The unramified computation of a Shimura integral for $\mathrm{SL}(2)\times \mathrm{GL}(2)$

Technical Summary: The Unramified Computation of a Shimura Integral for SL(2) × GL(2)

Problem Statement
This paper addresses the local unramified computation of a specific Rankin-Selberg integral of Shimura type, originally constructed by Ginzburg, Rallis, and Soudry [4]. The integral represents the partial tensor product $L$-function for a pair of irreducible automorphic cuspidal generic representations: one of $SL_2$ and the other of $GL_2$. Specifically, the authors focus on the low-rank case where the symplectic rank $n=1$ and the general linear rank $k=2$. The global integral, after unfolding, reduces to a local integral involving a Whittaker function for $SL_2$, a theta series associated with the Weil representation, and a Siegel Eisenstein series on the metaplectic double cover of $Sp_4$. The objective is to prove that this local integral equals the ratio of $L$-functions:
$$\frac{L(\pi_\nu \times \tau_\nu, s + 1/2)}{L(\tau_\nu, \text{Sym}^2, 2s + 1)}$$
where $\pi_\nu$ is an unramified representation of $SL_2(F)$ and $\tau_\nu$ is an unramified representation of $GL_2(F)$.

Methodology
The paper provides a direct and "intrinsic" proof of the unramified computation, distinct from the approach taken in the original work [4].

• Avoidance of Functional Equations: Unlike the proof in [4], which relies on the local functional equation for $GL_n \times GL_k$ (developed in [9]), this paper avoids such machinery entirely.
• Reliance on Casselman-Shalika Formulas: The core of the method utilizes the Casselman-Shalika formulas for unramified Whittaker functions on both $SL_2$ and $GL_2$ (Lemma 3.2).
• Explicit Unfolding and Integration: The authors perform an explicit calculation of the local integral by:
  1. Utilizing the Iwasawa decomposition for $SL_2(F)$.
  2. Applying the Weil representation $\omega_\psi$ of the semi-direct product $\widetilde{SL_2} \ltimes H$ (where $H$ is the Heisenberg group) to the Schwartz function.
  3. Decomposing the integration domain over the Heisenberg group variables into unramified ($OF$) and ramified ($F \setminus OF$) parts.
  4. Evaluating the resulting integrals using properties of the Weil factor $\gamma_\psi$ and character sums, specifically proving that certain higher-order terms vanish (Lemma 3.5).
• Series Summation: The final result is obtained by summing the resulting geometric series, which are identified with the generating functions of complete homogeneous symmetric polynomials, thereby recovering the explicit form of the $L$-functions.

Key Contributions and Results

• Theorem 3.1: The paper establishes that for unramified data and $\text{Re}(s) \gg 0$, the local integral $I(W^0_\pi, \Phi^0, \tilde{f}^0_{W(\tau, \psi_2), s})$ converges absolutely and equals the ratio $\frac{L(\pi \times \tau, s + 1/2)}{L(\tau, \text{Sym}^2, 2s + 1)}$.
• Isomorphism Correction: The authors note a difference in the isomorphism between the unipotent group $N_Q$ and the Heisenberg group $H$ compared to [4]. This difference results in the appearance of the section $\tilde{f}_{W(\tau, \psi_2), s}$ (involving the character $\psi_2(x) = \psi(2x)$) rather than $\tilde{f}_{W(\tau, \psi), s}$ in the unfolded integral.
• Direct Computation: The proof demonstrates that for the case $n = k-1$ (here $1 = 2-1$), the unipotent integration that typically complicates the general $n < k$ case is absent. This allows for a direct computation that mirrors techniques used for even-rank cases, without needing the heavy machinery of functional equations for general linear groups.

Significance and Claims
The paper claims to offer a "different and more intrinsic" proof of the unramified computation for this specific Shimura integral. By bypassing the local functional equation for general linear groups, the authors aim to provide a calculation that relies solely on the representation-theoretic properties of the groups involved (specifically the Casselman-Shalika formulas).

The authors modestly state that while they only treat the low-rank case $n=1, k=2$, they hope this calculation will "shed light on the techniques for direct unramified computation" in more general settings. They acknowledge that similar direct computations for orthogonal groups have been achieved by Kaplan [10, 11], but emphasize that their work avoids the functional equations used in the original $SL_2 \times GL_2$ proof by Ginzburg, Rallis, and Soudry. The paper does not propose new applications or future extensions beyond the scope of this specific computational proof.