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A spectral analysis of automorphic distributions and Poisson summation formulas. (English) Zbl 1066.11040

Define \(\mathcal{F}\) to be the Fourier transform on \(\mathbb{R}^{d}\) and \({\mathcal E}\) to be the Euler operator, \[ \mathcal{E}= \frac{1}{2\pi i} \left( \sum x_{j}\frac{\partial}{\partial x_{j}}+\frac{d}{2 }\right). \] If \(h\) is a Schwartz function on \(\mathbb{R}^{d}\), set \( t^{2\pi i\mathcal{E}}h(x)=t^{d/2}h(tx)\), for \(t>0.\) If \(\mathcal{S}\) is a tempered distribution, set \(\left\langle t^{2\pi i\mathcal{E}}\mathcal{S} ,h\right\rangle =\left\langle \mathcal{S},t^{-2\pi i\mathcal{E} }h\right\rangle .\) For a sequence of complex numbers \(\mathbf{a} =(a_{n})_{n\geq 1}\) such that the series \(D_{\mathbf{a}}(s)=\sum_{n\geq 1}\frac{a_{n}}{n^{s}}\) is nice, set \(D_{\mathbf{b}}(a)(s)=\frac{D_{ \mathbf{a}}(s)}{\zeta (s)},\) where \(\zeta (s)\) is Riemann’s zeta function. Define the Eisenstein distribution \(\mathfrak{E}_{\nu }^{d}\) for Schwartz functions \(h\) by \[ \left\langle \mathfrak{E}_{\nu }^{d},h\right\rangle =\sum_{m\in \mathbb{Z}^{d}-\{0\}}\int_{0}^{\infty }t^{-\nu +\frac{d}{2}}h(tm) \frac{dt}{t}. \] This converges for \(\text{Re}\nu <-d/2\) where it is an SL\((d,\mathbb{Z})\) -invariant, even, tempered distribution. For \(h(x)=e^{-\parallel x\parallel ^{2}}\) this is an Eisenstein series for SL\((d,\mathbb{Z}).\)
Theorem 2.5 of the paper under review says \(\mathfrak{E}_{\nu }^{d}\) extends as a tempered distribution to a meromorphic function of \(\nu \) whose only poles are at \(\nu =\pm d/2.\) One has \(\mathcal{F}\mathfrak{E}_{\nu }^{d}=\mathfrak{E}_{-\nu }^{d},\nu \neq \pm d/2.\) Proposition 2.6 says if \(r(m)\) is the greatest common divisor of the entries of \(m\in \mathbb{Z}^{d},\) and \(h\) is a Schwartz function on \(\mathbb{R}^{d}, \) then assuming the sequence \(\mathbf{a}=(a_{n})_{n\geq 1}\) has at most polynomial increase and the Dirichlet series \(D_{\mathbf{a}}(s)\) has at most polynomial increase on vertical strips and that the function \(D_{ \mathbf{b}}(a)(s)=\frac{D_{a}(s)}{\zeta (s)}\) extends as a meromorphic function in the plane, with no pole with real part \(\geq d/2,\) except possibly for a pole at \(s=d\), one has \[ 2\pi \sum_{m\in \mathbb{Z}^{d}-\{0\}}a_{r(m)}h(m)=\int_{- \infty }^{\infty }D_{\mathbf{b}}(\frac{d}{2}-i\lambda )\left\langle \mathfrak{E}_{i\lambda }^{d},h\right\rangle d\lambda +2\pi \text{Re} s_{s=d}\left( D_{\mathbf{b}}(s)\left\langle \mathfrak{E}_{\frac{d}{2} -s}^{d},h\right\rangle \right). \] The right hand side makes sense when \(h\) is slightly more general than a Schwartz function and can then serve to define the left hand side. Section 3 applies the theory to sequences \(a=(a_{n})_{n\geq 1}\) coming from Fourier coefficients of holomorphic modular forms \(f(z)\), \(z\) in the Poincaré upper half plane; i.e., \( f(z)=f(z+2)\) and \(f(-1/z)=\kappa \left( \frac{z}{i}\right) ^{wd}f(z),\kappa =\pm 1.\) Suppose \[ f(z)=f_{0}+\sum_{n\geq 1}f_{n}e^{\pi inz}. \] Set \(c_{0}=-f_{0}\) and for \(m\neq 0,\) set \(c_{m}=\sum\limits_{\underset{ n\geq 1} { n^w| m_j,all j}}f_{n}.\) Let \(\Phi \) be a radial Schwartz function (or the slightly weakened definition mentioned in Prop. 2.6 above). Define \(\Psi =\mathcal{K}_{d,w}\mathcal{F}\Phi ,\) where (with \(\mathcal{F}=\) Fourier transform and \(\mathcal{E}=\) Euler’s operator) and \(\mathcal{K}_{d,w}\) for Schwartz functions \(h\), given by \[ \mathcal{K}_{d,w}h(x)=2\pi \int_{0}^{\infty }t^{-wd}h(t^{-2w}x)J_{wd-1}(2wt)\,dt. \] Theorem 3.1 says then that \[ \sum_{m\in \mathbb{Z}^{d}}c_{m}\Phi (tm)=\kappa t^{-d}\sum_{m\in \mathbb{Z}^{d}}c_{m}\Psi \left( \frac{1}{t}m\right) . \] Section 4 extends the results to Maass wave forms. Section 5 finds the analog of the Gaussian for the transformation \(\mathcal{K}_{d,w}\mathcal{F},\) also analogs of Hermite functions. Section 6 gives general remarks on automorphic forms for \(GL(d,\mathbb{Z})\). In particular, it is noted that the present generalizations of Poisson summation are not related to non-Euclidean Poisson summation considered by the reviewer in her book “Harmonic analysis on symmetric spaces and applications. I, II”, Springer-Verlag (1985; Zbl 0574.10029), (1988; Zbl 0668.10033).

MSC:

11M41 Other Dirichlet series and zeta functions
11M36 Selberg zeta functions and regularized determinants; applications to spectral theory, Dirichlet series, Eisenstein series, etc. (explicit formulas)
46F99 Distributions, generalized functions, distribution spaces
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