Standard Spherical Hecke Algebra

• Standard spherical Hecke algebra is a commutative convolution algebra defined on bi-invariant, compactly supported functions over p-adic and Kac–Moody groups.
• The Satake isomorphism bridges the algebra with invariant functions on the coweight lattice, enabling precise harmonic analysis and explicit representation-theoretic calculations.
• Its structured double coset indexing and combinatorial properties underpin deep applications in number theory, quantum field theory, and arithmetic geometry.

The standard spherical Hecke algebra is a commutative convolution algebra constructed from bi-invariant compactly supported functions on a $p$-adic group (or appropriate generalizations, such as Kac-Moody groups over local fields), with deep ties to harmonic analysis, representation theory, number theory, quantum field theory, and arithmetic geometry.

1. Definition and Algebraic Construction

Given a (reductive or Kac–Moody) group %%%%1%%%% over a non-archimedean local field $F$ (with finite residue field of cardinality $q$), and a distinguished maximal open compact subgroup $K$ (often hyperspecial), the standard spherical Hecke algebra $\mathcal{H}(G,K)$ is defined as the convolution algebra of $K$-bi-invariant, compactly supported functions on $G$ with values in a commutative ring $R$. Convolution is taken via

$(f * g)(x) = \int_G f(y)g(y^{-1}x)\,dy$

or, more generally for Kac–Moody groups (Gaussent et al., 2012), via

$(f * g)(x, y) = \sum_z f(x, z) g(z, y)$

where $x, y, z$ are points in the hovel or double cosets $K\backslash G/K$.

The support is typically controlled using the Cartan decomposition: for reductive groups, the double coset space $K\backslash G/K$ is indexed by dominant coweights (or partitions for groups like $\mathrm{GL}_n$); for Kac–Moody groups, a semigroup $G^+$ is defined by positivity conditions in a combinatorial masure (hovel), and the double cosets $K\backslash G^+/K$ correspond to a semigroup $Y^{++}$ in the coweight lattice (Gaussent et al., 2012).

2. Key Structural Properties and Commutativity

The spherical Hecke algebra is always commutative. Classically, Gelfand’s trick uses the existence of an involution (antipode) on $G$ fixing $K$ and acting as inversion on a split torus (Gehrmann, 2020), which intertwines the convolution—yielding commutativity of the algebra. In derived or graded settings, this involution further produces graded commutativity, i.e.,

$h_1 * h_2 = (-1)^{ij} h_2 * h_1$

for $h_1, h_2$ in cohomological degrees $i, j$ (Gehrmann, 2020).

Structure constants $m_{\lambda,\mu}(\nu)$ in

$$c_\lambda * c_\mu = \sum_\nu m_{\lambda,\mu}(\nu) c_\nu$$

are computed by counting galleries (“triangles” or Hecke paths) in the hovel (building), and are polynomials in the residue field parameters $q_i, q'_i$ with integer coefficients governed by the geometry of the standard apartment (Gaussent et al., 2012).

3. The Satake Isomorphism and Harmonic Analysis

The Satake isomorphism identifies the spherical Hecke algebra $\mathcal{H}$ (with $R$ coefficients) with the ring of $W$-invariant functions on the coweight lattice group algebra $R[Y]$ (or a formal completion $R[[Y]]$). For Kac–Moody groups: $\mathcal{H}_R \xrightarrow{S} R[[Y]]^{W^v}$ with $$S(c_\lambda) = \sum_{\nu \le_{Q^\vee} \lambda} n_\lambda(\nu) e^\nu$$ (Gaussent et al., 2012).

This identification, disciplined by half-density twists (reflecting parameter measures), underpins harmonic analysis, connects to the dual group representation theory, and offers explicit formulae for spherical functions (Macdonald’s formula), Plancherel measure, and branching rules (see Section 4).

In motivic contexts, each Hecke algebra element and its associated orbital integral is encoded as a motivic constructible function, allowing transfer principles to prove identities (Langlands–Shelstad fundamental lemma, etc.) in broad settings (Casselman et al., 2016).

4. Representation-Theoretic and Combinatorial Formulae

The spherical Hecke algebra’s basis elements (characteristic functions on double cosets) interact under convolution with explicit structure constants. For finite groups (e.g., $\mathrm{GL}_n$ over a finite field), these correspond to Hall polynomials $h_{\lambda,\nu}^\mu$.

In double coset formulations, each coset is indexed by a combinatorial symmetric coweight (multiset of integers, block-diagonal exponents), and Pieri-type rules specify the effect of elementary multiplication:

• Pieri rule: $a_i - 1 \le b_i \le a_i + 1$ (Jin, 27 Jun 2024).
• Dual Pieri rule: $a_{i+1} \le b_i \le a_{i-1}$ for horizontal strip modifications.

Closed formulas for convolution coefficients $h_{\lambda,\nu}^\mu$ emerge from reduction algorithms that manipulate block-diagonal matrix representatives (Jin, 27 Jun 2024).

Alternator and character formulae link matrix coefficients (e.g., spherical vectors in principal series) to Weyl group sums and dual group characters, interpolating among Demazure, Weyl, and Casselman–Shalika formulas (Brubaker et al., 2015): $$\mathcal{A}\left(\pi^{\lambda + 2\rho_\epsilon^\vee - \rho^\vee} \prod ... \right)$$ with the alternator $\mathcal{A}$ summing over signed Weyl group actions.

5. Partition Functions, Motivic Transfer, and Fundamental Lemmas

Partition functions $P(G, V_R, \theta, E, q)$, defined as determinants like

$$P(E, q) = \prod_{\beta \in \Psi^+}(1 - q e^\beta)^{-1}$$

encode representation theoretic data (characters, weight multiplicities, L-functions) and specialize to Kostant's $q$-partition function. Twisted versions accommodate non-connected dual groups and endoscopic transfer, governing branching rules and inverse Satake coefficients (Casselman et al., 2016).

Motivic integration translates identities about orbital integrals, transfer factors, and Satake transforms into constructible functions, enabling powerful transfer results and uniform proofs (e.g., Langlands–Shelstad lemma, Kato–Lusztig inverse Satake, explicit branching rules) (Casselman et al., 2016).

Arithmetically, the spherical Hecke algebra mediates conjectural identities such as the arithmetic fundamental lemma (AFL): intersection numbers from Hecke correspondences on moduli (Rapoport–Zink spaces) are matched with derivatives of weighted orbital integrals under base change, with transfer factors (Li et al., 2023, Chen, 10 Feb 2025). Commutativity conjectures (essential for the well-posedness of Hecke operators on cycles) and large kernel conjectures for derivative maps are established in low-rank settings (Li et al., 2023).

6. Quantum Field Theory and Special Limits

In the context of $\mathcal{N}=2$ gauge theory, the spherical Hecke central (SHc) algebra acts on Nekrasov instanton partition functions, encoding both integrability and AGT correspondence. In the Nekrasov–Shatashvili (NS) limit (one equivariant parameter zero), the SHc generators become shift operators acting on Bethe roots, reproducing Bethe-like equations and TBA structures (Bourgine, 2014).

The algebraic structure of the instanton partition functions, viewed as sums over Young diagrams, admits an action of rescaled SHc generators. In this regime, phenomena such as Kanno–Matsuo–Zhang transformations and Mayer cluster expansions connect algebra action, diagram combinatorics, and nonlinear integral equations.

Central extensions of the spherical Hecke algebra become crucial for faithfully representing Virasoro and Heisenberg symmetries in gauge-conformal state constructions; these constructions are shown to be equivalent to conformal blocks in Liouville theory, with precise scaling and regular/irregular limits characterizing Argyres–Douglas theories (Rim et al., 2016).

7. Generalizations and Geometric Interpretations

The algebra extends naturally to:

• Spherical Hall algebras attached to arithmetic curves ($\overline{\operatorname{Spec}(\mathcal{O}_K)}$) for class number one number fields, which are shown to be isomorphic to Paley–Wiener shuffle algebras deformed by kernels constructed from Hecke $L$-functions (Li et al., 26 Nov 2024).
• Derived and graded versions, where cohomological structure and transfer maps (push–pull via Levi subgroups) create refined invariants for mod $p$ representation theory and automorphic cohomology (Ronchetti, 2018, Gehrmann, 2020).
• Infinite-dimensional Kac–Moody groups over local fields, where the hovel (ordered masure) and unequal parameters necessitate almost finite support and careful analysis of convolution structure (Gaussent et al., 2012).

Applications span the Langlands program, explicit harmonic analysis, trace formula comparisons, quantum field theory, intersection theory on arithmetic models, and the representation theory of infinite-dimensional and non-abelian groups.

Table: Key Structures in the Standard Spherical Hecke Algebra

Group/Space Type	Double Coset Indexing	Algebra Structure and Isomorphism
Reductive $G$ over $F$	Dominant coweights	Satake isomorphism: $\mathcal{H} \simeq R[Y]^W$
Kac–Moody $G$ over $F$	$Y^{++}$ in lattice	Satake: $H_R \simeq R[[Y]]^{W^v}$
$\mathrm{GL}_n$ over ring	Symmetric coweights	Block-diagonal, Pieri rules, Hall polynomials
Derived setting	Cohomology classes	Graded commutativity (Gehrmann, 2020)

Conclusion

The standard spherical Hecke algebra is a fundamental object encoding bi-invariant harmonic analysis, representation theory, and deep arithmetic/geometric phenomena. Its commutative (or graded-commutative) algebraic structure arises via convolution, governed by double coset combinatorics (Cartan decomposition, symmetric coweights, block-diagonal matrices, or lattice invariants). The Satake isomorphism relates this algebra to symmetric functions or completions of group algebras, facilitating explicit calculations and transfer of identities (fundamental lemma, branching formulas, etc.), and serving as a bridge across number theory, automorphic forms, dual group representation theory, and quantum field theory. Generalizations to derived, motivic, quantum, infinite-dimensional, and arithmetic settings continue to reveal new applications and research directions (Gaussent et al., 2012, Casselman et al., 2016, Gehrmann, 2020, Li et al., 2023, Li et al., 26 Nov 2024, Chen, 10 Feb 2025).