Worked solution: Ngô's proof via the Hitchin fibration and geometric support theorem (2008)

Step 3 of 9: Orbital integrals as point counts on affine Springer fibers

In plain words

An orbital integral over $\mathbb{F}_q((t))$ turns out to have a concrete geometric meaning: it counts (with the right weighting) lattices — grid-like sublattices of a space — that are left in place when $\gamma$ acts on them. The space of all such fixed lattices is called the affine Springer fiber of $\gamma$ .

This reinterprets a hard analytic integral as a counting problem on a concrete geometric object, exactly the kind of question algebraic geometry is built to answer.

\mathbf{O}_\gamma(\mathbf{1}_K) \ \propto \ \#\,\mathrm{Spr}_\gamma(\mathbb{F}_q), \qquad \mathrm{Spr}_\gamma = \{\text{lattices fixed by } \gamma\}

Detailed analysis

Over $F = \mathbb{F}_q((t))$ , an orbital integral for a suitable compact-open test function is related, after the standard normalization and truncation, to an $\mathbb{F}_q$ -point count on an affine Springer fiber $\mathrm{Spr}_\gamma$ . The full affine Springer fiber is generally an ind-scheme; finite-dimensional truncations or relevant quotients are the geometric objects whose cohomology enters the Grothendieck–Lefschetz trace formula. This converts the local integral into a cohomological counting problem.

Terms in this step

affine Springer fiber: The finite-dimensional algebraic variety parametrizing lattices in a vector space over $\mathbb{F}_q((t))$ that are stabilized by a given element $\gamma$ ; introduced by Kazhdan and Lusztig to study orbital integrals geometrically.

Knowledge used in this step

Algebraic geometry