Periodic Kazhdan–Lusztig Polynomials in Affine Theory

• Periodic Kazhdan–Lusztig polynomials are generalized invariants associated with affine Weyl groups that encode character data and extension spaces in representation theory.
• They are computed using combinatorial methods such as the doubled Bruhat graph and recursive inversion formulas, enabling explicit evaluations in low-rank cases.
• Their symmetry properties and Koszul duality offer deep insights into modular and affine representation theory, influencing conjectures like Lusztig’s.

A periodic Kazhdan–Lusztig polynomial is a generalization of the classical Kazhdan–Lusztig polynomial associated to the affine Weyl group. It encodes deep information about characters and extension spaces in modular and affine representation theory. The periodic variant is distinguished by the presence of a translation component in the affine Weyl group, and is governed by refined combinatorial and categorical structures that reflect periodic phenomena in equivariant representation categories over fields of positive characteristic and affine Kac–Moody contexts.

1. Structural Framework: Affine Weyl Groups and Bruhat Order

Let $G$ be a connected, simply-connected complex reductive group, with maximal torus $T$ and finite Weyl group $W = N_G(T)/T$. The root system is $\Phi \subset X^*(T)$, with positive roots $\Phi_+$ and simple reflections $s_\alpha$ for each $\alpha \in \Phi$. The untwisted affine Weyl group is $W_{\mathrm{aff}} = W \ltimes Q^\vee$, where $Q^\vee$ denotes the coroot lattice. $W_{\mathrm{aff}}$ acts affinely on the real Cartan subalgebra and is generated by simple and affine reflections.

The Bruhat order, a partial order, extends from $W$ to $W_{\mathrm{aff}}$, characterized by covering relations: for a real affine root $\beta$, $w < w s_\beta$ if and only if $w^{-1}(\beta) \in \Phi_{\mathrm{aff},+}$.

The length function $\ell: W_{\mathrm{aff}} \to \mathbb{N}$ and the associated dot-action $w \cdot \lambda := w(\lambda + \rho) - \rho$ are key elements in the categorical and combinatorial constructions underpinning periodic Kazhdan–Lusztig theory (Watanabe et al., 2016, Riche et al., 23 Nov 2025).

2. Definition of Periodic Kazhdan–Lusztig and $R$-Polynomials

Periodic Kazhdan–Lusztig polynomials $Q_{y, w}(q)$ (notation $P^\pi_{y,w}(q)$ or $P^{\mathrm{per}}_{y, w}(v)$ in the literature) are associated to pairs $y, w \in W_{\mathrm{aff}}$. They satisfy the recursion and axioms of parabolic Kazhdan–Lusztig theory:

• $Q_{y,w}(q) = 0$ unless $y \leq_{\infty} w$ in the semi-infinite Bruhat order.
• Normalization $Q_{y,y}(q) = 1$.
• Degree bound: $$\deg(Q_{y,w}(q)) \leq (\ell^{\infty}(w)-\ell^{\infty}(y)-1)/2$$.

Periodic $R$-polynomials $\mathcal{R}_{y,w}(q)$ are defined via the Hecke algebra $\mathcal{H}_{\mathrm{aff}}$, with bar involution $\Psi$:

$$\Psi(\widetilde{T}_w) = q^{-\ell^{\infty}(w)} \sum_{y \leq_{\infty} w} (-1)^{\ell^{\infty}(y,w)} \mathcal{R}_{y,w}(q) \widetilde{T}_y$$

Here, $\widetilde{T}_{x t_\lambda} = T_x X^\lambda$ denotes the Bernstein–Lusztig basis, and $\ell^{\infty}$ is the semi-infinite length function.

The polynomials $Q_{y, w}(q)$ are uniquely determined by the collection of $R$-polynomials via an inversion formula, recursively expressing $Q_{y,w}$ in terms of $\mathcal{R}_{y,w}$ and lower order $Q_{z,w}$ for $y <_{\infty} z < w$.

3. Combinatorial Computation: Doubled Bruhat Graph and Path Formula

The doubled Bruhat graph (DBG) is a graphical structure with vertices indexed by $W$. For each positive root $\beta \in \Phi_+$ and pair $y', w' \in W$ with $w' = y' s_\beta$, one creates directed edges:

• Bruhat edge: $y' < w'$, labeled $d = \frac{1}{2}(\ell(y', w') + 1)$.
• Quantum edge: $y' > w'$, labeled $$d = \frac{1}{2}(\ell(y', w') + 2\langle \rho, \beta^\vee \rangle + 1)$$.

A reflection order $\beta_1 \prec \beta_2 \prec \cdots$ is fixed on $\Phi_+$, and a label-increasing path of length $k$ from $y'$ to $w'$ is a directed sequence of edge traversals with non-decreasing root labels.

Given $y, w \in W_{\mathrm{aff}}$, one considers the set $P^\prec(y, w)$ of label-increasing paths in the semi-infinite Bruhat graph. Each path $\pi$ has:

• $\ell(\pi)$: total number of edges,
• $\deg(\pi)$: sum over the edges of degree increments measuring translation and reflection contributions,

and the periodic $R$-polynomial achieves the explicit combinatorial formula (Watanabe et al., 2016):

$$\mathcal{R}_{y, w}(q) = \sum_{\pi \in P^\prec(y, w)} q^{\deg(\pi)} (q - 1)^{\ell(\pi)}$$

This formula offers closed computations once all paths in $P^\prec(y, w)$ are enumerated, weighted by their combinatorial data.

4. Recursive Structure, Symmetries, and Koszul Duality

Periodic Kazhdan–Lusztig polynomials adhere to symmetries and dualities intrinsic to the affine Weyl group:

• $$P^{\mathrm{per}}_{x, y}(v) = P^{\mathrm{per}}_{x^{-1}, y^{-1}}(v)$$,
• $$P^{\mathrm{per}}_{x, y}(v) = P^{\mathrm{per}}_{w_0 y, w_0 x}(v)$$, with $w_0$ denoting the longest element in $W$.

The recursion for $Q_{y,w}(q)$ (and its coefficients $a^n_{y,w}$) mirrors combinatorial wall-crossing operations and semisimple algebraic functors (e.g., $\Theta_s$):

• Wall crossing functors correspond to steps in the Bruhat graph and manifest in distinguished triangles in derived categories.
• Via T-equivariant Koszul duality, modular category $\mathcal{O}$ (strongly $B$-equivariant $U(\mathfrak{g})$-modules) is matched to equivariant coherent sheaves and derived dg-categories, where convolution functors emulate combinatorial recursions.

Under this equivalence, the dimensions of extension groups $\dim\, \Ext^n_{Mod(\mathfrak{g},B)}(L(w\cdot 0), \nabla(y\cdot 0))$ coincide with the coefficients of $P^{\mathrm{per}}_{y,w}(v)$ (Riche et al., 23 Nov 2025).

5. Representation-Theoretic Interpretation and Examples

Periodic Kazhdan–Lusztig polynomials capture extension multiplicities in modular representation categories. In the principal block of $Mod(\mathfrak{g},B)$, under Lusztig’s conjecture (valid for $p \gg h$ where $h$ is the Coxeter number):

$\sum_{m \ge 0} \dim\, \Ext^m_{Mod(\mathfrak{g},B)}(L(w \cdot 0), \nabla(y \cdot 0)) v^m = P^{\mathrm{per}}_{y, w}(v)$

For each $n \geq 0$, $\dim\, \Ext^n_{Mod(\mathfrak{g},B)}(L(w \cdot 0), \nabla(y \cdot 0)) = a^n_{y, w}$, where $a^n_{y, w}$ is the coefficient of $v^n$ in $P^{\mathrm{per}}_{y, w}(v)$.

Explicit computations in rank-one ($A_1$), type $A_2$, or $B_2$ demonstrate the formula, with periodic KL values aligning with extension dimensions, and providing concrete data for modular and affine character formulas (Riche et al., 23 Nov 2025, Watanabe et al., 2016).

6. Applications and Implications in Modular and Affine Representation Theory

The periodic Kazhdan–Lusztig polynomials have significant implications for Lusztig’s modular conjecture and for the representation theory of affine Kac–Moody algebras at the critical level:

• Lusztig’s conjecture relates the character multiplicities of simple $G$-modules in positive characteristic (for $p \gg h$) to values $Q_{x,w}(1)$.
• Feigin–Frenkel conjecture asserts that multiplicities of simple constituents in restricted Verma modules are given by $Q_{y,w}(1)$.
• The closed combinatorial formula for $\mathcal{R}_{y,w}(q)$, and thus for $Q_{y,w}(q)$, enables concrete computations and checks in low-rank or small length cases.

A plausible implication is that similar formulas may hold for singular blocks and for modular Soergel bimodules or quantum group categories at roots of unity, potentially broadening the scope of periodic KL polynomial applicability.

7. Computational and Structural Perspectives

The combinatorial closed formula for periodic $R$-polynomials, together with recursive inversion algorithms, makes explicit computation of periodic Kazhdan–Lusztig polynomials feasible. This combinatorial toolkit is foundational for computer implementations, algorithmic verification, and experimental exploration in modular and affine representation theory, including semi-infinite sheaf-theoretic approaches and categorified geometric models.

The bar-involution orthogonality relations and associated Gram–Schmidt algorithms guarantee normalization and integrity in the recursive construction, ensuring that path-theoretic data translate rigorously into polynomial invariants and extension-multiplicity formulas. These results underpin recent developments in equivariant Koszul duality and the geometric theory of modular category $\mathcal{O}$ (Watanabe et al., 2016, Riche et al., 23 Nov 2025).