Poincaré series for non-Riemannian locally symmetric spaces

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• 作者：Fanny Kassel^a ; ¹ ; ^{fanny.kassel@math.univ-lille1.fr" class="auth_mail" title="E-mail the corresponding author} ; Toshiyuki Kobayashi^b ; ² ; ^{toshi@ms.u-tokyo.ac.jp" class="auth_mail" title="E-mail the corresponding author}
• 关键词：primary ; 22E40 ; 22E46 ; 58J50 ; secondary ; 11F72 ; 53C35
• 刊名：Advances in Mathematics
• 出版年：2016
• 出版时间：10 January 2016
• 年：2016
• 卷：287
• 期：Complete
• 页码：123-236
• 全文大小：1675 K

文摘

We initiate the spectral analysis of pseudo-Riemannian locally symmetric spaces Γ\G/H$\mathrm{\Gamma }\backslash G/H$, beyond the classical cases where H is compact (automorphic forms) or Γ is trivial (analysis on symmetric spaces).

For any non-Riemannian reductive symmetric space X=G/H$X=G/H$ on which the discrete spectrum of the Laplacian is nonempty, and for any discrete group of isometries Γ whose action on X   is “sufficiently proper”, we construct L²$L^2$-eigenfunctions of the Laplacian on X_Γ:=Γ\X$X_{\mathrm{\Gamma }}:=\mathrm{\Gamma }\backslash X$ for an infinite set of eigenvalues. These eigenfunctions are obtained as generalized Poincaré series, i.e.   as projections to X_Γ$X_{\mathrm{\Gamma }}$ of sums, over the Γ-orbits, of eigenfunctions of the Laplacian on X.

We prove that the Poincaré series we construct still converge, and define nonzero L²$L^2$-functions, after any small deformation of Γ inside G, for a large class of groups Γ. Thus the infinite set of eigenvalues we construct is stable under small deformations. This contrasts with the classical setting where the nonzero discrete spectrum varies on the Teichmüller space of a compact Riemann surface.

We actually construct joint L²$L^2$-eigenfunctions for the whole commutative algebra of invariant differential operators on X_Γ$X_{\mathrm{\Gamma }}$.