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On elliptic tempered characters. (English) Zbl 0822.22011
Let $$G$$ be a connected reductive algebraic group over a local field of characteristic zero (either real or $$p$$-adic). This paper considers the characters of elliptic tempered representations of $$G$$.
The author defines a family of virtual tempered characters, (each the trace of an induced representation twisted by an intertwining operator). Part of the local trace formula can be expanded in terms of these virtual characters, and this leads to an expression of the (noninvariant) trace formula as a local trace formula whose terms are all invariant.
If $$f$$ is a cuspidal function, $$\gamma$$ a $$G$$-regular element in the elliptic set of a Levi subgroup $$M$$, then the author shows how to expand the orbital integral $$I_ M(\gamma, f)$$ in terms of the elliptic tempered virtual characters. There is a parallel expansion for some related distributions in terms of truncated virtual characters.
These results are then used to establish orthogonality relations for the virtual tempered characters and also a local proof of Kazhdan’s Theorem that invariant orbital integrals are supported on characters. There are also applications to certain contour integrals and residues.
Reviewer: J.Repka (Toronto)

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