**Comparison of the two specializations of** **nonsymmetric Macdonald polynomials:**

**at** **t** **= 0 and at** **t** **=** **∞**

**t**

**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**_{i}**α**_{i}**∈****∆**^{+}_{af}**: (primitive) null root**
**α**_{i}**,** **i****∈****I**_{af}**=** **I****∪ {****0****}****: simple roots**

**P****=** ∑

**i****∈*** I* Z

**ϖ**

_{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* Z

**α**

^{∨}

_{i}

**Q**

^{∨}

^{,+}**=**∑

**i****∈*** I* Z

_{≥}**0**

**α**

^{∨}

_{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**_{β}**x,****x****∈****W**_{af}**, β****∈****∆**^{+}_{af}**,****with** **ℓ**^{∞}^{2}**(r**_{β}**x) =****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_{≩}**0****ϖ**_{i}**: level-zero dominant and regular**

**η****= (x**_{1}**>**^{∞}

**2** **x**_{2}**>**^{∞}

**2** **· · ·****>**^{∞}

**2** **x**_{s}**;**

**0 =** **a**_{0}**< a**_{1}**<****· · ·****< 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}**←**^{β}^{t}**y**_{t}_{−}_{1}^{β}**← · · ·**^{t}^{−}^{1}**←**^{β}^{2}**y**_{1}**←**^{β}^{1}**y**_{0}**=** **x**_{k+1}**with** **a**_{k}**⟨****y**_{l}_{−}_{1}**λ, β**_{l}^{∨}* ⟩ ∈* Z

**(1**

**≤**

^{∀}

**l**

**≤**

**t).****0 =** **a**_{0}**a**_{1}**a**_{2}**a**_{k}_{−}_{1}**a**_{k}**a**_{k+1}**a**_{s}_{−}_{1}**x**_{1}**λ****x**_{2}**λ****x**_{k}**λ x****k+1****λ****x**_{s}**λ**

**a**_{s}**= 1**

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

B_{0}^{∞}^{2}**(λ) : connected component of** B^{∞}^{2}**(λ) containing (e** **; 0 =** **a**_{0}**, a**_{1}**= 1)**
**For**

**η****= (x**_{1}**>**^{∞}

**2** **· · ·****>**^{∞}

**2** **x**_{s}**; 0 =** **a**_{0}**< a**_{1}**<****· · ·****< a**_{s}**= 1)** * ∈* B

^{∞}

^{2}**(λ)**

**above, we set**

**ι(η) :=****x**_{1}**∈****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**_{i}**ϖ**_{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**_{w}**v**_{λ}**}**^{w}_{∈}^{W}_{af}**⊂****V****(λ) such that**

**S**_{e}**v**_{λ}**=** **v**_{λ}**, and such that for all** **w****∈****W**_{af}**and** **i****∈****I**_{af}**,**

**if** **⟨****wλ, α**^{∨}_{i}**⟩ ≥****0, then** **E**_{i}**S**_{w}**v**_{λ}**= 0 and** **F**_{i}^{(}^{⟨}^{wλ,α}^{∨}^{i}^{⟩}^{)}**S**_{w}**v**_{λ}**=** **S**_{r}_{i}_{w}**v**_{λ}**,**
**if** **⟨****wλ, α**^{∨}_{i}**⟩ ≤****0, then** **F**_{i}**S**_{w}**v**_{λ}**= 0 and** **E**_{i}^{(}^{−⟨}^{wλ,α}^{∨}^{i}^{⟩}^{)}**S**_{w}**v**_{λ}**=** **S**_{r}_{i}_{w}**v**_{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**_{w}**u**_{λ}**}**^{w}_{∈}^{W}_{af}**⊂ B****(λ) such that**
**S**_{e}**u**_{λ}**=** **u**_{λ}**, and such that**

**if** **⟨****wλ, α**^{∨}_{i}**⟩ ≥****0, then** **e**_{i}**S**_{w}**u**_{λ}**= 0 and** **f**_{i}^{⟨}^{wλ,α}^{∨}^{i}^{⟩}**S**_{w}**u**_{λ}**=** **S**_{r}_{i}_{w}**u**_{λ}**,**
**if** **⟨****wλ, α**^{∨}_{i}**⟩ ≤****0, then** **f**_{i}**S**_{w}**u**_{λ}**= 0 and** **e**^{−⟨}_{i}^{wλ,α}^{∨}^{i}^{⟩}**S**_{w}**u**_{λ}**=** **S**_{r}_{i}_{w}**u**_{λ}**for all** **w****∈****W**_{af}**and** **i****∈****I**_{af}**.**

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

**i****∈****I**

**m**_{i}**ϖ**_{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** **c**_{0}**= (ρ**^{(i)}**)**_{i}_{∈}_{I}**of partitions**

**such that the length of the partition** **ρ**^{(i)}**is** ≨ **m**_{i}**(**^{∀}**i****∈****I****);**

**for** **c**_{0}**= (ρ**^{(i)}**)**_{i}_{∈}_{I}**∈****Par(λ), we set** **|****c**_{0}**|****:=** ∑

**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** **c**_{0}**= (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}^{−}

**0****u**_{∞}**⊗****τ**_{λ}**⊗****u**_{−∞}**∈ B****(****∞****)** * ⊗* {

**τ***}*

_{λ}**⊗ B****(****−∞****),**
**for some c**_{0}**= (ρ**^{(i)}**)**_{i}_{∈}_{I}**∈****Par(λ) (or, c**_{0}**= (ρ**^{(i)}**)**_{i}_{∈}_{I}**∈****Par(λ));**

**also, we have**

**S**_{c}^{−}

**0****u**_{∞}**⊗****τ**_{λ}**⊗****u**_{−∞}**≡****S**_{c}^{−}

**0****v**_{λ}**(mod** **q).**

**Remark**

**The elements** **S**_{c}_{0}**, where** **c**_{0}**= (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**_{i}**ϖ**_{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}^{−}

**0****u**_{∞}**⊗****τ**_{λ}**⊗****u**_{−∞}**) =** **η**^{c}^{0}**for all c**_{0}**∈****Par(λ).**

**Here, for each c**_{0}**∈****Par(λ), the element** **η**^{c}^{0}* ∈* B

^{∞}

^{2}**(λ) is an extremal**

**element of the form:**

**η**^{c}^{0}**= (t**_{ξ}_{1}**, . . . , t**_{ξ}_{s}_{−}_{1}**, t**_{ξ}_{s}**=** **e****; 0 =** **a**_{0}**, . . . , 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_{≥}^{0}**α**^{∨}_{i}

**for all 1** **≤****k****≤****s****−****1;** **ξ**_{s}**:= 0 by convention.**

**Demazure submodules of** **V****(λ)**
**λ****=** ∑

**i****∈****I****m**_{i}**ϖ**_{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**_{x}**v**_{λ}**⊂****V****(λ),**

**where** **S**_{x}**v**_{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**_{i}**ϖ**_{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**_{i}**ϖ**_{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**

Z**α**_{i}**; we set**

**gr-ch(V**_{e}^{−}**(λ)) :=** ∑

**γ****∈****Q**

**(dim**_{C}_{(q)}**V**_{e}^{−}**(λ)**_{λ+γ}_{−}_{kδ}**)e**^{λ+γ}**q**^{−}^{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(p}^{J}^{)}**t**^{ℓ(dir(p}^{J}^{))}**(t**^{−}^{1}**−****t)**^{|}^{J}^{|}

∏

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

∏

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

**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**_{i}**ϖ**_{i}**∈****P**_{+}**is such that** **m***_{i}* ≩

**0 (**

^{∀}

**i**

**∈**

**I****).**

**For the unit element** **e****∈****W****, we set**
**W**_{e}**(λ) :=** **V**_{e}^{−}**(λ)**

/ ∑

**c**_{0}**∈****Par(λ)****\****(****∅****)**_{i}_{∈}_{I}

**U**_{q}^{−}**(g**_{af}**)S**_{c}^{−}

**0****v**_{λ}**;**

**recall that Par(λ) is the set of** **I****-tuples c**_{0}**= (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}^{−}

**0****v**_{λ}**∈****V****(λ) is an extremal vector of weight** **λ****− |****c**_{0}**|****δ.**

**We denote the quotient map by**

**cl :** **V**_{e}^{−}**(λ)** ↠ **W**_{e}**(λ).**

**Remark** **W**_{e}**(λ) has the crystal basis**

{**η*** ∈* B

**0**

^{∞}

^{2}**(λ)**

**|**

**κ(η)**

**∈***}*

**W**

**.****Now, for** **η****= (x**_{1}**, . . . , x**_{s}**;** **a**_{0}**, . . . , a**_{s}**)** * ∈* B

^{∞}

^{2}**(λ), we set**

**cl(η) := (cl(x**

_{1}**), . . . ,**

**cl(x**

_{s}**) ;**

**a**

_{0}

**, . . . , 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**_{β}**x****in the semi-infinite Bruhat graph if and only if**
**(1)** **k****= 0,** **w**^{−}^{1}**α****∈****∆**^{+}**, and** **ℓ(wr**_{w}**−****1****α****) =** **ℓ(w) + 1, or**

**(2)** **k****= 1,** **w**^{−}^{1}**α****∈****∆**^{+}**, and** **ℓ(wr**_{w}**−****1****α****) =** **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*** ∈* B

**0**

^{∞}

^{2}**(λ)**

**|**

**κ(η)**

**∈***}*

**W*** ⊂* B

^{∞}

^{2}**(λ).**

**Remark**

**As** **U**_{q}**(g)-modules,**

**W**_{e}**(λ)** **∼****=** ⊗

**i****∈****I****W****(ϖ**_{i}**)**^{⊗}^{m}^{i}**,**
**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) * ⊕* C

**c).****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**_{i}**ϖ**_{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}**(u**_{0}**→****u**_{1}**) +** **· · ·****+ wt**_{λ}**(u**_{k}_{−}_{1}**→****u**_{k}**),**
**by taking a shortest directed path**

**u****=** **u**_{0}**→****u**_{1}**→ · · · →****u**_{k}_{−}_{1}**→****u**_{k}**=** **v****in the QBG.**

**For** **η*** ∈* B

**(λ)**

_{cl}**of the form:**

**η****= (w**_{1}**, . . . , w**_{s}**; 0 =** **a**_{0}**, a**_{1}**, . . . , a**_{s}**= 1),**
**we set** **κ(η) :=****w**_{s}**∈****W****, and**

**wt(η) :=**

**s****−****1**

∑

**i=0**

**(a**_{i+1}**−****a**_{i}**)w**_{i+1}**λ****∈****P****;**
**we also set**

**deg**_{w}_{◦}**(η) :=** **−**

∑**s**

**i=1**

**a**_{i}**wt**_{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**^{deg}^{w}^{◦}^{(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****= (w**_{1}**,****· · ·****, w**_{s}**; 0 =** **a**_{0}**, a**_{1}**,****· · ·****, a**_{s}**= 1),**
**we set**

**Deg(η) :=**

**s****−****1**

∑

**i=1**

**a**_{i}**wt**_{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).**