Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 17 (2021), 051, 48 pages      arXiv:1912.00050      https://doi.org/10.3842/SIGMA.2021.051

Spectra of Compact Quotients of the Oscillator Group

Mathias Fischer and Ines Kath
Institut für Mathematik und Informatik der Universität Greifswald, Walther-Rathenau-Str. 47, D-17489 Greifswald, Germany

Received September 28, 2020, in final form April 24, 2021; Published online May 13, 2021

Abstract
This paper is a contribution to harmonic analysis of compact solvmanifolds. We consider the four-dimensional oscillator group ${\rm Osc}_1$, which is a semi-direct product of the three-dimensional Heisenberg group and the real line. We classify the lattices of ${\rm Osc}_1$ up to inner automorphisms of ${\rm Osc}_1$. For every lattice $L$ in ${\rm Osc}_1$, we compute the decomposition of the right regular representation of ${\rm Osc}_1$ on $L^2(L\backslash{\rm Osc}_1)$ into irreducible unitary representations. This decomposition allows the explicit computation of the spectrum of the wave operator on the compact locally-symmetric Lorentzian manifold $L\backslash {\rm Osc}_1$.

Key words: Lorentzian manifold; wave operator; lattice; solvable Lie group.

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References

1. Auslander L., Kostant B., Polarization and unitary representations of solvable Lie groups, Invent. Math. 14 (1971), 255-354.
2. Berndt B.C., Evans R.J., The determination of Gauss sums, Bull. Amer. Math. Soc. (N.S.) 5 (1981), 107-129.
3. Brezin J., Harmonic analysis on compact solvmanifolds, Lecture Notes in Math., Vol. 602, Springer-Verlag, Berlin - Heidelberg, 1977.
4. Corwin L., Greenleaf F.P., Character formulas and spectra of compact nilmanifolds, J. Funct. Anal. 21 (1976), 123-154.
5. Fischer M., Lattices of oscillator groups, J. Lie Theory 27 (2017), 85-110, arXiv:1303.1970.
6. Fujiwara H., Ludwig J., Harmonic analysis on exponential solvable Lie groups, Springer Monographs in Mathematics, Springer, Tokyo, 2015.
7. Hecke E., Lectures on the theory of algebraic numbers, Graduate Texts in Mathematics, Vol. 77, Springer-Verlag, New York - Berlin, 1981.
8. Howe R., On Frobenius reciprocity for unipotent algebraic groups over $Q$, Amer. J. Math. 93 (1971), 163-172.
9. Howe R., Topics in harmonic analysis on solvable algebraic groups, Pacific J. Math. 73 (1977), 383-435.
10. Huang H., Lattices and harmonic analysis on some 2-step solvable Lie groups, J. Lie Theory 13 (2003), 77-89.
11. Kirillov A.A., Merits and demerits of the orbit method, Bull. Amer. Math. Soc. (N.S.) 36 (1999), 433-488.
12. Kirillov A.A., Lectures on the orbit method, Graduate Studies in Mathematics, Vol. 64, Amer. Math. Soc., Providence, RI, 2004.
13. Medina A., Revoy P., Les groupes oscillateurs et leurs réseaux, Manuscripta Math. 52 (1985), 81-95.
14. Newman M., Integral matrices, Pure and Applied Mathematics, Vol. 45, Academic Press, New York - London, 1972.
15. Pukánszky L., Characters of algebraic solvable groups, J. Funct. Anal. 3 (1969), 435-494.
16. Raghunathan M.S., Discrete subgroups of Lie groups, Ergebnisse der Mathematik und ihrer Grenzgebiete, Vol. 68, Springer-Verlag,Berlin - Heidelberg, 1972.
17. Richardson L.F., Decomposition of the $L^{2}$-space of a general compact nilmanifold, Amer. J. Math. 93 (1971), 173-190.
18. Siegel C.L., Über das quadratische Reziprozitätsgesetz in algebraischen Zahlkörpern, Nachr. Akad. Wiss. Göttingen Math.-Phys. Kl. II 1960 (1960), 1-16.
19. Streater R.F., The representations of the oscillator group, Comm. Math. Phys. 4 (1967), 217-236.
20. Wolf J.A., Harmonic analysis on commutative spaces, Mathematical Surveys and Monographs, Vol. 142, Amer. Math. Soc., Providence, RI, 2007.
21. Zeghib A., The identity component of the isometry group of a compact Lorentz manifold, Duke Math. J. 92 (1998), 321-333.