Topological View of Cherednik's Inner Product

• The paper demonstrates that Cherednik's inner product can be interpreted through geometric constructions in symplectic topology and equivariant cohomology of moduli spaces.
• It establishes a canonical correspondence between the algebraic constant term formula in DAHA and the intersection pairings observed in affine Springer and Hitchin fibers.
• The approach unifies algebraic and topological methods by matching Floer theoretic pairings in cotangent bundles with the abstract Shapovalov form, paving the way for further generalizations.

The topological interpretation of Cherednik's inner product focuses on realizing the algebraic sesquilinear pairing—central in the theory of (double-affine) Hecke algebras—via geometric constructions rooted in symplectic topology and the equivariant cohomology of certain moduli spaces. This approach elucidates the nature of the inner product as a nondegenerate, symmetric form, and bridges the abstract representation theory with tangible enumerative invariants of spaces such as cotangent bundles and affine Springer fibers (Oblomkov et al., 2014, Gao et al., 9 Nov 2025).

1. Cherednik Algebras and Their Polynomial Representations

The double-affine Hecke algebra (DAHA) $\mathbb{H}_{\kappa}$ of type $A_{\kappa-1}$ is generated by the finite Hecke generators $T_1,\ldots,T_{\kappa-1}$ and Laurent generators $X_1^{\pm 1},\ldots,X_{\kappa}^{\pm 1}$, $Y_1^{\pm 1},\ldots,Y_{\kappa}^{\pm 1}$, with relations encoding the combinatorics of braids and multiplicative "double-affine" structure. The polynomial representation $P_{\kappa}$ is an induced module constructed as

$$P_{\kappa} = \mathbb{H}_{\kappa} \otimes_{\mathbb{H}_{fin}\otimes \mathbb{Z}[Y_i]} \mathbb{Z}[s^{\pm 1}, c^{\pm 1}][X_1^{\pm 1},...,X_{\kappa}^{\pm 1}],$$

with the finite subalgebra acting by specified scalars.

Cherednik's inner product $\langle-, - \rangle_{Cher}$ is defined algebraically by three core properties:

• Star–bilinearity: $$\langle r \cdot f, g \rangle = r \langle f, g \rangle = \langle f, r^* \cdot g \rangle$$ for ring scalars $r$ under involution $* : s \mapsto s^{-1}$, $c\mapsto c^{-1}$.
• Unitarity: $\langle Hf, g \rangle = \langle f, H^*g \rangle$, for $H^*$ the adjoint under involution.
• Normalization: $\langle 1,1 \rangle = 1$.

A core formula expresses $\langle f,g \rangle_{Cher}$ as a constant term pairing involving the $q$-Pochhammer symbol:

$$(\star)\qquad \langle f,g \rangle_{Cher} = \text{CT}_x \left[ f(x^{-1})g(x) \prod_{i\neq j} \frac{(x_i/x_j;q)_{\infty}}{(t x_i/x_j;q)_{\infty}} \right],$$

where $\text{CT}_x$ extracts the degree zero term.

2. Geometric Realization via Affine Springer and Hitchin Fibers

The representation-theoretic modules over Cherednik algebras find concrete geometric realizations within the equivariant cohomology of affine Springer fibers and Hitchin moduli spaces (Oblomkov et al., 2014). For a regular semisimple $\gamma \in \mathfrak{g}(F)$, the affine Springer fiber

$$Sp_{\gamma} = \{ gI \in \mathrm{Fl} : \mathrm{Ad}(g^{-1})\gamma \in \mathrm{Lie}\,I \}$$

admits an action of the torus $\mathbb{C}^*_{loop} \times \mathbb{C}^*_{dil}$ and centralizer $S_\gamma$. Its equivariant cohomology $H^*_{\mathbb{C}^*}(Sp_\gamma)$ forms a module over $\mathbb{Q}[u]$, paralleling the role of central elements in the Cherednik setting.

The perverse Leray filtration $P_\bullet$ on the cohomology of Hitchin fibers (moduli spaces of $G$-Higgs bundles $(E, \varphi)$) provides a grading matched to the polynomial representation grading, and the product formula $M_a \cong [Sp_\gamma / S_\gamma]$ relates global and local geometry.

3. The Intersection Pairing and Frobenius Structure

The cup product on $H^*(Sp_\gamma)^{S_\gamma}$ yields a graded algebra with the associated graded $Gr^P_*(H^*(Sp_\gamma)^{S_\gamma})$. The top degree piece $Gr^P_{2N}$ is one-dimensional, and the extraction of the coefficient of the fundamental class $[Sp_\gamma]$ defines a symmetric, nondegenerate pairing:

$(x, y) := \ell(x \cup y),$

where $\ell(x)$ is the coefficient of $[Sp_\gamma]$ in $x \cup 1$.

This intersection pairing satisfies the contravariance properties:

• $(\eta \cdot x, y) = -(x, \eta \cdot y)$,
• $(\xi \cdot x, y) = (x, \xi \cdot y)$,
• $(w \cdot x, w \cdot y) = (x,y)$, directly mirroring the algebraic symmetries of the Cherednik form.

A key result is that the cup product pairing, after specialization $u = 1$ of the equivariant parameter, coincides with the Shapovalov form $J(-,-)$ on the spherical module $L_\nu(triv)$, thus identifying the algebraic inner product with the top-degree intersection pairing perverse-gradedly.

4. Floer-Theoretic Interpretation: Holomorphic Strips in Cotangent Bundles

An alternative topological interpretation is achieved via higher-dimensional Heegaard Floer homology (HDHF). Within the cotangent bundle $T^*T^2$, for fixed basepoints $q_1,\ldots,q_\kappa$ and a conormal Lagrangian $L_\alpha$ associated to a homotopically nontrivial loop, the wrapped Floer homology

$$HW(T^*T^2, \{q_i\}, \alpha) \cong P_\kappa \otimes_{\mathbb{Z}[s^{\pm 1},\ldots]} \mathbb{Z}\llbracket \hbar \rrbracket,$$

identifies Floer generators with monomials in the polynomial representation.

The Floer pairing is defined as

$$\langle x, y \rangle_{Floer} = \sum_{u \in \mathcal{M}(x,y)} (-1)^{\mu(u)} q^{\omega(u)},$$

with $\mathcal{M}(x,y)$ the moduli space of rigid holomorphic strips $u$, $\mu(u)$ the Maslov index, and $q^{\omega(u)}$ the Novikov-weight determined by the symplectic area. Only index-zero strips contribute in this setting.

Explicit matching of algebraic and Floer pairings is demonstrated: for the isomorphism $\Phi : P_\kappa \cong HW(T^*T^2, \{q_i\}, \alpha)$,

$$\langle \Phi(f), \Phi(g) \rangle_{Floer} = \langle f, g \rangle_{Cher},$$

with normalization and adjoint properties rigidly preserved. The Floer-side pairing is realized via intersection theory, and the algebraic constant term formula emerges naturally from enumerative strip counting.

5. Key Lemmas and Formal Properties

The identification of Cherednik's inner product with geometric and symplectic pairings relies on a set of formal correspondences:

• Adjoint Property: For any DAHA generator $H$,

$$\langle H\cdot x, y \rangle_{Floer} = \langle x, H^*\cdot y \rangle_{Floer},$$

corresponding to strip reversal and algebraic involution.

• Hecke Relations: Boundary degenerations in Floer theory equate to the quadratic Hecke relations, e.g.,

$T_i^2 = (s - s^{-1})T_i + 1.$

• Nondegeneracy: The chain-level Floer pairing is nondegenerate on homology under Novikov coefficients, mirroring the algebraic situation.

6. Contextual Significance and Broader Connections

The topological approach unifies Cherednik’s inner product with intersection theory on moduli spaces and enumerative counts of holomorphic curves. The symplectic interpretation clarifies previously opaque algebraic constructions—such as constant term and contour integral definitions—and situates the inner product within the broader framework of Lagrangian Floer theory and Fukaya categories. Explicit geometric realization via cotangent bundles and surface multisections (as in the work of Auroux, Nadler–Zaslow, and Honda–Tian–Yuan) connects representation theory to symplectic geometry in a mathematically rigorous manner (Gao et al., 9 Nov 2025).

A plausible implication is that further generalizations of this interpretation may systematize the topology-algebra correspondence for other types of Hecke algebras and moduli problems, reinforcing the geometric representation-theoretic paradigm.

7. Dimension Formulas and Rank-Two Examples

Explicit formulas for the dimensions of spherical modules $L_\nu(triv)$ in both algebraic and geometric realizations are established (Oblomkov et al., 2014). For $\nu = d/m$ with $(d,m) = 1$ and rank $r$,

$$\dim L_\nu(triv) = d^r \sum_{w \in W_\nu \backslash W / W_P} \dim H^*(Hess^w_P)^{Ann(\lambda^w)}.$$

For rank two cases:

• Type A$_2$, $\nu = d/3$, $\dim = d^2$,
• Type C$_2$, $\nu = d/4$, $\dim = 4d^2$,
• Type $^2$A$_2$, $\nu = d/2$, $\dim = 3d^2$,

with the cup-pairing reproducing the positive-definite Shapovalov form on the respective spherical modules.

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The identification of Cherednik's inner product with geometric pairings—via intersection theory and holomorphic strip counts—demonstrates that the otherwise abstract algebraic form arises intrinsically from fundamental topological and symplectic structures. The connection is explicit, canonical, and robust under a range of geometric contexts.