Let G be a connected, real, semisimple Lie group with finite center, and K a maximal compact subgroup of G. In this paper, we derive K-equivariant asymptotics for heat traces with remainder estimates on compact Riemannian manifolds carrying a transitive and isometric G-action. In particular, we compute the leading coefficient in the Minakshishundaram-Pleijel expansion of the heat trace for Bochner-Laplace operators on homogeneous vector bundles over compact locally symmetric spaces of arbitrary rank.
In this paper we construct explicitly a square integrable residual automorphic representation of the special orthogonal group SO 2n , through Eisenstein series. We show that this representation comes from an elliptic Arthur parameter ψ and appears in the space L 2 (SO 2n (Q)\SO 2n (A Q )) with multiplicity one. Next, we consider parameters whose Hecke matrices, at the unramified places, have eigenvalues bigger (in absolute value), than those of the parameter constructed before. The main result is, that these parameters cannot be cuspidal. We establish bounds for the eigenvalues of Hecke operators, as consequences of Arthur's conjectures for SO 2n . Next, we calculate the character and the twisted characters for the representations that we constructed. Finally, we find the composition of the global and local Arthur's packets associated to our parameter ψ. All the results in this paper are true if we replace Q by any number field F .
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