Dimensions of Spaces of Automorphic Forms

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


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.


Dirac Operator Automorphic Form Discrete Subgroup Discrete Series Maximal Compact Subgroup 
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© Birkhäuser Boston 2006

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