Dirac Operators in Representation Theory pp 145-151 | Cite as

# Dimensions of Spaces of Automorphic Forms

Chapter

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## 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
Assume that rank

*G*be a connected semisimple noncompact Lie group with finite center. Let*K*⊂*G*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*L*^{2}(Γ\*G*) is decomposed discretely with finite multiplicities:$$
L^2 (\Gamma \backslash G) \cong \mathop \oplus \limits_{\pi \in \hat G} m(\Gamma ,\pi )\mathcal{H}_\pi .
$$

*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