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

# Dimensions of Spaces of Automorphic Forms

Chapter

## 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
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## Copyright information

© Birkhäuser Boston 2006