Fourier-Eisenstein transform and Plancherel formula for rational binary quadratic forms. (English) Zbl 0776.11018
Let \({\mathbf X}\) be the space of non-degenerate rational symmetric matrices of size 2 and let \(G:=\bigl\{g\in\text{GL}_ 2(\mathbb{Q})\mid\text{det }g>0\bigr\}\), \(\Gamma:=\text{SL}_ 2(\mathbb{Z})\). The group \(G\) acts on \({\mathbf X}\) by \(g* x=(\text{det }g)^{-1}\cdot gx^ t g\). Let \({\mathcal C}^ \infty(\Gamma\backslash{\mathbf X})\) be the space of \(\Gamma\)-invariant \(\mathbb{C}\)-valued functions on \({\mathbf X}\) and its subspace \({\mathcal S}(\Gamma\backslash{\mathbf X})\) be the space of functions whose supports consist of a finite number of \(\Gamma\)-orbits. The Hecke algebra \({\mathcal H}(G,\Gamma)\) of the group \(G\) with respect to \(\Gamma\) acts naturally on these spaces.
The purpose of this paper is to analyze the structure of \({\mathcal S}(\Gamma\backslash{\mathbf X})\) as \({\mathcal H}(G,\Gamma)\)-module through an integral transform with the kernel function of the Eisenstein series \(E(x;s_ 1,s_ 2)\), which is called the Fourier-Eisenstein transform. The authors decompose the space \({\mathcal S}(\Gamma\backslash{\mathbf X})\) into the direct sum \(\bigoplus_ \chi{\mathcal S}(\Gamma\backslash{\mathbf X})_ \chi\) and define the integral transform \(F_ \chi\) from \({\mathcal S}(\Gamma\backslash{\mathbf X})_ \chi\) to \({\mathcal R}\). Here \(\mathcal R\) is the ring \({\mathcal R}:=\mathbb{C}[x_ 2,x_ 3,\dots,x_ p,\dots]\) with \(x_ p:=p^ t+p^{-t}\) and \(p\) runs over all rational primes. They prove that \(F_ \chi\) gives an isomorphism of \({\mathcal H}(G,\Gamma)\)-modules. When we define structures of pre-Hilbert spaces of \({\mathcal S}(\Gamma\backslash{\mathbf X})_ \chi\) and \(\mathcal R\) via suitable inner products, the map \(F_ \chi\) can be extended to an isometry between the completions of these spaces. This result may be considered as the Plancherel formula for the Fourier-Eisenstein transform \(F_ \chi\). An explicit form of the inverse transformation of \(F_ \chi\) follows quite easily from the Plancherel formula. Furthermore, using the main result, we can determine all \({\mathcal H}(G,\Gamma)\)-eigenfunctions in \({\mathcal C}^ \infty(\Gamma\backslash{\mathbf X})\).
The purpose of this paper is to analyze the structure of \({\mathcal S}(\Gamma\backslash{\mathbf X})\) as \({\mathcal H}(G,\Gamma)\)-module through an integral transform with the kernel function of the Eisenstein series \(E(x;s_ 1,s_ 2)\), which is called the Fourier-Eisenstein transform. The authors decompose the space \({\mathcal S}(\Gamma\backslash{\mathbf X})\) into the direct sum \(\bigoplus_ \chi{\mathcal S}(\Gamma\backslash{\mathbf X})_ \chi\) and define the integral transform \(F_ \chi\) from \({\mathcal S}(\Gamma\backslash{\mathbf X})_ \chi\) to \({\mathcal R}\). Here \(\mathcal R\) is the ring \({\mathcal R}:=\mathbb{C}[x_ 2,x_ 3,\dots,x_ p,\dots]\) with \(x_ p:=p^ t+p^{-t}\) and \(p\) runs over all rational primes. They prove that \(F_ \chi\) gives an isomorphism of \({\mathcal H}(G,\Gamma)\)-modules. When we define structures of pre-Hilbert spaces of \({\mathcal S}(\Gamma\backslash{\mathbf X})_ \chi\) and \(\mathcal R\) via suitable inner products, the map \(F_ \chi\) can be extended to an isometry between the completions of these spaces. This result may be considered as the Plancherel formula for the Fourier-Eisenstein transform \(F_ \chi\). An explicit form of the inverse transformation of \(F_ \chi\) follows quite easily from the Plancherel formula. Furthermore, using the main result, we can determine all \({\mathcal H}(G,\Gamma)\)-eigenfunctions in \({\mathcal C}^ \infty(\Gamma\backslash{\mathbf X})\).
Reviewer: M.Muro (Yanagido)
MSC:
| 11E45 | Analytic theory (Epstein zeta functions; relations with automorphic forms and functions) |
| 42B10 | Fourier and Fourier-Stieltjes transforms and other transforms of Fourier type |
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