2006/09/04 by Ehud Hrushovski, David Kazhdan, Hrushovski, Ehud +1 · 1 citation
Mathematics · #03C60 #11S80 #20C08 #Advanced Topics in Algebra #Advanced Topology and Set Theory #Algebraic Geometry (math.AG) #FOS: Mathematics #Functional Equations Stability Results #Logic (math.LO) #Representation Theory (math.RT) #math.AG #math.LO #math.RT #msc:03C60 #msc:11S80 #msc:20C08
paper · pdf · doi:10.48550/arxiv.math/0609115
33 pages, with an appendix by Nir Avni. v.2 minor spelling mistake. v.3 Some typos corrected, notably in Lemma 3.12 and Theorem 3.22; Cor. 3.5 is stated more strongly with same proof; a couple of remarks have been added
openalex publication_date 2006/09/04 · arxiv created 2013/09/03 · arxiv updated 2013/09/04 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
In \citeHK, an integration theory for valued fields was developed with a Grothendieck group approach. It was shown that the semiring of semi-algebraic sets with measure preserving morphisms is isomorphic to a certain semiring formed out of twisted varieties over the residue field and rational polytopes over the value group. With a view to representation-theoretic applications, we require a simpler description of the possible values of the integration, and in particular natural homomorphisms into fields. In the present paper we obtain such results after tensoring with Q. Since this operation trivializes the full semiring, we restrict to bounded sets. We show that the resulting Q-algebra is generated by its one-dimensional part. In the "geometric" case, i.e. working over an elementary submodel as a base, we determine the structure precisely. As a corollary we obtain useful canonical homomorphisms in the general case. In the appendix we define the Iwahori Hecke algebra of SL2 over an algebraically closed valued field using motivic integration (to replace the Haar measure) and study its structure.