Explicit asymptotic expansions for tame supercuspidal characters

We combine the ideas of a Harish-Chandra–Howe local character expansion, which can be centred at an arbitrary semisimple element, and a Kim–Murnaghan asymptotic expansion, which so far has been considered only around the identity. We show that, for most smooth, irreducible representations (those containing a good, minimal K-type), Kim–Murnaghan-type asymptotic expansions are valid on explicitly defined neighbourhoods of nearly arbitrary semisimple elements. We then give an explicit, inductive recipe for computing the coefficients in an asymptotic expansion for a tame supercuspidal representation. The only additional information needed in the inductive step is a fourth root of unity, which we expect to be useful in proving stability and endoscopic-transfer identities.

Components of Affine Springer Fibers

Let G be a connected split reductive group over a field of characteristic zero or sufficiently large characteristic, $\gamma _0\in (\operatorname{Lie}\mathbf{G})((t))$ be any topologically nilpotent regular semisimple element, and $\gamma =t\gamma _0$. Using methods from p-adic orbital integrals, we show that the number of components of the Iwahori affine Springer fiber over $\gamma$ modulo $Z_{\mathbf{G}((t))}(\gamma )$ is equal to the order of the Weyl group.

The real Chevalley involution

The Chevalley involution of a connected, reductive algebraic group over an algebraically closed field takes every semisimple element to a conjugate of its inverse, and this involution is unique up to conjugacy. In the case of the reals we prove the existence of a real Chevalley involution, which is defined over $\mathbb{R}$, takes every semisimple element of $G(\mathbb{R})$ to a $G(\mathbb{R})$-conjugate of its inverse, and is unique up to conjugacy by $G(\mathbb{R})$. We derive some consequences, including an analysis of groups for which every irreducible representation is self-dual, and a calculation of the Frobenius Schur indicator for such groups.

Quadratic unipotent blocks in general linear, unitary and symplectic groups

An irreducible ordinary character of a finite reductive group is called quadratic unipotent if it corresponds under Jordan decomposition to a semisimple element

Real Moment Map and Hyperkähler Geometry Techniques: The Cotangent Bundle to the 2-Sphere and SL(2, ℂ)-Adjoint Orbits

Going back to Kirwan and others, there is an established theory that uses moment map techniques to study actions by complex reductive groups on Kähler manifolds. Work of P. Heinzner, G. Schwarz, and H. Stötzel has extended this theory to actions by real reductive groups. In this paper, we apply these techniques to actions of the real group SU(1,1) ⊂ SL(2, ℂ) on a certain complex manifold of dimension two. More precisely, because of the SU(2)-invariant hyperkähler structure on this manifold, we are able to study a family of actions which includes and “interpolates” two well-known actions of SL(2, ℂ): the adjoint action on the orbit of a semisimple element of 𝔰𝔩(2, ℂ), and the action of SL(2, ℂ) on the cotangent bundle of the flag variety of SL(2, ℂ).

The displacement function and the return map

This chapter studies the displacement function dᵧ on X that is associated with a semisimple element γ‎ ∈ G. If φ‎″, t ∈ R denotes the geodesic flow on the total space X of the tangent bundle of X, the critical set X(γ‎) ⊂ X of dᵧ can be easily related to the fixed point set Fᵧ ⊂ X of the symplectic transformation γ‎⁻¹φ‎₁ of X. The chapter studies the nondegeneracy of γ‎⁻¹φ‎₁ − 1 along Fᵧ. More fundamentally, this chapter gives important quantitative estimates on how much φ‎ ½ differs from φ‎ ˗½γ‎ away from Fᵧ. These quantitative estimates are based on Toponogov's theorem.

Evaluation of supertraces for a model operator

This chapter evaluates the supertrace of the heat kernel of a hypoelliptic operator acting over p × g, given a semisimple element γ‎ ∈ G. It begins by introducing a hypoelliptic operator, its heat kernel, and a corresponding supertrace Jᵧ(Y₀ᵗ), if Y₀ᵗ ∈ t(γ‎). Then, by a conjugation of the hypoelliptic operator, the chapter obtains a simpler operator where p × p and t have been decoupled. This new operator splits naturally into a scalar part and a matrix part. Hereafter, the chapter evaluates the trace of the heat kernel of the scalar part, and computes the supertrace of the matrix part of the heat kernel. This chapter concludes with an explicit formula for Jᵧ(Y₀ᵗ).