Relative Character Asymptotics Beyond Stability for $\mathrm{PGL}_2 \times \mathrm{GL}_1$

1. Problem Statement and Context

The Core Question:
The paper addresses the asymptotic behavior of relative characters for representations of real and $p$-adic Lie groups. Specifically, it investigates the distribution $H_{\pi, \chi}(a)$ associated with a pair $(G, H) = (\text{PGL}_2(F), \text{GL}_1(F))$ where $F$ is a non-archimedean local field. The relative character is defined as:
$$H_{\pi, \chi}(a) = \sum_{v \in \mathcal{B}(\pi)} \ell_\chi(\text{Op}(a)v) \ell_\chi(v)$$
where $\pi$ is a unitary representation of $G$, $\chi$ is a character of $H$, $\ell_\chi$ is a linear functional (period integral), and $\text{Op}(a)$ is an operator defined via the Fourier transform of a test function $a$ on the dual Lie algebra $\mathfrak{g}^*$.

The Challenge:
Previous work by Nelson and Venkatesh established asymptotic formulas for relative characters in the stable regime. In this regime, the relative coadjoint orbit $\mathcal{O}_{\pi, \sigma}$ is a smooth manifold (specifically an $H$-torsor), and the asymptotics are governed by an integral over this orbit.
However, in non-stable regimes (where "conductor dropping" occurs, i.e., the conductors of $\pi$ and $\chi$ are close or equal), the relative coadjoint orbit becomes singular (e.g., a cross shape with a singularity at the origin) or fails to be a single orbit. The geometry becomes too complex for the standard microlocal analysis used in the stable case. The paper aims to compute these asymptotics in the non-stable setting for $\text{PGL}_2 \times \text{GL}_1$.

2. Methodology

The author employs a combination of non-archimedean microlocal analysis, representation theory, and explicit computation of local functional equations.

A. Test Functions and Wavepackets

Instead of general test functions, the paper focuses on non-archimedean wavepackets. These are functions $a_\tau$ on $\mathfrak{g}^*$ that are characteristic functions of small balls around a "frequency" $T\tau$ (where $T = \varpi^{-N}$ is a scaling parameter).

• Microlocalization: The operator $\text{Op}(a_\tau)$ projects onto vectors that are "microlocalized" at $\tau$, meaning they behave like eigenvectors for the action of the Lie algebra near the identity.
• Iwahori Factorization: The key technical innovation is decomposing the operator $\text{Op}(a)$ using the Iwahori factorization of the congruence subgroup $K(N) = K_- K_0 K_+$. This allows the operator to be factored into directional components:
  $$\text{Op}(a) \approx \text{Op}_-(a_+) \text{Op}_0(a_0) \text{Op}_+(a_-)$$
  where each component acts on specific subspaces of the Lie algebra (negative unipotent, diagonal, positive unipotent).

B. Kirillov Model and Functional Equations

The calculation is performed within the Kirillov model of the representation $\pi$, which realizes $\pi$ as a space of functions on $F^\times$.

• Kirillov Model: This model diagonalizes the action of the unipotent subgroup $N$, simplifying the action of $\text{Op}_+$.
• Weyl Element and GL2 Functional Equation: To handle the negative unipotent direction ($\text{Op}_-$), the author utilizes the Weyl element $w = \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix}$ and the GL2 $\times$ GL1 local functional equation. This relates the action of $\text{Op}_-$ to $\text{Op}_+$ via the gamma factor $\gamma(\pi \otimes \chi)$.
• Explicit Calculation: The author explicitly computes the action of these operators on specific vectors $v_\chi^R$ (approximations of the period functional) for both supercuspidal and principal series representations. This involves detailed analysis of epsilon factors, Gauss sums, and conductor exponents.

C. Phase Space Integration

The final step involves computing the integral of the test function $a$ over a specific geometric locus in the dual Lie algebra, termed the hyperbola $\text{Hyp}(\pi, \chi)$. The author shows that the explicit operator calculations match the integral over this locus.

3. Key Contributions

1. Breaking the Stability Hypothesis: The primary contribution is the derivation of relative character asymptotics in a regime where the conductor of the Rankin-Selberg $L$-function $L(s, \pi \times \chi)$ drops significantly. This is the first such result for $\text{PGL}_2 \times \text{GL}_1$ in the non-archimedean setting that does not require the "stable pair" condition (where eigenvalues of infinitesimal characters are disjoint).
2. Geometric Interpretation of Singular Orbits: The paper identifies that even when the relative coadjoint orbit is singular (a cross shape in the stable case analysis, or a hyperbola in the non-stable case), the asymptotics are still governed by a well-defined geometric object: a hyperbola in $\mathfrak{g}^*$.
   • The locus is defined as $\text{Hyp}(\pi, \chi) = \{(x, y, z) : x = \alpha_\chi, yz = \alpha_{\pi, \chi}\}$.
   • Here, $\alpha_{\pi, \chi}$ is a parameter derived from the conductors and epsilon factors of the representations.
3. Explicit Formula for Non-Stable Asymptotics: The paper provides an exact formula (Theorem 1.3) for the relative character $H_{\pi, \chi}(a)$ as an integral over this hyperbola, valid for test functions supported on scales determined by the conductors.

4. Main Results

Theorem 1.3 (Main Theorem):
Let $\pi$ be a principal series or supercuspidal unitary representation of $\text{PGL}_2(F)$ and $\chi$ a unitary character of $\text{GL}_1(F)$. Under specific conditions on the conductors (allowing for significant conductor dropping), and for a test function $a$ supported on elements with specific coordinate bounds, the relative character satisfies:
$$H_{\pi, \chi}(a) = \int_{\text{Hyp}(\pi, \chi)} a(\xi) d\xi$$
where $\text{Hyp}(\pi, \chi)$ is a hyperbola in the dual Lie algebra.

• Principal Series Case: If $\pi = \chi_0 \boxplus \chi_0^{-1}$, the hyperbola is determined by the product of conductors of $\chi_0 \chi^{-1}$ and $\chi_0^{-1} \chi^{-1}$.
• Supercuspidal Case: The hyperbola is determined by the conductor of the associated representation of the quadratic extension $E/F$.

Table of Results (Proposition 5.8):
The paper provides a detailed table (Table 1) describing the value of the inner product (and thus the relative character) based on the valuation of the coordinates $\tau_y$ and $\tau_z$ of the test function's frequency.

• If $\tau_y, \tau_z \in \mathcal{O}$ (integers), the value is proportional to $(2N - c(\pi, \chi))$.
• If one or both coordinates are non-integers (large conductors), the value involves terms like $1/|X_r|$ and conditions on the sum of valuations relative to the conductor $c(\pi, \chi)$.

5. Significance

• Subconvexity Applications: The asymptotics of relative characters are a crucial tool in the subconvexity problem for $L$-functions. Recent breakthroughs (e.g., by Nelson) rely on these asymptotics. However, existing methods often require the "non-conductor dropping" assumption to ensure stability. This paper removes that restriction, potentially allowing for the proof of subconvexity bounds in cases where conductors are close, which is a major open problem in analytic number theory.
• Geometric Understanding of Singularities: The work clarifies how relative character asymptotics behave when the underlying geometry (the relative coadjoint orbit) is singular. It suggests that the "correct" geometric object governing the asymptotics is not necessarily a smooth orbit, but a specific algebraic variety (the hyperbola) that persists even when stability fails.
• Non-Archimedean Microlocal Analysis: The paper advances the toolkit for non-archimedean microlocal analysis by successfully applying wavepacket decompositions and Iwahori factorizations to compute relative periods, bridging the gap between the real and $p$-adic theories.

In summary, Hammonds extends the theory of relative character asymptotics beyond the stable locus, providing explicit formulas for $\text{PGL}_2 \times \text{GL}_1$ that accommodate significant conductor dropping, thereby opening new avenues for subconvexity research and geometric representation theory in the $p$-adic setting.