Quantum wreath products and $p$-adic general linear group

1. Problem Statement and Motivation

The paper addresses the representation theory of the $p$-adic general linear group $G = GL_d(F)$ and its metaplectic covers $\tilde{G}$, specifically focusing on the pro-$p$ level.

• Context: The study of $p$-adic groups often relies on Hecke algebras associated with compact subgroups. While the spherical and Iwahori levels are well-understood (via the Satake isomorphism and affine Hecke algebras), the pro-$p$ Iwahori level (associated with the pro-$p$ Iwahori subgroup $I^{(1)}$) presents significant challenges.
• The Gap:
  • For the linear group $G$, the pro-$p$ Hecke algebra is known to be isomorphic to the affine Yokonuma algebra.
  • For the metaplectic cover $\tilde{G}$, the structure of the pro-$p$ Hecke algebra $\tilde{H}^{(1)}$ and its Gelfand–Graev modules (which encode Whittaker models) is much more complex.
  • Existing methods fail to provide a transparent description of these algebras and modules, particularly because the "multiplicity one" property for Whittaker models fails in the metaplectic setting, and the support of the Hecke algebra is reduced in a non-trivial way.
• Specific Goals:
  1. Describe the structure of the pro-$p$ metaplectic Hecke algebra $\tilde{H}^{(1)}$.
  2. Identify the structure of the pro-$p$ Gelfand–Graev module $\tilde{W}_{I^{(1)}}$ as a module over $\tilde{H}^{(1)}$.
  3. Construct the corresponding pro-$p$ Schur algebra $\tilde{S}^{(1)}$ and determine its basis and multiplication rules.
  4. Investigate the failure of the local Shimura correspondence at the pro-$p$ level.

2. Methodology: Quantum Wreath Products

The authors introduce and utilize the theory of Quantum Wreath Products (specifically of skew polynomial type) to solve these problems. This approach shifts the problem from analytic $p$-adic methods to algebraic structures involving skew polynomials and generalized braid relations.

• Quantum Wreath Products ($B \wr H(d)$): An algebra generated by a base algebra $B^{\otimes d}$ and Hecke-like generators $H_1, \dots, H_{d-1}$, subject to braid relations, quadratic relations with coefficients in $B^{\otimes d}$, and "wreath" relations describing the interaction between $H_i$ and $B^{\otimes d}$.
• Skew Polynomial Base: The authors define the base algebra $A$ as a ring of skew Laurent polynomials:
  $$A = F \rtimes_n \mathbb{C}[x^{\pm 1}]$$
  where $F = \mathbb{C}[t]/(t^m - 1)$ (with $m=q-1$) and the skew commutativity is defined by $x^{\pm 1}f = \xi^{\pm 2} f x^{\pm 1}$ for a root of unity $\xi$.
• Key Technical Innovation: They develop a theory of skew polynomial quantum wreath products ($A \wr H(d)$). Unlike standard polynomial wreath products, the quadratic relations in this setting involve complicated degree-one coefficients. The authors prove that these relations split into linear factors (Lemma 2.1.11), allowing for the definition of "trivial" and "sign" modules.
• Wreath Modules: They utilize the concept of wreath modules (introduced in prior work by the authors) to construct specific representations. By starting with a base module and inducing via a twisted Hecke algebra, they construct the antispherical module for the quantum wreath product.

3. Key Contributions and Results

A. Isomorphism of Algebras (Theorem A)

The authors establish a fundamental isomorphism between the metaplectic pro-$p$ Hecke algebra $\tilde{H}^{(1)}$ and the skew polynomial quantum wreath product $A \wr H(d)$:
$$\tilde{H}^{(1)} \cong A \wr H(d)$$

• Significance: This provides a transparent algebraic presentation of $\tilde{H}^{(1)}$, replacing the complex Bernstein–Lusztig relations found in previous $p$-adic literature with the structured relations of the quantum wreath product.
• PBW Basis: As a consequence, they derive an explicit PBW basis for $\tilde{H}^{(1)}$, confirming a conjecture regarding its structure.

B. Identification of Gelfand–Graev Modules (Theorem C)

The paper identifies the pro-$p$ component of the metaplectic Gelfand–Graev module $\tilde{W}_{I^{(1)}}$ with the antispherical wreath module of the quantum wreath product:
$$\tilde{W}_{I^{(1)}} \cong V(1) \wr \text{sgn}$$

• Mechanism: The authors rescale the basis of the Gelfand–Graev module to normalize Gauss sums, then show that the action of the Hecke algebra generators matches the action of the quantum wreath product on the antispherical module.
• Generalization: This generalizes the known result for the affine Hecke algebra (where the Gelfand–Graev module corresponds to the antispherical module) to the metaplectic pro-$p$ setting.

C. Pro-$p$ Schur Algebras (Theorem D)

Using the identification of the module, the authors define the pro-$p$ Schur algebra $\tilde{S}^{(1)} = \text{End}_{\tilde{H}^{(1)}}(\tilde{W}_{I^{(1)}})$.

• Result: They prove an isomorphism $\tilde{S}^{(1)} \cong S(1, d)$, where $S(1, d)$ is the wreath Schur algebra associated with the quantum wreath product.
• Explicit Structure: They provide an explicit basis (of Dipper-James type) and multiplication formulas for $\tilde{S}^{(1)}$. This solves the problem of understanding the structure of these algebras, which was previously inaccessible.

D. Failure of Local Shimura Correspondence (Corollary B)

The paper investigates the relationship between the metaplectic and non-metaplectic levels via the local Shimura correspondence.

• Iwahori Level: At the Iwahori level, the metaplectic Hecke algebra $\tilde{H}_I$ is isomorphic to the non-metaplectic affine Hecke algebra $H_I$ (Savin's correspondence).
• Pro-$p$ Level: The authors show that at the pro-$p$ level, the algebras $\tilde{H}^{(1)}$ and $H^{(1)}$ are non-isomorphic for $n \geq 2$.
• Reason: The centers of the two algebras differ because the base algebras involve different skew polynomial structures ($F \rtimes_n \mathbb{C}[x^{\pm 1}]$ vs. $\mathbb{C}[x^{\pm 1}]$). This provides an algebraic proof of the failure of the local Shimura correspondence at the pro-$p$ level.

E. Recovery of Iwahori Results

By using an idempotent $\epsilon_I$ to "descend" from the pro-$p$ level to the Iwahori level, the authors recover the known isomorphism:
$$\tilde{S}_I \cong (S_q)_q(\bar{n}, d)$$
where $(S_q)_q(\bar{n}, d)$ is the quantum affine Schur algebra. This confirms that their new framework is consistent with existing results while offering a unified perspective.

4. Significance and Impact

1. Unified Framework: The paper unifies the study of affine Hecke algebras, affine Yokonuma algebras, and metaplectic Hecke algebras under the single umbrella of quantum wreath products.
2. New Tools for $p$-adic Representation Theory: By translating difficult $p$-adic analytic problems into algebraic problems involving skew polynomials and wreath products, the authors provide a "transparent" method to derive results that were previously inaccessible.
3. Resolution of Open Problems: The paper explicitly solves the structural problems (P1, P2, P3) regarding the pro-$p$ Gelfand–Graev modules and Schur algebras, which were open questions in the literature (specifically referencing the work of Gao–Gurevich–Karasiewicz).
4. Foundation for Future Work: The construction of the pro-$p$ Schur algebra and its basis opens the door to studying the representation theory of these algebras, potentially leading to new insights into the mod-$p$ Langlands program and the classification of supersingular representations.
5. Clarification of Metaplectic Phenomena: The work clarifies exactly how and why the metaplectic cover alters the structure of Hecke algebras at the pro-$p$ level, distinguishing it clearly from the Iwahori level where the correspondence holds.

In summary, this paper successfully bridges the gap between the representation theory of $p$-adic groups and the algebraic theory of quantum groups by introducing skew polynomial quantum wreath products, thereby providing a complete structural description of pro-$p$ Hecke algebras and their modules for both linear and metaplectic groups.