Global Hecke-Baxter Operator

• Global Hecke-Baxter operator is an adelic convolution operator in a spherical Hecke algebra that unifies non-Archimedean Hecke correspondences with an Archimedean Baxter kernel to yield completed L-factors.
• It factorizes into local Euler products, acting diagonally on GL₂-Eisenstein series with eigenvalues directly corresponding to local L-factors.
• Its construction mirrors Baxter operators in integrable systems, highlighting a structural analogy that connects automorphic harmonic analysis with quantum integrability.

Searching arXiv for the specified paper and closely related Hecke–Baxter literature. Use the arXiv search tool to retrieve papers by id (Gerasimov et al., 14 Jul 2025, Gerasimov et al., 2024, Chicherin et al., 2012), and (Gerasimov et al., 2012), and related results on Hecke-Baxter operators. The global Hecke-Baxter operator is an element of a global Hecke algebra that combines non-Archimedean Hecke correspondences with an Archimedean Baxter-type convolution kernel into a single object whose eigenvalues on automorphic data are global $L$-factors. In the $GL_2$ setting, it is constructed in the spherical Hecke algebra attached to the double-coset space $GL_2(\mathbb{Z})\backslash GL_2(\mathbb{R})/O_2$, and acts diagonally on $GL_2$-Eisenstein series with eigenvalue given by the corresponding completed global $L$-factor (Gerasimov et al., 14 Jul 2025). More broadly, the expression “global Hecke-Baxter operator” sits at the intersection of two established formalisms: the Hecke-theoretic description of local and adelic $L$-factors, and the Baxter-operator formalism of quantum integrable systems, where local $R$- or $L$-operators are assembled into commuting monodromy and transfer objects (Gerasimov et al., 2024).

1. Definition in the $GL_2$ spherical Hecke algebra

For $G=GL_2(\mathbb{R})$ and $GL_2$0, the double-coset space

$GL_2$1

is viewed as parameterizing “metricized” two-tori over $GL_2$2. The spherical global Hecke algebra

$GL_2$3

is the convolution algebra of compactly supported bi-invariant functions, or distributions, on $GL_2$4 which are left-invariant under $GL_2$5 and right-invariant under $GL_2$6. Its convolution product is

$GL_2$7

with $GL_2$8 a fixed Haar measure on $GL_2$9. In this setting $GL_2(\mathbb{Z})\backslash GL_2(\mathbb{R})/O_2$0 is commutative since $GL_2(\mathbb{Z})\backslash GL_2(\mathbb{R})/O_2$1 is a Gelfand pair (Gerasimov et al., 14 Jul 2025).

The classical Hecke operator $GL_2(\mathbb{Z})\backslash GL_2(\mathbb{R})/O_2$2 is obtained from the double coset

$GL_2(\mathbb{Z})\backslash GL_2(\mathbb{R})/O_2$3

and acts by

$GL_2(\mathbb{Z})\backslash GL_2(\mathbb{R})/O_2$4

The non-Archimedean generating series is

$GL_2(\mathbb{Z})\backslash GL_2(\mathbb{R})/O_2$5

which converges for $GL_2(\mathbb{Z})\backslash GL_2(\mathbb{R})/O_2$6. The Archimedean Baxter kernel is

$GL_2(\mathbb{Z})\backslash GL_2(\mathbb{R})/O_2$7

with associated convolution operator

$GL_2(\mathbb{Z})\backslash GL_2(\mathbb{R})/O_2$8

The global Baxter operator is then defined as

$GL_2(\mathbb{Z})\backslash GL_2(\mathbb{R})/O_2$9

Equivalently, it may be described as first applying the discrete average

$GL_2$0

and then the Archimedean Baxter integral (Gerasimov et al., 14 Jul 2025).

In kernel form,

$GL_2$1

This formula makes explicit that the global operator is not merely a formal product of local objects, but also a concrete convolution kernel on the global double-coset space (Gerasimov et al., 14 Jul 2025).

2. Local factorization and Euler-product structure

A central structural fact is that the global Hecke algebra and its Baxter element factorize over the places $GL_2$2 of $GL_2$3: $GL_2$4 At the Archimedean place,

$GL_2$5

while for each finite prime $GL_2$6,

$GL_2$7

On each local spherical principal series representation $GL_2$8 of $GL_2$9, $L$0 acts by multiplication by the local factor $L$1, and $L$2 acts in the Archimedean principal series by $L$3 (Gerasimov et al., 14 Jul 2025).

This local-to-global factorization is part of a broader Hecke-Baxter formalism for $L$4. For $L$5, $L$6, the local Archimedean kernel is

$L$7

and the corresponding convolution operator preserves the one-dimensional $L$8-fixed subspace in each spherical principal series $L$9. Its eigenvalue on the normalized spherical vector is

$L$0

At finite primes, the local spherical Hecke operator acts by the standard unramified local factor

$L$1

and the global Hecke-Baxter operator is

$L$2

on the restricted tensor-product spherical vector $L$3 (Gerasimov et al., 2024).

This identifies the global Hecke-Baxter operator with an adelic convolution operator whose spectrum is multiplicative across places. A plausible implication is that the defining feature of “globality” in this context is not merely a long-chain composition, but the Euler-product assembly of local Hecke-Baxter data into a single operator whose eigenvalue recovers a completed $L$4-function.

3. Spectral action on Eisenstein series

For $L$5, let $L$6 be the unitary spherical principal series with spectral parameters $L$7, induced from

$L$8

If $L$9 is the unique $R$0-fixed vector and $R$1 the unique $R$2-invariant distribution in the dual, then the Eisenstein series is

$R$3

In classical coordinates $R$4, $R$5, one has

$R$6

The global Hecke-Baxter operator acts diagonally: $R$7 where

$R$8

Equivalently,

$R$9

The completed factor satisfies the functional equation

$L$0

reflecting the involutive symmetry of both the Eisenstein series and the Baxter kernel under the Cartan involution $L$1 (Gerasimov et al., 14 Jul 2025).

In this spectral description, the operator is characterized by its diagonal action on spherical automorphic vectors. That point aligns with the local Archimedean formulation in higher rank, where the Hecke-Baxter kernel is specified by the property that it acts on an $L$2-fixed vector in a $L$3-principal series representation via multiplication by the local Archimedean $L$4-factor canonically attached to the representation (Gerasimov et al., 2024).

4. Relation to Baxter operators in integrable systems

In quantum integrable systems, Baxter operators are usually assembled from local $L$5- or $L$6-operators. For an $L$7-site periodic spin chain with $L$8-symmetry, one introduces the monodromy

$L$9

and constructs the Baxter $GL_2$0-operator as a trace over an auxiliary infinite-dimensional space,

$GL_2$1

One then has

$GL_2$2

together with the global $GL_2$3-$GL_2$4 relation

$GL_2$5

with explicit structure functions $GL_2$6 (Chicherin et al., 2012).

The Hecke condition is crucial in that construction. At the local level,

$GL_2$7

and this Hecke-type relation ensures that when neighboring $GL_2$8-factors are permuted in the monodromy of $GL_2$9, scalar factors cancel, so that $G=GL_2(\mathbb{R})$0 is single-valued and globally well-defined. It also guarantees the commutativity properties needed for $G=GL_2(\mathbb{R})$1 (Chicherin et al., 2012).

An analogous assembly principle appears in the dynamical Yang-Baxter setting. Starting from a local dynamical Hecke operator $G=GL_2(\mathbb{R})$2 satisfying a Hecke quadratic relation and a braid relation with dynamical shifts, one Baxterizes it by

$G=GL_2(\mathbb{R})$3

obtaining a dynamical $G=GL_2(\mathbb{R})$4-matrix. On $G=GL_2(\mathbb{R})$5, the operators

$G=GL_2(\mathbb{R})$6

satisfy dynamical braid relations and define a representation of the global dynamical Hecke algebra, often called the Hecke-Baxter algebra. The corresponding transfer matrix

$G=GL_2(\mathbb{R})$7

satisfies $G=GL_2(\mathbb{R})$8 (Ren, 2023).

These constructions justify a careful distinction. In integrable spin chains, “global” refers to the chain-wide composition of local data into monodromy, transfer, or $G=GL_2(\mathbb{R})$9-operators. In the arithmetic $GL_2$00 setting, “global” refers to the adelic or Euler-product combination of local Hecke-Baxter operators across all places. The common feature is an assembly of local Hecke-type ingredients into a commuting global operator, but the ambient categories are different.

5. Algebraic realizations and higher-rank analogues

The Hecke-Baxter terminology also appears in purely algebraic Baxterization frameworks. In the fused Hecke algebra $GL_2$01, one defines spectral-parameter-dependent elements

$GL_2$02

with explicit coefficients

$GL_2$03

and these satisfy the braided Yang-Baxter equation

$GL_2$04

After passing to the local representation

$GL_2$05

their images $GL_2$06 obey the ordinary Yang-Baxter equation and yield a commuting transfer matrix

$GL_2$07

by the usual train argument (Crampe et al., 2020).

A related fusion picture occurs for Hecke-type $GL_2$08-matrices on reducible or composite quantum-group representations. Starting from a Hecke-type solution

$GL_2$09

satisfying the Yang-Baxter and Hecke relations, one constructs descendant $GL_2$10-matrices, extended Lax operators, the monodromy

$GL_2$11

and the transfer matrix

$GL_2$12

The consistency of the global construction is ensured by the quantum-group invariance and Hecke-algebra relations built into the local data (Khachatryan, 2018).

In harmonic-analytic settings, the Baxter operator formalism for Macdonald polynomials provides another higher-rank analogue. The operators $GL_2$13 and $GL_2$14 commute with the Macdonald-Ruijsenaars Hamiltonians and satisfy functional relations

$GL_2$15

The paper further states that the generating series and Baxter operators generate a commutative subalgebra isomorphic to the spherical Hecke algebra of $GL_2$16 at the Archimedean place, and conjectures a global Baxter operator

$GL_2$17

acting by the completed $GL_2$18-function

$GL_2$19

on global Whittaker or Macdonald-type functions (Gerasimov et al., 2012).

6. Interpretive issues, misconceptions, and open directions

A common misconception is to identify every global Hecke-Baxter operator with a transfer matrix in the quantum inverse scattering method. The arithmetic $GL_2$20 operator is instead an element of a global Hecke algebra attached to

$GL_2$21

defined by convolution and diagonalized by Eisenstein series, with eigenvalue equal to a completed global $GL_2$22-factor (Gerasimov et al., 14 Jul 2025). By contrast, the spin-chain and fused-Hecke constructions produce monodromy or $GL_2$23-operators on tensor-product state spaces, with commutativity derived from Yang-Baxter or RTT relations (Chicherin et al., 2012).

Another possible source of confusion is the role of the Hecke relation itself. In the quantum-chain literature, the Hecke-type relation is a local quadratic or unitarity constraint that enables Baxterization, factor permutation, and global commutativity. In the arithmetic literature, “Hecke” refers to spherical Hecke algebras, double cosets, and convolution operators. The shared terminology is therefore structural rather than literal: in both settings a Hecke-algebraic input organizes a commuting family of operators, but the local relations and representation spaces differ. This suggests a genuine analogy, not an identity.

The recent arithmetic program makes this analogy explicit. In the formalism of quantum integrable systems, the Baxter operator $GL_2$24 has eigenvalues whose zeros satisfy Bethe ansatz equations. By analogy, the global Baxter eigenvalue $GL_2$25 is described as an entire function of $GL_2$26 whose zeros encode the arithmetic spectrum, and it is conjectured that “an arithmetic Bethe ansatz” may be formulated for the zeros of $GL_2$27, possibly through functional relations of the form

$GL_2$28

with explicit $GL_2$29 built from local data. No closed system of Bethe equations is yet written in the cited work (Gerasimov et al., 14 Jul 2025).

A further development is the realization of the Archimedean Hecke-Baxter operator via Heisenberg-group extensions. For $GL_2$30, the kernel $GL_2$31 is identified with a generalized Whittaker function for an extension of $GL_2$32 by a Heisenberg Lie group, and can also be lifted to a matrix element for an extension of $GL_2$33 by a Heisenberg Lie group (Gerasimov et al., 2024). This does not alter the spectral characterization of the operator, but it enlarges the representation-theoretic framework in which the Archimedean factor is understood.

Taken together, these results present the global Hecke-Baxter operator as a precise meeting point of automorphic harmonic analysis, spherical Hecke algebras, and Baxter-type integrability. In the arithmetic $GL_2$34 case it is already a concrete global convolution operator with eigenvalue $GL_2$35 on Eisenstein series; in higher-rank and Macdonald-type settings, the same principle appears as an established local formalism together with a conjectural or adelic global completion (Gerasimov et al., 2012).