Completed Iwahori-Hecke Algebra

• Completed Iwahori–Hecke algebra is a topologically enlarged convolution algebra that accommodates infinite sums in analytic and representation-theoretic contexts.
• It unifies various completions—Schwartz, C*-algebra, geometric, and algebraic—across p-adic, affine, and Kac–Moody group settings.
• Key algebraic structures such as the Satake isomorphism, centers, and quantum metric properties are generalized to support deep links with harmonic analysis and categorical representation theory.

A completed Iwahori-Hecke algebra is a topologically enlarged or formally completed version of the Iwahori–Hecke algebra, the convolution algebra of Iwahori-bi-invariant functions on a p-adic group, or their analogues for Kac-Moody groups. Completions are constructed to accommodate infinite sums arising from analytic, representation-theoretic, or geometric contexts where ordinary compact support is too restrictive, such as for affine, double-affine, or Kac–Moody settings. The central structures, Satake isomorphisms, and centers of completed Iwahori–Hecke algebras often generalize finite-dimensional theory and support deep connections with harmonic analysis, invariant theory, and categorical representation theory.

1. Constructions and Notions of Completion

There are several frameworks for completion, reflecting different contexts:

• Classical p-adic groups: For G a reductive p-adic group, the Iwahori–Hecke algebra $H = C_c(I \backslash G / I)$ is first constructed as the space of compactly supported, Iwahori–bi-invariant functions with convolution. Its completions include:
  • Schwartz completion $S(G)^s$: The algebra obtained by completing relative to a family of seminorms that measure decay with respect to a length function, making it a Fréchet algebra suitable for harmonic analysis on smooth representations (Solleveld, 2016).
  • Reduced $C^*$-algebra completion $C_r^*(G)^s$: The norm closure of $H$ in the algebra of bounded operators on $L^2(G)$, used for studying unitary representations and operator K-theory.
• Kac–Moody and affine settings: For split Kac–Moody groups over non-archimedean local fields, there are two main approaches:
  • Geometric completion $\widehat{\mathcal{H}}$: The algebra of Iwahori–bi-invariant functions on $G$ (or a suitable semigroup, e.g., $G^+$ in the affine Kac–Moody loop group setting) with coefficients supported on subsets of the (co)weight lattice $Y^+$ that are “Weyl almost finite.” Here, for $E \subset Y$,

  $$E\textrm{ is almost finite }\iff \exists\textrm{ finite }J\subset Y \textrm{ such that }E\subset \bigcup_{\nu\in J}(\nu - Q_+^\vee),$$

  and $E$ is Weyl almost finite if its $W_0$-saturation is almost finite (Hébert et al., 20 Oct 2025). - Algebraic completion $\widetilde{\mathcal{H}}$: The completion given algebraically via the Bernstein–Lusztig presentation, allowing infinite sums in the generators, without enforcing geometric support conditions. The elements are general formal series $\sum_{\lambda} h_\lambda T_\lambda$, with $h_\lambda$ in finite sums over $W_0$ (Hébert et al., 20 Oct 2025).

For p-adic loop groups and double affine Hecke algebras, the completion involves passing to "semi-infinite" completions, enabling the convolution algebra to be well-defined on infinite-dimensional spaces indexed by the Tits cone or similar semi-groups (Braverman et al., 2014, Muthiah, 2015).

2. Centers, Satake Isomorphisms, and Invariant Rings

A central structural property of (completed) Iwahori–Hecke algebras is the determination of their centers and the role of the Satake isomorphism:

• Bernstein's presentation: For $H_I$, the Iwahori–Matsumoto algebra, every element is written as a linear combination of elements $\theta_x T_w$ ($x$ in the coweight lattice, $w$ in $W$), with $\theta_x$ normalized translation elements. The crucial commutation relation (involving simple reflections $s$ and $\theta_x$) controls noncommutativity:

$$T_s \theta_x - \theta_{s(x)} T_s = (1-q^{-1})\frac{\theta_x - \theta_{s(x)}}{1-\theta_{-a^\vee}}$$

where $a^\vee$ is the simple coroot (Savin, 2012).

• Satake isomorphism: Passing to the double-coset algebra $C_c(I\backslash G/K)$ for a maximal compact $K$, there is a module isomorphism with the group ring $C[X]$, and the Satake map induces an isomorphism

$\mathcal{H}_K \simeq C[X]^W.$

For completed algebras, the same structure persists: the center of the completed Iwahori–Hecke algebra remains isomorphic to the completed spherical Hecke algebra, i.e., the $W$-invariants in the completed group algebra $C[[X]]^W$.

• Kac–Moody setting: The center of the completed algebra $\widehat{\mathcal{H}}$ is isomorphic to Looijenga's invariant ring $C[[Y^+]]^{W_0}$, matching the spherical Hecke algebra under the Satake isomorphism for Kac–Moody groups (Hébert et al., 20 Oct 2025). Notably, if $W_0$ is infinite, purely algebraic completions without geometric support can have a trivial, or much too small, center compared to the completion with Weyl almost finite support.

3. Structure and Presentation: Bases, Order, and Filtration

Across all settings, the completed Iwahori–Hecke algebra admits concrete presentations and bases:

• Double coset basis: In the p-adic loop group setting, the completed convolution algebra $H(G^+, I)$ is described using a basis of indicator functions supported on double cosets $I x I$, indexed by a semigroup $W_T = W \ltimes T$ (T the Tits cone) (Muthiah, 2015). The structure constants in this basis are polynomials in the residue field cardinality $q$.
• Length filtrations: Many completions are equipped with a length filtration induced by the word length in the Weyl group or its double affine analogues, yielding natural quantum metric structures when viewed as operator algebras (Klisse et al., 11 Aug 2025). In the right-angled Coxeter system case, a Haagerup-type inequality involving the length filtration and a combinatorial condition on the Coxeter diagram (absence of induced squares) determines metric properties and compactness.
• Order structures: The set $W_T$ admits a "double affine Bruhat order"—initially a preorder, shown to be a partial order whenever the length filtration is well-behaved; this partial order is essential for structural understanding and for the development of Kazhdan–Lusztig theory in these double affine settings (Muthiah, 2015).
• Combinatorial presentations: In both finite and infinite cases, completed Iwahori–Hecke (and related Hecke endomorphism) algebras admit presentations by generators and relations, often reflecting filtrations or cell decompositions associated with the underlying Coxeter system or Weyl group (Du et al., 2015, Hébert et al., 20 Oct 2025).

4. Analytic and Geometric Features: Quantum Metrics and Topological Completions

The completed Iwahori–Hecke algebra underpins rich analytic and geometric structures:

• Quantum metric spaces: For certain right-angled Coxeter systems of finite rank, the completed Iwahori–Hecke algebra (in the $C^*$-algebra norm) with a canonical filtration satisfies Rieffel’s compact quantum metric space criteria if and only if the Coxeter diagram’s complement contains no induced squares (Klisse et al., 11 Aug 2025). The resulting spectral triple is defined using the Dirac operator corresponding to the length filtration and yields a compact quantum metric space $(C^*_{r,q}(W), L_S^{(q)})$, where $L_S^{(q)}$ is defined via commutators with the Dirac operator.
• Gromov–Hausdorff propinquity and deformation: As $q\to 1$, the quantum metric structure on the Iwahori–Hecke algebra converges to that of the group $C^*$-algebra $C^*_r(W)$ in Latrémolière’s quantum Gromov–Hausdorff propinquity. This establishes topological and metric continuity of the deformation from group algebras to Hecke algebras (Klisse et al., 11 Aug 2025).
• Plancherel theory and K-theory: The Schwartz and $C^*$-completions admit Plancherel isomorphisms with algebras of smooth or continuous $W$-invariant operator-valued functions on a torus, allowing the analytic machinery of harmonic analysis and $K$-theory to be extended to the completed Iwahori–Hecke context. Morita equivalence often passes to these completions, simplifying categorical and K-theoretic computations for reductive and affine cases (Solleveld, 2016).

5. Representation Theory and Categorical Aspects

The completed Iwahori–Hecke algebra is fundamental in modular, categorical, and geometric representation theory:

• Principal series representations and Satake parameters: For Kac–Moody groups, principal series for the completed algebra are induced from one-dimensional characters of the torus subalgebra. They decompose as direct sums (or generalizations) of weight spaces indexed by the (vectorial) Weyl group, with explicit intertwining operators constructed in the completed algebra (Hébert, 2018).
• Centers and block theory: Interpolating completions via universal centralizing algebras connect to block classification and the paper of centers in stable and deformed settings. For type $A$, the stable center interpolates the centers of all $H_n(q)$, is isomorphic to a symmetric function algebra via evaluation at Jucys–Murphy elements, and block decompositions correspond to content-matching multisets (Ryba, 2022).
• Ext-algebras and derived completions: In the modular and derived context, the pro-$p$ Iwahori–Hecke Ext-algebra $E^*$ (graded by extension length) is finitely presented as a noncommutative algebra and encodes the derived structure of the category of smooth mod-$p$ representations. Its explicit finite presentation is critical for constructing functors and spectral sequences linking smooth representations to module categories for completed Hecke algebras (Bodon, 2 May 2024).
• Categorical actions and functorial liftings: The completed Iwahori–Hecke algebra acts categorically via twisting, shuffling, and Zuckerman functors on graded categories $O$, leading to explicit graded character formulae and capturing the combinatorial structure of Kazhdan–Lusztig theory in a categorical setting (Fang et al., 5 Sep 2024).

6. Variants, Comparison of Completions, and Future Directions

The construction of a completed Iwahori–Hecke algebra is sensitive to choices of support, presentation, and topologies:

• Geometric vs algebraic completions: The geometric completion $\widehat{\mathcal{H}}$ (with Weyl almost finite support) yields the expected large center matching Looijenga's invariant ring and supports a Satake isomorphism to the spherical Hecke algebra, crucial for the Kac–Moody and double affine contexts (Hébert et al., 20 Oct 2025). The algebraic completion $\widetilde{\mathcal{H}}$ defined via pure formal series in generators can significantly under-approximate the center when $W_0$ is infinite.
• Functional-analytic vs formal completions: Schwartz and $C^*$-completions in the analytic setting are designed for harmonic analysis and operator theory, while formal completions support the manipulation of infinite linear combinations on algebraic grounds.
• Extension to higher rank and generalizations: Techniques and results for completions in rank one (e.g., explicit Ext-algebra presentations) serve as a model for future development in higher rank and for understanding derived and block-theoretic phenomena in the representation theory of $p$-adic and Kac–Moody groups.
• Further applications: Completed Iwahori–Hecke algebras open perspectives in noncommutative geometry, topological invariants of operator algebras, categorification programs, as well as explicit calculations in modular and derived representation theory. Their rich structure bridges the worlds of harmonic analysis, algebraic and geometric representation theory, and quantum metric geometry.

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The completed Iwahori–Hecke algebra serves as a unifying structure that retains the algebraic, analytic, and categorical features of the Iwahori–Hecke algebra in infinite-dimensional, topological, categorical, and noncommutative geometric contexts. Its construction, presentation, and central properties—especially regarding the center, Satake isomorphism, and quantum metric structures—are foundational across modern arithmetic, representation-theoretic, and operator-algebraic research.