The value ring of geometric motivic integration and the Iwahori Hecke algebra of SL_2

Ehud Hrushovski; David Kazhdan

Home/Archive/arXiv

arXiv · math/0609115

The value ring of geometric motivic integration and the Iwahori Hecke algebra of SL_2

Ehud Hrushovski·David Kazhdan

View original source ↗Read source document ↗Export metadata ↓

Abstract

In \cite{HK}, 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 $SL_2$ over an algebraically closed valued field using motivic integration (to replace the Haar measure) and study its structure.

Explore related subjects

math.AG math.LO math.RT

Keep this discovery

BibTeX RIS

Ehud Hrushovski, David Kazhdan. 2006-09-04. The value ring of geometric motivic integration and the Iwahori Hecke algebra of SL_2. https://arxiv.org/abs/math/0609115

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Perverse Euler Characteristics of Hermitian Locally Symmetric Spaces

We prove that finite-volume locally Hermitian symmetric spaces of noncompact type have nonnegative perverse Euler characteristics. To show this, we obtain a nefness result for the logarithmic cotangent bundle of a smooth toroidal compactification. Combining this with a positivity criterion for Euler characteristics of perverse sheaves, we deduce the nonnegativity result. We further prove that the inequality is strict for perverse sheaves with full support. As applications, we get nonnegativity results for perverse Euler characteristics on various moduli spaces.

math.AG↗

arXiv2026

Coupled Pklt Tuples and Varieties of Pklt Type

We introduce asymptotic multiplier ideal sheaves and log canonical thresholds associated with tuples of pseudoeffective divisors on a projective klt pair. We prove that the threshold of a coupled potentially klt tuple is computed by a quasi-monomial valuation. For varieties of potentially klt type, we prove that every big divisor admits a birational Zariski decomposition with semiample positive part. We also prove finite generation of multisection rings of big divisors and give a criterion for a variety of potentially klt type to be a Mori dream space.

math.AG↗

arXiv2026

Graded Betti numbers of general curves of large degree

Let $C$ be a smooth projective complex curve of genus $g$ and gonality $k$, and $L$ be a very ample line bundle on $C$. When $L$ has sufficiently large degree, the vanishing and nonvanishing of the Koszul cohomology groups $K_{p,q}(C,L)$ have been determined previously, but the exact values of the graded Betti numbers $\kappa_{p,q}(C, L)$ remain largely unknown. In this paper, we give explicit closed formulas for all graded Betti numbers $\kappa_{p,q}(C, L)$ when the Brill--Noether locus $W_k^1(C)$ has the expected dimension and $H^1(C, L \otimes \omega_C^{-1})=0$. Consequently, we determine the complete Betti table for a general curve when $\deg L \geq 4g-3$ or when $\deg L \geq 3g-3$ and $L$ is general. We also explicitly compute the Boij--S\"{o}derberg coefficient of the section ring $R(C, L)$ governing asymptotic purity, and show eventual monotonicity of the remaining coefficients: they decrease for hyperelliptic curves and increase under a natural generic reducedness assumption on the relevant Brill--Noether loci.

math.AG↗