A spectral mean value theorem for \(\text{GL}(3)\). (English) Zbl 1208.11070
Let \(\{\phi_j\}\) be an orthonormal basis of Maaß forms for \(\text{SL}(3,{\mathbb Z})\), and \(W_j^{(1,1)}(y)\) be the first Fourier coefficient of \(\phi_j\). The paper gives a bound on the averages of \(|W_j^{(1,1)}(y)|^2\) over certain subsets of the cuspidal spectrum of \(\text{SL}(3)\), when \(y\) is a fixed diagonal matrix that is sufficiently dominant. The proof is an elegant application of Kuznetsov formula in \(\text{SL}(3)\) case, which is much more complex than the formula in \(\text{SL}(2)\) case.
Reviewer: Zhengyu Mao (Newark)
MSC:
| 11F72 | Spectral theory; trace formulas (e.g., that of Selberg) |
| 11F37 | Forms of half-integer weight; nonholomorphic modular forms |
Software:
GL(n)packReferences:
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