Spectrum of semisimple locally symmetric spaces and admissibility of spherical representations

Salah Mehdi (1) , M Olbrich (2)

1 IECL - Institut Élie Cartan de Lorraine
2 FSTC - Faculty of Science, Technology and Communication [Luxembourg]

We consider compact locally symmetric spaces Γ\G/H where G/H is a non-compact semisimple symmetric space and Γ is a discrete subgroup of G. We discuss some features of the joint spectrum of the (commutative) algebra D(G/H) of invariant differential operators acting, as unbounded operators, on the Hilbert space L^2(Γ\G/H) of square integrable complex functions on Γ\G/H. In the case of the Lorentzian symmetric space SO_0 (2, 2n)/SO_0 (1, 2n), the representation theoretic spectrum is described explicitly. The strategy is to consider connected reductive Lie groups L acting transitively and co-compactly on G/H, a cocompact lattice Γ ⊂ L, and study the spectrum of the algebra D(L/L ∩ H) on L^2(Γ\L/L ∩ H). Though the group G does not act on L^2(Γ\G/H), we explain how (not necessarily unitary) G-representations enter into the spectral decomposition of D(G/H) on L^2(Γ\G/H) and why one should expect a continuous contribution to the spectrum in some cases. As a byproduct, we obtain a result on the L-admissibility of G-representations. These notes contain the statements of the main results, the proofs and the details will appear elsewhere.

Domaines: Mathématiques [math]

Salah Mehdi, M Olbrich. Spectrum of semisimple locally symmetric spaces and admissibility of spherical representations. Lie Groups, Number Theory and Vertex Algebras : Conference on Representation Theory XVI, Dražen Adamović, Andrej Dujella, Antun Milas, Pavle Pandžić, Jun 2019, Dubrovnik, Croatia. pp.55-64. ⟨hal-03336291⟩