(June 23, 2008)
Spectral Theory for SL2(Z)nSL2(R)/SO2(R) Paul Garrett
[email protected]
http://www.math.umn.edu/˜garrett/
• Pseudo-Eisenstein series • Fourier-Laplace-Mellin transforms • Recollection of facts about Eisenstein series • First decomposition of pseudo-Eisenstein series • Interlude: the constant function • Toward an isometry
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We will restrict out attention to the simplest possible case, namely G = SL2 ( ), Γ = SL2 ( ), and right K = SO(2)-invariant functions on Γ\G. That is, we neglect the finite primes and holomorphic automorphic forms. Let N be the subgroup of G consisting of upper-triangular unipotent matrices, and P the parabolic subgroup consisting of all upper-triangular matrices. For simplicity we give K total measure 1, rather than 2π. This will account for some discrepancies between formulas below and their analogues in other sources. For a locally integrable function f on Γ\G the constant term cP f of f (along P ) is defined to be Z cP f (g) = f (ng) dn N ∩Γ\N
As usual, a (locally integrable) function f on Γ\G is a cuspform if for almost all g ∈ G cP f (g) = 0
1. Pseudo-Eisenstein series While cuspforms are mysterious, a completely not-mysterious type of automorphic form is constructed directly from functions ϕ ∈ Cc∞ (N \G) by forming the incomplete theta series or pseudo-Eisenstein series [1] X Ψϕ (g) = ϕ(γg) P ∩Γ\Γ
[1.0.1] Remark: Note that P ∩ Γ differs from N ∩ Γ just by {±12 }, which are both in the center of Γ and are in K. Since our interest for the moment is only in right K-invariant functions, everything here will be invariant under {±12 }. [1.0.2] Lemma: The series for an incomplete theta series is absolutely and uniformly convergent for g in compacts, and yields a function in Cc∞ (Γ\G).
Proof: Given ϕ ∈ Cc∞ (N \G/K, let C be a compact set in G so that N · C contains the support of ϕ. Fix a compact subset Co of G in which g ∈ G is constrained to lie. Then a summand ϕ(γg) is non-zero only if γg ∈ N · C, which is to say γ ∈ Γ ∩ N · C · g −1 which requires that γ ∈ Γ ∩ N · C · Co−1 [1] In 1966 Godement called these incomplete theta series, but more recently Moeglin-Waldspurger strengthened the
precedent of calling them pseudo-Eisenstein series
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Paul Garrett: Spectral Theory for SL2 ( )\SL2 ( )/SO2 ( ) (June 23, 2008) If this held, then (N ∩ Γ) · γ ⊂ Γ ∩ N · C · Co−1 and then (N ∩ Γ) · γ ∈ (N ∩ Γ)\Γ ∩ (N ∩ Γ)\(N · C · Co−1 ) The second term on the right-hand side is compact, since (N ∩ Γ)\N is compact. The first term on the right-hand side is discrete, since N ∩ Γ is a closed subgroup. Thus, the right hand side is compact and discrete, so is finite. Thus, the series is in fact locally finite, and defines a smooth function on Γ\G/K. To show that it has compact support in Γ\G, proceed similarly. That is, for a summand ϕ(γg) to be non-zero, it must be that g ∈Γ·C That implies Γ·g ⊂Γ·C and Γ · g ∈ Γ\(Γ · C) The right-hand side is compact, being the continuous image of a compact set under the continuous map G → Γ\G, proving the compact support. ///
[1.0.3] Remark: We will make incessant use of the lemma that for a countably-based locally compact Hausdorff topological group G, and for a Hausdorff space X on which G acts transitively, X is homeomorphic to the quotient of G by the isotropy group of a chosen point in X. And we will use standard integration theory on quotients such as Γ\G and N \G, etc. Let h, i be the complex hermitian form Z hf1 , f2 i =
f1 (g) f2 (g) dg Γ\G
with respect to a fixed right G-invariant measure on Γ\G.
[1.0.4] Proposition: A locally integrable automorphic form f is a cuspform if and only if hf,