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Constant terms in powers of a Laurent polynomial. (English) Zbl 0916.22007
The following is a conjecture of O. Mathieu: Let \(K\) be a connected real compact Lie group. Let \(f\) and \(g\) be \(K\)-finite functions on \(K\). Assume for all \(n \geq 1\) that the constant term of \(f^{n}\) vanishes, i.e. \[ \int_{K} f^{n}(k) \;dk = 0 . \] Then all but finitely many of the constant terms \(\{ \int_{K} f^{n}(k) g(k) \;dk \}\) also vanish. The authors prove this when \(K\) is a real torus with complexification \(T\). The all-important case when \(T = {\mathbf G}_{m}\) involves some clever arguments with Newton polygons.

22E30 Analysis on real and complex Lie groups
32A17 Special families of functions of several complex variables
30E99 Miscellaneous topics of analysis in the complex plane
Full Text: DOI
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