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Unipotent orbital integrals of spherical functions on $$p$$-adic $$4\times 4$$ symplectic groups. (English) Zbl 0765.22006
Let $${\mathbf G}$$ denote a connected reductive algebraic group defined over a $$p$$-adic field $$F$$ of characteristic zero. Among the $${\mathbf G}(F)$$-invariant distributions on $${\mathbf G}(F)$$, unipotent orbital integrals play an important role. However, not much is known about them. In this paper, we make a detailed study of the complex vector space consisting of restrictions of unipotent orbital integrals to the convolution algebra of spherical functions on $${\mathbf G}(F)$$, where $${\mathbf G}={\mathbf {GSp}}(4)$$, $${\mathbf {Sp}}(4)$$, or $${\mathbf{PSp}}(4)$$. In particular, we obtain a concrete description of this space as a space of linear forms on the image space of the Satake isomorphism. This is the problem of Fourier transform of unipotent orbital integrals for spherical functions. We then use this result to obtain matchings between unipotent orbital integrals of spherical functions on $${\mathbf G}(F)$$ and on its “unramified” endoscopic groups. These results can be applied to the “fundamental lemma” for $${\mathbf G}(F)$$.
Reviewer: M.Assem (Ottawa)

##### MSC:
 22E35 Analysis on $$p$$-adic Lie groups 11F72 Spectral theory; trace formulas (e.g., that of Selberg) 43A90 Harmonic analysis and spherical functions 22E50 Representations of Lie and linear algebraic groups over local fields
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