Spectra of Compact Quotients of the Oscillator Group

Mathias Fischer   ; Ines Kath

Symmetry, integrability and geometry: methods and applications, Tome 17 (2021)

Résumé

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

Détail

Keywords: Lorentzian manifold, wave operator, lattice
Mots-clés : solvable Lie group.

Bibliographie

[1] Auslander L., Kostant B., “Polarization and unitary representations of solvable Lie groups”, Invent. Math., 14 (1971), 255–354 | DOI | MR | Zbl

[2] Berndt B. C., Evans R. J., “The determination of Gauss sums”, Bull. Amer. Math. Soc. (N.S.), 5 (1981), 107–129 | DOI | MR | Zbl

[3] Brezin J., Harmonic analysis on compact solvmanifolds, Lecture Notes in Math., 602, Springer-Verlag, Berlin – Heidelberg, 1977 | DOI | MR | Zbl

[4] Corwin L., Greenleaf F. P., “Character formulas and spectra of compact nilmanifolds”, J. Funct. Anal., 21 (1976), 123–154 | DOI | MR | Zbl

[5] Fischer M., “Lattices of oscillator groups”, J. Lie Theory, 27 (2017), 85–110, arXiv: 1303.1970 | MR | Zbl

[6] Fujiwara H., Ludwig J., Harmonic analysis on exponential solvable Lie groups, Springer Monographs in Mathematics, Springer, Tokyo, 2015 | DOI | MR | Zbl

[7] Hecke E., Lectures on the theory of algebraic numbers, Graduate Texts in Mathematics, 77, Springer-Verlag, New York – Berlin, 1981 | DOI | MR | Zbl

[8] Howe R., “On Frobenius reciprocity for unipotent algebraic groups over $Q$”, Amer. J. Math., 93 (1971), 163–172 | DOI | MR | Zbl

[9] Howe R., “Topics in harmonic analysis on solvable algebraic groups”, Pacific J. Math., 73 (1977), 383–435 | DOI | MR | Zbl

[10] Huang H., “Lattices and harmonic analysis on some 2-step solvable Lie groups”, J. Lie Theory, 13 (2003), 77–89 | MR | Zbl

[11] Kirillov A. A., “Merits and demerits of the orbit method”, Bull. Amer. Math. Soc. (N.S.), 36 (1999), 433–488 | DOI | MR | Zbl

[12] Kirillov A. A., Lectures on the orbit method, Graduate Studies in Mathematics, 64, Amer. Math. Soc., Providence, RI, 2004 | DOI | MR | Zbl

[13] Medina A., Revoy P., “Les groupes oscillateurs et leurs réseaux”, Manuscripta Math., 52 (1985), 81–95 | DOI | MR | Zbl

[14] Newman M., Integral matrices, Pure and Applied Mathematics, 45, Academic Press, New York – London, 1972 | MR | Zbl

[15] Pukánszky L., “Characters of algebraic solvable groups”, J. Funct. Anal., 3 (1969), 435–494 | DOI | MR | Zbl

[16] Raghunathan M. S., Discrete subgroups of Lie groups, Ergebnisse der Mathematik und ihrer Grenzgebiete, 68, Springer-Verlag, Berlin – Heidelberg, 1972 | MR | Zbl

[17] Richardson L. F., “Decomposition of the $L^{2}$-space of a general compact nilmanifold”, Amer. J. Math., 93 (1971), 173–190 | DOI | MR

[18] Siegel C. L., “Über das quadratische {R}eziprozitätsgesetz in algebraischen Zahlkörpern”, Nachr. Akad. Wiss. Göttingen Math.-Phys. Kl. II, 1960 (1960), 1–16 | MR | Zbl

[19] Streater R. F., “The representations of the oscillator group”, Comm. Math. Phys., 4 (1967), 217–236 | DOI | MR | Zbl

[20] Wolf J. A., Harmonic analysis on commutative spaces, Mathematical Surveys and Monographs, 142, Amer. Math. Soc., Providence, RI, 2007 | DOI | MR | Zbl

[21] Zeghib A., “The identity component of the isometry group of a compact Lorentz manifold”, Duke Math. J., 92 (1998), 321–333 | DOI | MR | Zbl