2021/04/10 by Mehdi, Salah, Olbrich, Martin
#FOS: Mathematics #Representation Theory (math.RT)
paper · doi:10.48550/arxiv.2104.04740
We consider compact locally symmetric spaces Γ\backslash 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 L2(Γ\backslash G/H) of square integrable complex functions on Γ\backslash G/H. In the case of the Lorentzian symmetric space SO0(2,2n)/SO0(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 L2(Γ\backslash L/L∩ H). Though the group G does not act on L2(Γ\backslash G/H), we explain how (not necessarily unitary) G-representations enter into the spectral decomposition of D(G/H) on L2(Γ\backslash 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.