Comparison of the two specializations of nonsymmetric Macdonald polynomials: at t = 0 and at t = 1

Comparison of the two specializations of nonsymmetric Macdonald polynomials:

at t = 0 and at t = ∞

Satoshi Naito

(Tokyo Institute of Technology) This talk is based on joint works

Basic notation g_af : untwisted affine Lie algebra over C h_af ⊂ g_af : Cartan subalgebra

∆⁺_af ⊂ (h_af)^∗ : positive affine roots c = ∑

i∈I_af a^∨_i α^∨_i ∈ g_af : canonical central element α^∨_i , i ∈ I_af = I ∪ {0} : simple coroots

δ = ∑

i∈I_af a_iα_i ∈ ∆⁺_af : (primitive) null root α_i, i ∈ I_af = I ∪ {0} : simple roots

P = ∑

i∈I Zϖ_i: classical weight lattice

E_i, F_i, i ∈ I_af = I ∪ {0} : Chevalley generators for g_af ϖ = Λ − a^∨Λ , i ∈ I : level-zero fundamental weights

Λ_i, i ∈ I_af = I ∪ {0} : affine fundamental weights W = ⟨r_i | i ∈ I⟩ : finite Weyl group

r_i, i ∈ I : simple reflections

W_af = W ⋉ Q^∨ : affine Weyl group Q^∨ = ∑

i∈I Zα^∨_i Q^∨,+ = ∑

i∈I Z_≥0α^∨_i ρ = 1

2

∑

α∈∆⁺

α = ∑

i∈I ϖ_i ∈ P : Weyl vector

∆⁺ ⊂ h^∗ : positive roots of the finite-dim. subalgebra g (⊂ g )

Semi-infinite Bruhat graph ℓ^∞2 (x), x ∈ W_af : semi-infinite length, defined by

ℓ^∞2 (wt_µ) := ℓ(w) + 2⟨ρ, µ⟩ for w ∈ W and µ ∈ Q^∨

Semi-infinite Bruhat graph is a ∆⁺_af-labeled, directed graph with vertex set W_af whose edges are of the form:

x −→^β r_βx, x ∈ W_af, β ∈ ∆⁺_af, with ℓ^∞2 (r_βx) = ℓ^∞2 (x) + 1.

For x, y ∈ W_af, x ≤^∞₂ y ⇐⇒^def

∃directed path from x to y in the semi-infinite Bruhat graph;

x <^∞

2 y ⇐⇒^def x ≤^∞₂ y and x ̸= y

Semi-infinite LS paths λ ∈ P₊ = ∑

i∈I

Z_≩0ϖ_i : level-zero dominant and regular

η = (x₁ >^∞

2 x₂ >^∞

2 · · · >^∞

2 x_s ;

0 = a₀ < a₁ < · · · < a_s = 1),

where x_k ∈ W_af and a_k ∈ Q, is a semi-infinite LS path of shape λ if for all 1 ≤ k ≤ s − 1, there exists a directed path

x_k = y_t ←^βt y_t−1 ^β← · · ·^t−1 ←^β2 y₁ ←^β1 y₀ = x_k+1 with a_k⟨y_l−1λ, β_l^∨⟩ ∈ Z (1 ≤ ^∀l ≤ t).

0 = a₀ a₁ a₂ a_k−1 a_k a_k+1 a_s−1 x₁λ x₂λ x_kλ xk+1λ x_sλ

a_s = 1

B^∞2 (λ) : the set of all semi-infinite LS paths of shape λ

B₀^∞2 (λ) : connected component of B^∞2 (λ) containing (e ; 0 = a₀, a₁ = 1) For

η = (x₁ >^∞

2 · · · >^∞

2 x_s ; 0 = a₀ < a₁ < · · · < a_s = 1) ∈ B^∞2 (λ) above, we set

ι(η) := x₁ ∈ W_af : initial direction of η, κ(η) := x_s ∈ W_af : final direction of η.

Remark

Extremal weight modules and their crystal bases λ = ∑

i∈I

m_iϖ_i ∈ P₊, m_i ∈ Z_≥⁰ : level-zero dominant U_q(g_af) : quantum affine algebra

V (λ) : extremal weight module of extremal weight λ over U_q(g_af);

this is a module generated by a vector v_λ over U_q(g_af)

with the relation that v_λ is “extremal of weight λ” in the sense:

∃{S_wv_λ}^w_∈^W_af ⊂ V (λ) such that

S_ev_λ = v_λ, and such that for all w ∈ W_af and i ∈ I_af,

if ⟨wλ, α^∨_i ⟩ ≥ 0, then E_iS_wv_λ = 0 and F_i^{(⟨wλ,α∨i ⟩)}S_wv_λ = S_riwv_λ, if ⟨wλ, α^∨_i ⟩ ≤ 0, then F_iS_wv_λ = 0 and E_i^{(−⟨wλ,α∨i ⟩)}S_wv_λ = S_riwv_λ.

B(λ) : crystal basis of V (λ)

u_λ ∈ B(λ) : extremal element corresponding to v_λ;

this element is “extremal of weight λ” in the following sense:

∃{S_wu_λ}^w_∈^W_af ⊂ B(λ) such that S_eu_λ = u_λ, and such that

if ⟨wλ, α^∨_i ⟩ ≥ 0, then e_iS_wu_λ = 0 and f_i^{⟨wλ,α∨i ⟩}S_wu_λ = S_riwu_λ, if ⟨wλ, α^∨_i ⟩ ≤ 0, then f_iS_wu_λ = 0 and e^−⟨_i ^{wλ,α∨i ⟩}S_wu_λ = S_riwu_λ for all w ∈ W_af and i ∈ I_af.

Connected components of the crystal basis B(λ) λ = ∑

i∈I

m_iϖ_i ∈ P₊, m_i ∈ Z≥0 : level-zero dominant

V (λ) : extremal weight module of extremal weight λ over U_q(g_af) v_λ ∈ V (λ) : (generating) extremal vector of weight λ

U_q(g_af) : quantum affine algebra U_q⁺(g_af) : positive part of U_q(g_af)

B(−∞) ∋ u_−∞ : crystal basis of U_q⁺(g_af) U_q⁻(g_af) : negative part of U_q(g_af)

B(∞) ∋ u_∞ : crystal basis of U_q⁻(g_af)

B(λ) : crystal basis of V (λ)

u_λ ∈ B(λ) : extremal element corresponding to v_λ

Par(λ) : the set of I-tuples c₀ = (ρ⁽ⁱ⁾)_i∈I of partitions

such that the length of the partition ρ⁽ⁱ⁾ is ≨ m_i (^∀i ∈ I);

for c₀ = (ρ⁽ⁱ⁾)_i∈I ∈ Par(λ), we set |c₀| := ∑

i∈I |ρ⁽ⁱ⁾|, where |ρ⁽ⁱ⁾| is the size of the partition ρ⁽ⁱ⁾ for i ∈ I. Par(λ) : the set of I-tuples c₀ = (ρ⁽ⁱ⁾)_i∈I of partitions

such that the length of the partitions ρ⁽ⁱ⁾ is ≤ m_i (^∀i ∈ I)

Fact (Kashiwara, Beck-Nakajima) As crystals,

B(λ) ⊂ B(∞) ⊗ { τ_λ}

⊗ B(−∞).

Moreover, every extremal element in B(λ) is connected to an extremal element of the form:

S_c⁻

0u_∞ ⊗ τ_λ ⊗ u_−∞ ∈ B(∞) ⊗ { τ_λ}

⊗ B(−∞), for some c₀ = (ρ⁽ⁱ⁾)_i∈I ∈ Par(λ) (or, c₀ = (ρ⁽ⁱ⁾)_i∈I ∈ Par(λ));

also, we have

S_c⁻

0u_∞ ⊗ τ_λ ⊗ u_−∞ ≡ S_c⁻

0v_λ (mod q).

Remark

The elements S_c0, where c₀ = (ρ⁽ⁱ⁾)_i∈I are I-tuples of partitions,

are the “purely imaginary” PBW-type basis elements in U_q⁺(g_af), and S_c⁻

0 := S_c^∨

0, where the C(q)-algebra automorphism ^∨ of U_q(g_af) is given by

E_i^∨ := F_i, F_i^∨ := E_i, (q^h)^∨ = q^−h, and the C-algebra automorphism ¯ of U_q(g_af) is given by

E_i := E_i, F_i := F_i, q^h := q^−h, q := q⁻¹.

Realization of the crystal B(λ) Assume that

λ = ∑

i∈I

m_iϖ_i ∈ P₊ is such that m_i ≩ 0 (^∀i ∈ I).

Theorem

We have an isomorphism

Φ_λ : B(λ) →^∼ B^∞2 (λ) of crystals such that

Φ_λ(S_c⁻

0u_∞ ⊗ τ_λ ⊗ u_−∞) = η^c0 for all c₀ ∈ Par(λ).

Here, for each c₀ ∈ Par(λ), the element η^c0 ∈ B^∞2 (λ) is an extremal element of the form:

η^c0 = (t_ξ1, . . . , t_ξs−1, t_ξs = e ; 0 = a₀, . . . , a_s = 1), s ≥ 1, with ξ_k ∈ Q^∨ (1 ≤ k ≤ s − 1), such that

ξ_k − ξ_k+1 ∈ ∑

i∈I a_k⟨λ, α^∨_i ⟩∈Z

Z_≥⁰α^∨_i

for all 1 ≤ k ≤ s − 1; ξ_s := 0 by convention.

Demazure submodules of V (λ) λ = ∑

i∈I m_iϖ_i ∈ P₊ : level-zero dominant and regular U_q⁻(g_af) : negative part of U_q(g_af)

For each x ∈ W_af, we set

V_x⁻(λ) := U_q⁻(g_af)S_xv_λ ⊂ V (λ),

where S_xv_λ ∈ V (λ) is an extremal vector of weight xλ.

Remark

V_x⁻(λ) ∼= V_e⁻(xλ) ⊂ V (xλ).

Fact (Kashiwara)

For each x ∈ W_af, V_x⁻(λ) has the crystal basis B_x⁻(λ) = (S_x^∗)⁻¹(

B(xλ) ∩ (B(∞) ⊗ τ_xλ ⊗ u_−∞)) , where S_x^∗ : B(λ) → B^∼ (xλ) is an isomorphism of crystals.

Characterization of B_x⁻(λ) Assume that λ = ∑

i∈I

m_iϖ_i ∈ P₊ is such that m_i ≩ 0 (^∀i ∈ I).

For each x ∈ W_af, we set

B_≥^∞2_x(λ) := {

η ∈ B^∞2 (λ) | κ(η) ≥^∞₂ x} .

Theorem

For each x ∈ W_af,

Φ_λ(B_x⁻(λ)) = B_≥^∞2_x(λ),

where Φ_λ : B(λ) →^∼ B^∞2 (λ) is the isomorphism above of crystals.

Relation with symmetric Macdonald polynomials Assume that λ = ∑

i∈I

m_iϖ_i ∈ P₊ is such that m_i ≩ 0 (^∀i ∈ I).

Write V_e⁻(λ) as:

V_e⁻(λ) = ⊕

γ∈Q k∈Z_≥0

V_e⁻(λ)_λ+γ−kδ,

where Q = ∑

i∈I

Zα_i ; we set

gr-ch(V_e⁻(λ)) := ∑

γ∈Q

(dim_C(q) V_e⁻(λ)_λ+γ−kδ)e^λ+γq^−k.

Theorem

gr-ch(V_e⁻(λ)) = P_λ(x ; q⁻¹, 0)

∏

i∈I m_i

∏

r=1

(1 − q^−r) ,

with x = e^λ+γ, where P_λ(x ; q, 0) denotes the specialization at t = 0 of the symmetric Macdonald polynomial P_λ(x;q, t).

Remark

Here we have used our previous result that the “graded character”

gr-ch(W_e(λ)) of the local Weyl module W_e(λ) (in the notation below) is identical to the specialization P_λ(x ; q⁻¹, 0) at t = 0 of the symmetric Macdonald polynomial P_λ(x ; q⁻¹, t).

Remark

(the Ram-Yip formula)

The nonsymmetric Macdonald polynomial E_w◦λ(x ; q, t) is equal to the following:

∑

p_J

x^wt(pJ)t^ℓ(dir(pJ))(t⁻¹ − t)^|J|

∏

j∈J₋ q^deg(βj∨)t^{⟨2ρ,−cl(β∨j ⟩}

∏

j∈J(1 − q^{deg(β∨j )}t^{⟨2ρ,−cl(βj∨)⟩}) ,

where p_J runs over specific finite sequences of elements in W_af corre- sponding to certain finite sets J determined by λ;

wt(p_J) ∈ P , dir(p_J) ∈ W, cl(β_j) ∈ −∆⁺ and deg(β_j^∨) ∈ Z_≩⁰ for j ∈ J,

Relation with level-zero fundamental representations Assume that λ = ∑

i∈I

m_iϖ_i ∈ P₊ is such that m_i ≩ 0 (^∀i ∈ I).

For the unit element e ∈ W, we set W_e(λ) := V_e⁻(λ)

/ ∑

c₀∈Par(λ)\(∅)_i∈I

U_q⁻(g_af)S_c⁻

0v_λ;

recall that Par(λ) is the set of I-tuples c₀ = (ρ⁽ⁱ⁾)_i∈I of partitions such that the length of the partition ρ⁽ⁱ⁾ is ≤ m_i for all i ∈ I,

and S_c⁻

0v_λ ∈ V (λ) is an extremal vector of weight λ − |c₀|δ.

We denote the quotient map by

cl : V_e⁻(λ) ↠ W_e(λ).

Remark W_e(λ) has the crystal basis

{η ∈ B0^∞2 (λ) | κ(η) ∈ W} .

Now, for η = (x₁, . . . , x_s ; a₀, . . . , a_s) ∈ B^∞2 (λ), we set cl(η) := (cl(x₁), . . . , cl(x_s) ; a₀, . . . , a_s),

where cl : W_af ↠ W is a (surjective) homomorphism given by:

cl(wt_µ) = w for w ∈ W, µ ∈ Q^∨. Note that

{ | ∈ B^∞ ∈ }

Note

For x = wt_µ ∈ W_af and β = α + kδ ∈ ∆⁺_af,

x −→^β r_βx in the semi-infinite Bruhat graph if and only if (1) k = 0, w⁻¹α ∈ ∆⁺, and ℓ(wr_w−1α) = ℓ(w) + 1, or

(2) k = 1, w⁻¹α ∈ ∆⁺, and ℓ(wr_w−1α) = ℓ(w) − 2⟨ρ, w⁻¹(α^∨)⟩ + 1.

We set

B(λ)_cl := {

cl(η) | η ∈ B^∞2 (λ)} , the set of quantum LS paths of shape λ.

Then, as a U_q(g)-crystal, B(λ)_cl is isomorphic to {η ∈ B0^∞2 (λ) | κ(η) ∈ W}

⊂ B^∞2 (λ).

Remark

As U_q(g)-modules,

W_e(λ) ∼= ⊗

i∈I W(ϖ_i)^⊗mi, where

W(ϖ_i) : i-th level-zero fundamental representation of U_q^′(g_af);

U_q(g) ⊂ U_q^′(g_af) = U_q((C[t, t⁻¹] ⊗_C g) ⊕ Cc).

Proposition

For the unit element e ∈ W, the graded character gr-ch(W_e(λ)) of W_e(λ) is identical to the specialization E_w◦λ(x ; q⁻¹, 0) at t = 0 of the nonsymmetric Macdonald polynomial E_w◦λ(x ; q⁻¹, t), where w_◦ ∈ W denotes the longest element.

Remark We have

E_w◦λ(x ; q⁻¹, 0) = P_λ(x ; q⁻¹, 0), where w_◦ ∈ W is the longest element.

Quantum Bruhat graph

QBG : quantum Bruhat graph associated with W and ∆⁺; this is a labeled, directed graph with

vertex set W,

edges : u →^β v, u, v ∈ W and β ∈ ∆⁺, where u →^β v means that

(1) v = ur_β and ℓ(v) = ℓ(u) + 1 (Bruhat edge), or

(2) v = ur_β and

ℓ(v) = ℓ(u) − 2⟨ρ, β^∨⟩ + 1 (quantum edge).

Specialization at t = ∞ of nonsymmetric Macdonald polynomials Let λ = ∑

i∈I m_iϖ_i ∈ P₊ be level-zero dominant and regular, i.e., m_i ≩ 0 for all i ∈ I.

For an edge u →^β v in the QBG, we set

wt_λ(u → v) :=

{0 (for a Bruhat edge),

⟨λ, β^∨⟩ (for a quantum edge).

Also, for u, v ∈ W, we set

wt_λ(u ⇒ v) := wt_λ(u₀ → u₁) + · · · + wt_λ(u_k−1 → u_k), by taking a shortest directed path

u = u₀ → u₁ → · · · → u_k−1 → u_k = v in the QBG.

For η ∈ B(λ)_cl of the form:

η = (w₁, . . . , w_s; 0 = a₀, a₁, . . . , a_s = 1), we set κ(η) := w_s ∈ W, and

wt(η) :=

s−1

∑

i=0

(a_i+1 − a_i)w_i+1λ ∈ P ; we also set

deg_w◦(η) := −

∑s

i=1

a_iwt_λ(w_i+1 ⇒ w_i), where w_s+1 := w_◦ ∈ W (the longest element in W).

Now, we define:

gch_w◦(B(λ)_cl) := ∑

η∈B(λ)_cl

q^degw◦(η)e^wt(η).

Theorem

In the notation and setting above, we have

E_w◦λ(x ; q, ∞) = gch_w◦(B(λ)_cl).

Comparison with the specialization at t = 0 For η ∈ B(λ)_cl of the form:

η = (w₁, · · · , w_s; 0 = a₀, a₁, · · · , a_s = 1), we set

Deg(η) :=

s−1

∑

i=1

a_iwt_λ(w_i+1 ⇒ w_i).

Remark We have

Also, we set

gr-ch(B(λ)_cl) := ∑

η∈B(λ)_cl

q^−Deg(η)e^wt(η).

Theorem

In the notation and setting above, we have

gr-ch(B(λ)_cl) = gr-ch(W_e(λ)) = E_w◦λ(x ; q⁻¹, 0).