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The poles of Igusa zeta integrals and the unextendability of semi-invariant distributions. (English) Zbl 1481.11112

The author’s summary: “We investigate the relationship between the poles of Igusa zeta integrals and the unextendability of semi-invariant distributions. Under some algebraic conditions, we obtain an upper bound for the order of the poles of Igusa zeta integral, and by using the order of the poles we give a criterion on the unextendability of semi-invariant distributions. A key ingredient of our method is the idea of generalized semi-invariant distributions”.
The paper contains also an appendix “Extension of Intertwining Operators and Residues” (by Shachar Carmeli and Jiuzu Hong). Its authors’ summary is as follows.
“In this appendix, we give a homological algebra interpretation of the main result of this paper by relating residues and 1-cocycles. We work with smooth representations of \(l\)-groups in this appendix. It would be interesting to see which of the results presented here hold in the archimedean case as well”.

MSC:

11S80 Other analytic theory (analogues of beta and gamma functions, \(p\)-adic integration, etc.)
11S40 Zeta functions and \(L\)-functions
20G05 Representation theory for linear algebraic groups
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