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Dimensions of Spaces of Automorphic Forms

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Part of the Mathematics: Theory & Applications book series (MTA)

Abstract

In this chapter we prove a formula for dimensions of spaces of automorphic forms which sharpens the result of Langlands and Hotta-Parthasarathy [L], [HoP2]. Let G be a connected semisimple noncompact Lie group with finite center. Let KG be a maximal compact subgroup of G, and let Γ ⊂ G be a discrete subgroup. Assume that Γ\G is compact and that Γ acts freely on G/K. Then X = Γ\G/K is a compact smooth manifold. Furthermore, the action of G by right translation on the Hilbert space L2(Γ\G) is decomposed discretely with finite multiplicities:
$$ L^2 (\Gamma \backslash G) \cong \mathop \oplus \limits_{\pi \in \hat G} m(\Gamma ,\pi )\mathcal{H}_\pi . $$
Assume that rank G is equal to rank K. We calculate the multiplicity m(Γ, π) for a discrete series representation π.

Keywords

Dirac Operator Automorphic Form Discrete Subgroup Discrete Series Maximal Compact Subgroup 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Birkhäuser Boston 2006

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