeta / the \(L\)) — the abstract harmonic the LCA / the adele the global.
Check¶
Show that \(\widehat{\mathbb{T}}=\mathbb{Z}\) and \(\widehat{\mathbb{Z}}=\mathbb{T}\), and explain the Pontryagin duality \(\hat{\hat G}=G\).
The \(\widehat{\mathbb{T}}=\mathbb{Z}\): the characters \(\chi:\mathbb{T}\to\mathbb{T}\) are the \(\chi_n(e^{i\theta})=e^{in\theta}\) for \(n\in\mathbb{Z}\) (the continuous homomorphisms from \(\mathbb{T}\) to \(\mathbb{T}\), the \(n\)-th Fourier mode), so \(\widehat{\mathbb{T}}=\mathbb{Z}\). The \(\widehat{\mathbb{Z}}=\mathbb{T}\): the characters \(\chi:\mathbb{Z}\to\mathbb{T}\) are the \(\chi_z(n)=z^n\) for \(z\in\mathbb{T}\) (the \(z\) the value at \(1\in\mathbb{Z}\), the \(\chi(n)=\chi(1)^n=z^n\)), so \(\widehat{\mathbb{Z}}=\mathbb{T}\). The Pontryagin: \(\hat{\hat G}=G\) (the double dual, the \(\widehat{\widehat{\mathbb{T}}}=\widehat{\mathbb{Z}}=\mathbb{T}\), the \(\widehat{\widehat{\mathbb{Z}}}=\widehat{\mathbb{T}}=\mathbb{Z}\), the canonical isomorphism \(G\to\hat{\hat G}\) the \(g\mapsto(\chi\mapsto\chi(g))\)). So the \(\widehat{\mathbb{T}}=\mathbb{Z}\), \(\widehat{\mathbb{Z}}=\mathbb{T}\), the Pontryagin \(\hat{\hat G}=G\).
Explain the Tate thesis: the zeta function on the adele via the Fourier / the Poisson summation.
The adele \(\mathbb{A}=\mathbb{R}\times\prod_p'\mathbb{Q}_p\) (the restricted product, the global). The Fourier on \(\mathbb{A}\): the \(\hat f(\chi)=\int_\mathbb{A}f\overline{\chi}\,dx\) (the self-dual \(\widehat{\mathbb{A}}=\mathbb{A}/\mathbb{Q}\), the \(\mathbb{Q}\subset\mathbb{A}\) the discrete cocompact). The Poisson on the adele: \(\sum_{x\in\mathbb{Q}}f(x)=\sum_{\xi\in\mathbb{Q}}\hat f(\xi)\) (the Poisson summation, the \(\mathbb{Q}\) the discrete / the \(\mathbb{A}/\mathbb{Q}\) the compact, the Poisson the self-dual). The Tate thesis: the zeta \(\zeta(s)=\sum_{\mathfrak{a}}N(\mathfrak{a})^{-s}\) the adele the integral \(\int_{\mathbb{A}^\times}f(x)|x|^s\,d^\times x\) (the Tate, the zeta the adele the Fourier / the Poisson, the analytic NT / L-functions). The key: the zeta the adele the Fourier / the Poisson, the Tate the global the \(L\)-function.
Pitfalls¶
- The LCA the abelian (the non-abelian the Peter–Weyl / the representation, the representation theory) — the Pontryagin the LCA, the non-abelian the Peter–Weyl.
- The Tate the self-dual adele (the \(\widehat{\mathbb{A}}=\mathbb{A}/\mathbb{Q}\), the Poisson on the adele) — the adele the self-dual, the \(\mathbb{Q}\) the discrete / the \(\mathbb{A}/\mathbb{Q}\) the compact.
- The \(p\)-adic the local (the \(\mathbb{Q}_p\) the LCA, the Fourier / the zeta the local, the prismatic the \(p\)-adic) — the \(p\)-adic the local, the adele the global.
- The Peter–Weyl the compact non-abelian (the \(L^2(G)=\bigoplus V_\pi\otimes V_\pi^*\), the representation the Fourier the generalization) — the compact the Peter–Weyl, the non-compact the non-Peter–Weyl.
- The Haar the unique up to scalar (the left-invariant the Radon, the measure theory) — the Haar the unique, the scalar the normalization.
Refs¶
- Pontryagin, Topological Groups (the duality); Haar, "Der Massbegriff" (the measure)
- Peter–Weyl, "Die Vollständigkeit der Primitiven Darstellungen" (1927); the compact non-abelian
- Tate, "Fourier analysis in number fields and Hecke's zeta-functions" (1950 thesis); the adele / the zeta
- Hewitt–Ross, Abstract Harmonic Analysis I–II; Folland, A Course in Abstract Harmonic Analysis
Next¶
FA index · Foundations · Restriction & Strichartz · Oscillatory & Carleson · Hardy & BMO · Analytic NT / L-functions · Algebraic NT