Eigenvalues and eigenspaces for the twisted Dirac operator over 𝑆𝑈(𝑁,1) and 𝑆𝑝𝑖𝑛(2𝑁,1)

Galina, Esther; Vargas, Jorge

doi:10.1090/S0002-9947-1994-1189792-6

Eigenvalues and eigenspaces for the twisted Dirac operator over $\textrm {SU}(N,1)$ and $\textrm {Spin}(2N,1)$
HTML articles powered by AMS MathViewer

by Esther Galina and Jorge Vargas PDF

Trans. Amer. Math. Soc. 345 (1994), 97-113 Request permission

Abstract:

Let X be a symmetric space of noncompact type whose isometry group is either $SU(n,1)$ or $Spin(2n,1)$. Then the Dirac operator D is defined on ${L^2}$-sections of certain homogeneous vector bundles over X. Using representation theory we obtain explicitly the eigenvalues of D and describe the eigenspaces in terms of the discrete series.

References

M. F. Atiyah, Elliptic operators, discrete groups and von Neumann algebras, Colloque “Analyse et Topologie” en l’Honneur de Henri Cartan (Orsay, 1974) Astérisque, No. 32-33, Soc. Math. France, Paris, 1976, pp. 43–72. MR 0420729
Michael Atiyah and Wilfried Schmid, A geometric construction of the discrete series for semisimple Lie groups, Invent. Math. 42 (1977), 1–62. MR 463358, DOI 10.1007/BF01389783
Alain Connes and Henri Moscovici, The $L^{2}$-index theorem for homogeneous spaces of Lie groups, Ann. of Math. (2) 115 (1982), no. 2, 291–330. MR 647808, DOI 10.2307/1971393
Wilfried Schmid, Die Randwerte holomorpher Funktionen auf hermitesch symmetrischen Räumen, Invent. Math. 9 (1969/70), 61–80 (German). MR 259164, DOI 10.1007/BF01389889
Joseph A. Wolf, Essential self-adjointness for the Dirac operator and its square, Indiana Univ. Math. J. 22 (1972/73), 611–640. MR 311248, DOI 10.1512/iumj.1973.22.22051
Nolan R. Wallach, Harmonic analysis on homogeneous spaces, Pure and Applied Mathematics, No. 19, Marcel Dekker, Inc., New York, 1973. MR 0498996