Inducing Whittaker Functions from Higher Ranks

Technical Summary: Inducing Whittaker Functions from Higher Ranks

Problem Statement
Whittaker functions serve as fundamental components in the theory of automorphic forms and $L$-functions, specifically acting as the archimedean components of unitary cuspidal automorphic representations of $GL(n, \mathbb{A})$. While the propagation of these functions from lower to higher ranks (or between adjacent ranks) has been studied extensively—most notably by Ishii and Stade, who derived formulas relating Whittaker functions on $GL(n, \mathbb{R})$ to those on $GL(n-1, \mathbb{R})$ and $GL(n-2, \mathbb{R})$—a direct construction allowing for a "backpropagation" from an arbitrary higher rank $n$ to a lower rank $m$ (where $2 \le m < n$) in a single step was lacking. The central problem addressed is whether a Whittaker function for $SL(n, \mathbb{Z})$, when restricted to a block-diagonal embedding of $SL(m, \mathbb{Z})$, retains the structural properties of a Whittaker function for the lower rank group, and if so, under what conditions on the Langlands parameters.

Methodology
The paper employs a direct restriction method based on the block-diagonal embedding of the generalized upper half-planes. Let $h_n = GL(n, \mathbb{R}) / (O(n, \mathbb{R}) \cdot \mathbb{R}^\times)$ denote the generalized upper half-plane. The author defines an embedding $h_m \hookrightarrow h_n$ via the map:
$$xy \mapsto \begin{pmatrix} xy & 0 \\ 0 & I_{n-m} \end{pmatrix}$$
where $xy \in h_m$ and $I_{n-m}$ is the identity matrix of size $n-m$.

Given Jacquet's Whittaker function $W^{(n)}_{\alpha, N}$ for $SL(n, \mathbb{Z})$ with Langlands parameters $\alpha = (\alpha_1, \dots, \alpha_n)$ and character tuple $N = (N_1, \dots, N_{n-1})$, the author constructs a candidate function $V^{(m)}_{\alpha, N}: h_m \to \mathbb{C}$ by restricting $W^{(n)}_{\alpha, N}$ to this embedded subspace:
$$V^{(m)}_{\alpha, N}(xy) := W^{(n)}_{\alpha, N}\left( \begin{pmatrix} xy & 0 \\ 0 & I_{n-m} \end{pmatrix} \right)$$

To prove that $V^{(m)}_{\alpha, N}$ is a valid Whittaker function for $SL(m, \mathbb{Z})$, the paper verifies the three defining conditions outlined in Definition 2.7:

1. Equivariance: The function must transform by a specific unitary character under the unipotent radical $U(m, \mathbb{R})$.
2. Eigenfunction Property: The function must be a simultaneous eigenfunction of the Casimir operators (invariant differential operators) for $GL(m, \mathbb{R})$.
3. Moderate Growth/Square-Integrability: The function must satisfy growth conditions ensuring square-integrability on the Siegel set.

Key Contributions and Results
The main result, Theorem 1.2 (also stated as Theorem 3.1), establishes that the restriction $V^{(m)}_{\alpha, N}$ is indeed a Whittaker function for $SL(m, \mathbb{Z})$ provided a specific linear condition is met by the Langlands parameters of the original function.

• The Condition: The parameters $\alpha = (\alpha_1, \dots, \alpha_n)$ must satisfy:
  $$\sum_{i=1}^m \alpha_i = \frac{m(m-n)}{2}$$
  This condition ensures that the shifted parameters for the induced function, defined as $\beta_i = \alpha_i + \frac{n-m}{2}$ for $1 \le i \le m$, satisfy the normalization requirement $\sum_{i=1}^m \beta_i = 0$ necessary for valid Langlands parameters on $GL(m, \mathbb{R})$.

• The Induced Parameters: Under the above condition, the induced function $V^{(m)}_{\alpha, N}$ possesses:

  • Langlands Parameters: $\left( \alpha_i + \frac{n-m}{2} \right)_{1 \le i \le m}$.
  • Character Tuple: $(N_1, \dots, N_{m-1})$, which corresponds to the truncation of the original character tuple $N$ to its first $m-1$ entries.

• Proof Mechanism:

  • Equivariance (Lemma 4.1): Follows directly from the block-diagonal structure, as the unipotent elements of $U(m, \mathbb{R})$ embed naturally into $U(n, \mathbb{R})$ without affecting the identity block.
  • Eigenfunction Property (Lemma 4.2): The Casimir operators $D^{(m)}_\ell$ for $GL(m, \mathbb{R})$ act on the restricted function identically to how the corresponding operators $D^{(n)}_\ell$ act on the original function, restricted to the block. The shift in parameters arises from the difference in the $\rho$-shifts between $\mathfrak{gl}(n, \mathbb{R})$ and $\mathfrak{gl}(m, \mathbb{R})$.
  • Square-Integrability (Lemma 4.3): The proof relies on a non-trivial bound for $SL(n, \mathbb{Z})$-Whittaker functions established by Goldfeld, Muthuvel, and Wang [GMW21], demonstrating that the decay properties of the higher-rank function are sufficient to ensure integrability on the lower-rank Siegel set.

Significance
The paper claims to provide a "complementary result" to the propagation formulas of Ishii and Stade. While previous work focused on recursive steps reducing rank by 1 or 2, this construction allows for a direct "backpropagation" from $GL(n, \mathbb{R})$ to $GL(m, \mathbb{R})$ for any $m < n$ in a single step. The result is described as "natural" because it relies on the standard block-diagonal embedding and the restriction of the canonical Jacquet Whittaker function.

The paper emphasizes that the linear condition on the Langlands parameters is not merely a technical artifact but has a representation-theoretic interpretation: it is the precise condition required to normalize the induced parameters for the lower-rank group. The work is situated within the classical framework of automorphic forms on $GL(n, \mathbb{R})$ and does not propose new experimental applications or future implications beyond the theoretical construction of these induced functions.