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Eisenstein series with non-unitary twists. (English) Zbl 1405.11069

A Selberg type trace formula has been established for compact locally symmetric spaces \(\Gamma\backslash G/K\) with a non-unitary finite-dimensional twist \(\chi:\Gamma\to\text{GL}(V)\) where \(G\) is of Lie type [W. Müller, Int. Math. Res. Not. 2011, No. 9, 2068–2109 (2011; Zbl 1214.11067)] or totally disconnected {[the authors, Math. Z. 284, No. 3–4, 1199–1210 (2016, Zbl 1364.11101)]. For non-cocompact lattices, the first step to such a formula is the study of Eisenstein series \(E_c(s,z)\) (\(s\in\mathbb C\), \(z\in G/K\), \(c\) any cusp). The authors investigate here the particular framework of \(G=\mathrm{PSL}_2(\mathbb R)\), \(\mathbb H=G/K\) the hyperbolic plane and a non-cocompact lattice \(\Gamma\): for convergence purpose, they suppose that \(\chi(\gamma)\) is unitary for every parabolic element \(\gamma\in\Gamma\).
The Eisenstein series are defined similarly to the untwisted case, essentially with a twisting factor \(\chi(\gamma\)) added in each term. Their convergence on the half-plane \(\{\mathrm{Re} (s)>1+\alpha\}\) is deduced through refined bounds on the operator norms \(\|\chi(\gamma)\|\). There exists one \(\alpha>0\) such that the asymptotic estimates hold: \(\|\chi(\gamma)\|=\mathcal O(\|\gamma\|_F^{2\alpha}) =\mathcal O_{z,w\in\mathbb H}(\exp(\alpha d_{\mathbb H}(z,\gamma w)))\) where \(\|\gamma\|_F=\sqrt{a^2+b^2+c^2+d^2}\) is the Frobenius norm of \(\begin{pmatrix}{a}&b\\c&d\end{pmatrix}\in\text{PSL}_2(\mathbb R)\).
Moreover, as corollaries, the authors deduce the spectral decomposition of the cusp forms space and prove the meromorphic continuation of the Eisenstein series with their functional equation as it is done for the untwisted Laplacian (see [H. Iwaniec Spectral methods of automorphic forms. 2nd ed. Providence, RI: AMS (2002; Zbl 1006.11024)] for example).}

MSC:

11F72 Spectral theory; trace formulas (e.g., that of Selberg)
11F12 Automorphic forms, one variable
11F03 Modular and automorphic functions
11F70 Representation-theoretic methods; automorphic representations over local and global fields
22E50 Representations of Lie and linear algebraic groups over local fields
22E46 Semisimple Lie groups and their representations
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References:

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[2] A. Deitmar and F. Monheim, A trace formula for non-unitary representations of a uniform lattice, Math. Z. 284 (2016), no. 3-4, 1199-1210. · Zbl 1364.11101
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[4] H. Iwaniec, Spectral Methods of Automorphic Forms, second edition, Graduate Studies in Mathematics, 53, American Mathematical Society, Providence, RI, 2002. · Zbl 1006.11024
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[6] W. M¨uller, A Selberg trace formula for non-unitary twists, Int. Math. Res. Not. IMRN 2011, no. 9, 2068-2109. · Zbl 1214.11067
[7] M. A. Shubin, Pseudodifferential Operators and Spectral Theory, translated from the 1978 Russian original by Stig I. Andersson, second edition, Springer-Verlag, Berlin, 2001. · Zbl 0980.35180
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