Representation Theory of Symmetric Spaces over Finite or Local Fields

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We studied representations and harmonic analysis of symmetric spaces associated to reductive groups over finite or p-adic fields as a generalization of the representation theory of these groups. In the case of symmetric spaces over p-adic fields, we succeeded in constructing new examples of relatively cuspidal representations. We established criteria for tempered representations in the form analogous to the case of relatively cuspidal representations or square-integral representations. This is a natural extension of the criteria for group case. In the finite fields case, we studied a construction of relatively cuspidal representations on symmetric spaces by cohomological induction.