Kac-Moody Satake Isomorphism

• Kac-Moody Satake isomorphism is a generalization of the classical Satake correspondence, extending duality concepts to infinite-dimensional groups with completed support conditions.
• It employs both Weyl almost finite and Bernstein–Lusztig completions to manage function spaces on Kac-Moody groups, reflecting distinct geometric and algebraic methodologies.
• This framework establishes a robust center, identified with Looijenga’s invariant ring, which underpins advances in harmonic analysis, automorphic forms, and geometric representation theory.

The Kac-Moody Satake isomorphism concerns a far-reaching generalization of the classical Satake correspondence, which relates spherical Hecke algebras of split reductive groups over local fields to representation rings (or function algebras) of their Langlands dual groups, to the setting of (split and almost-split) Kac-Moody groups. This theory is foundational in understanding harmonic analysis, representation theory, and the emerging geometric and quantum structures associated with infinite-dimensional Kac-Moody symmetries.

1. Conceptual Framework and Satake Isomorphism for Kac-Moody Groups

In the classical context, the Satake isomorphism identifies the spherical Hecke algebra $\mathcal{H}_s$ (bi-invariant functions with convolution) for a reductive group over a local non-archimedean field with $\mathbb{C}[Y]^W$, the algebra of Weyl-invariant regular functions on the coweight lattice $Y$ of a maximal torus. For a split Kac-Moody group $G$ over a non-Archimedean local field, the analogous objects require a substantial generalization:

• Spherical Hecke Algebra: This is defined not just for a usual maximal compact subgroup, but via bi-invariant functions on $G$ with almost finite support, indexed by the dominant coweight cone $Y^{++}$, due to the infinite-dimensional nature of $G$ (Gaussent et al., 2012).
• Satake Isomorphism: The principal result is that the spherical Hecke algebra $H$ is isomorphic to the subalgebra $R[[Y]]^{W^v}$ of the completed group algebra over the coweight lattice $Y$, consisting of $W^v$-invariant Laurent series with "almost finite" support (Abdellatif et al., 2017, Hébert et al., 20 Oct 2025). This completion, often called Looijenga’s invariant ring, replaces the role of the representation ring or the ring of symmetric polynomials in the finite-dimensional case.
• Completions of Iwahori-Hecke Algebras: For Kac-Moody groups, the standard Iwahori-Hecke algebra must also be completed, since functions with finite support are insufficient due to the non-local compactness of $G$ (Abdellatif et al., 2017, Hébert et al., 20 Oct 2025). The resulting completed Iwahori-Hecke algebra admits a large center isomorphic to Looijenga’s invariant ring, which is in turn related to the spherical Hecke algebra via the Kac-Moody Satake isomorphism.

2. Functional and Algebraic Constructions: Completions and Support Conditions

Two main approaches to the completion of Iwahori-Hecke algebras have emerged:

• Weyl Almost Finite Support ($\widehat{\mathcal{H}}$): This method defines the completed algebra as the space of Iwahori-bi-invariant functions on $G$ whose support, modulo the Weyl group action, lies in a finite union of translates of the positive coroot cone. This ensures the construction is firmly rooted in the geometry and group-theoretic functions on $G$ (Hébert et al., 20 Oct 2025).
• Algebraic Bernstein-Lusztig Completion ($\widetilde{\mathcal{H}}$): The approach by Abdellatif–Hebert instead completes by summing over the coweight lattice with "almost finite support"—i.e., for a finite set $J$, support inside $J - Q^\vee_+$—applied directly to the Bernstein–Lusztig basis, disregarding any geometric realization as functions on the group.

A key technical result is that, in the presence of an infinite Weyl group, these two completions are generally not comparable by inclusion. There exist subsets (counterexamples in affine types) that are almost finite in the sense of Abdellatif–Hebert but not Weyl almost finite, and vice versa (Hébert et al., 20 Oct 2025).

Table: Comparison of Completions

Completion Variant	Definition Basis	Support Condition
$\widehat{\mathcal{H}}$	Functions on $G$	Weyl almost finite (geometric)
$\widetilde{\mathcal{H}}$	Bernstein–Lusztig algebraic presentation	u–almost finite (coordinatewise in $Y$)

3. Centers and the Looijenga Invariant Ring

The motivation for considering such completions is to obtain a sufficiently large center, which, via the Satake isomorphism, can be identified with the spherical Hecke algebra:

• Center of Completed Iwahori-Hecke Algebra: The center $\mathcal{Z}(\widehat{\mathcal{H}})$ is isomorphic to $R\llbracket Y\rrbracket^{W^v}$—the Looijenga invariant ring of Weyl-invariant formal Laurent series with almost finite support (Abdellatif et al., 2017, Hébert et al., 20 Oct 2025). This realizes the abstract commutative algebra that encodes the Satake parameters in the Kac-Moody setting.
• Satake Isomorphism Implementation: The completion defined via Weyl almost finite support produces, functorially, an isomorphism between the spherical Hecke algebra and the center of the completed Iwahori-Hecke algebra, thereby extending the classical link between bi-invariant harmonic analysis and dual (Langlands) group theory.

This generalizes the familiar isomorphism in the finite-dimensional case:

$\mathcal{H}_s \cong \mathbb{C}[Y]^W$

to the infinite-dimensional setting:

$$\mathcal{Z}(\widehat{\mathcal{H}}) \cong R\llbracket Y\rrbracket^{W^v}$$

4. Geometric and Representation-Theoretic Implications

These algebraic and functional frameworks have direct implications for geometry and representation theory:

• Hecke Algebras from Masures: The key geometric objects are masures (ordered hovels), which are stratified generalizations of Bruhat–Tits buildings apt for Kac-Moody groups. The geometric structure underpins the organization of double coset spaces and supports the analytic notion of almost finite support (Gaussent et al., 2012, Abdellatif et al., 2017).
• Spherical and Parahoric Hecke Algebras: Restrictions of the Satake isomorphism may be constructed for any "spherical" facet in the masure, with the appropriate completion of support to ensure well-defined convolution products even in infinite settings (Abdellatif et al., 2017).
• Link to Dual Groups: The Satake isomorphism, generalized in the Kac–Moody setting, still relates to the (categorical or representation-theoretic) side via the Langlands dual group, in the sense that $R[[Y]]^{W^v}$ can be identified with formal character functions on the torus of the Langlands dual group, subject to Weyl invariance.
• Compatibility with Geometric Satake and Categorification: The theory connects to geometric Satake equivalence for affine and Kac–Moody groups (Bouthier et al., 13 Oct 2025), with hyperbolic localization and the identification of intersection cohomology complexes with highest weight modules of the dual group.

5. Comparative Analysis and Future Directions

• Comparison of Completions: While the algebraic Bernstein–Lusztig approach is algebraically efficient, the geometric function-theoretic approach (Weyl almost finite support) is more directly interpretable in terms of harmonic analysis and geometric representation theory (Hébert et al., 20 Oct 2025). In finite rank or well-behaved types, the completions can coincide, but in general, one cannot universally relate them by inclusion.
• Role of Support Conditions: The choice of support condition directly affects the admissibility of infinite series in the completed algebra and thus which elements (and, via duality, which objects on the dual group side) are visible in the harmonic analysis.
• Center and Automorphic Aspects: The existence of a "large center" matching the spherical Hecke algebra is specifically designed to support automorphic and harmonic analysis developments for Kac–Moody groups, paralleling the critical role of the center in the classical setting.
• Further Applications: The algebraic architecture developed here is anticipated to interact strongly with advancements in quantum and geometric representation theory (for example, quantum Satake (Golyshev et al., 2011)) and in applications to moduli problems and categorification schemes (Brundan et al., 2019, Bouthier et al., 13 Oct 2025).

6. Summary

The Kac–Moody Satake isomorphism, as formulated through appropriate completions of the Iwahori–Hecke algebra and identification of their centers with the spherical Hecke algebra, extends the fundamental duality between harmonic analysis on $p$-adic (or loop) groups and the representation theory of dual groups to the infinite-dimensional context. The precise nature of the algebraic completion—specifically, the choice of support condition—determines the scope and compatibility with geometric, functional, and representation-theoretic structures, leading to a framework robust enough to encode the depth and complexity of the Kac–Moody setting (Abdellatif et al., 2017, Hébert et al., 20 Oct 2025).