Abstract
We introduce a seminorm on a space of C ∞ functions on a reductive symmetric space G/H. We show that the C ∞, K-finite eigenfunctions of the algebra of all G-invariant differential operators on G/H for which this seminorm is finite as well as for their derivatives from the left, are tempered. Reciprocally, we explicite this seminorm for the K-finite tempered eigenfunctions with a regular character. Our work generalises and specifies the results of H. Midorikawa [7] in the group case.
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