# The Subregular Germ of Orbital Integrals

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*Thomas C. Hales*

Langlands theory predicts deep relationships between representations of different reductive groups over a local or global field. The trace formula attempts to reduce many such relationships to problems concerning conjugacy classes and integrals over conjugacy classes (orbital integrals) on \(p\)-adic groups. It is possible to reformulate these problems as ones in algebraic geometry by associating a variety \(Y\) to each reductive group. Using methods of Igusa, the geometrical properties of the variety give detailed information about the asymptotic behavior of integrals over conjugacy classes. This monograph constructs the variety \(Y\) and describes its geometry. As an application, the author uses the variety to give formulas for the leading terms (regular and subregular germs) in the asymptotic expansion of orbital integrals over \(p\)-adic fields. The final chapter shows how the properties of the variety may be used to confirm some predictions of Langlands theory on orbital integrals, Shalika germs, and endoscopy.

#### Table of Contents

# Table of Contents

## The Subregular Germ of Orbital Integrals

- Contents v6 free
- Introduction ix10 free
- I. Basic Constructions 114 free
- II. Coordinates and Coordinate Relations 1326
- 1. The Coordinates x(W,(β) 1326
- 2. The Coordinates ω(β) 1326
- 3. The Extension of ω(β) to Y" 1427
- 4. The Coordinate Ring 1629
- 5. A Computation of t[sup(-1)]n[sup(-1)]tn 1831
- 6. A Technical Lemma 2033
- 7. Application to G[sub(2)] 2134
- 8. The Functions n[sub(γ)] 2740
- 9. The Fundamental Divisors on Y[sub(Γ)] 2942

- III. Groups of Rank Two 3346
- IV. The Subregular Spurious Divisor 5164
- V. The Subregular Fundamental Divisor 7790
- VI. Rationality and Characters 96109
- VII. Applications to Endoscopic Groups 110123
- References 133146
- Notation and Conventions 135148