Partial tropicalization of G ∗ with real form τ

In this section, we would like to apply the construction in previous section to the positive complex Poisson variety(G^∗,iπG^∗,ΘG^∗,ΦG^∗)with real formτ.

Recall that we have the real formτ onG^∗is given by τ:G^∗ →G^∗ : (b, b−)7→

b−

−T, b^−T .

and we know that((G^∗)^τ,Φ^τ_G∗,Θ^τ_G∗)is isomorphic to(B−,ΦBK,ΘB−)as positive variety via the projectionpr₂. Letϑ=θ0×θ1:G^rm×G^mm→H×U₋∼=B−be a toric chart ofB−, which extend to aτ-compatible chartθe:=η◦(θ₀×θ₁×θ₁)forG^∗, whereηis given by Eq (7.4). Note thatpr₂◦θ^τ =θ0×θ1. Also note that(G^∗)^τ is given by

n

(b, b−)∈G^∗ |b−=b^−T o

, which is isomorphic toAU−⊂B−viapr₂. ThusPT(G^∗,θ,eΦG^∗, τ) ∼= (B−,ΦBK, ϑ)^t

R(0)×(S¹)^m viapr₂and we get the following detropicalization map:

E_ϑ,s:R^r+m×(S¹)^m→AU−

(λ−r, . . . , λm, ϕ1, . . . , ϕm)7→ϑ

e^sλ−r, . . . , e^sλ−1, e^sλ1+iϕ1, . . . , e^sλm+iϕm

. Next we compute the constant Poisson bracketπ_∞^θ ofPT(G^∗, θ)for a specific toric chartθ.

Leti ∈R(w0, e)be a double reduced word for(w0, e). Recall that for the seedσ(i),we have the following cluster variables onG^w0,e⊂B−by (4.10):

∆_k := ∆ukω_ik,ω_ik,fork∈[−r,−1]∪[1, m],

whereu_k=s_i1· · ·s_ikfork∈[1, m]andu_k=efork∈[−r,−1]. Note that∆_k’s determine a birational isomorphism

G^w0,e →(C^×)^m+r : g7→(∆−r(g), . . . ,∆_m(g)). whose inverse gives a toric chart ofB−

σ(i) : (C^×)^m+r→G^w0,e ,→B−. (7.15) Denote byθ(i) theτ-compatible chart forG^∗ extendingσ(i)as in previous discussion. Then one compute:

Theorem 7.7.1. The Poisson bivectorπ∞^θ(i)onPT(G^∗,iπ_G^∗,Φ_G^∗, θ(i), τ)is given by

{λ_k, ϕp}= 0, fork>p; (7.16)

{λ_k, ϕp}= (ukωik, upωip)−(ωik, ωip), fork < p; (7.17) {λ_k, λ_p}={ϕ_k, ϕ_p}= 0. for allk, p.

Proof. Denote by{·,·}_logthe log-canonical part ofiπG^∗. By Eq (7.5), the number2i$k,pis 2i{(∆_k)2,(∆p◦τ)1}_log

(∆_k)₂(∆_p◦τ)₁ = (ω_ik, ω_ip)−(u_kω_ik, u_pω_ip).

By Theorem 5.2.4 and the fact thatpr₂is anti-Poisson, we know2iπ_k,pis 2i{(∆_k)2,(∆p)2}_log

(∆k)2(∆p)2

= (u_kω_ik, u_pω_ip)−(ω_ik, ω_ip), wherek6p.

Take this into the definition, one gets desired formulas. ♦

7.7. PARTIAL TROPICALIZATION OFG^∗WITH REAL FORMτ 79 Proposition 7.7.2. For the Poisson manifoldPT(G^∗) = PT(G^∗,iπ_G^∗,Φ_G^∗, θ(i), τ)with the Poisson structureπ^θ(i)∞ , we have

(1) The functionsλ_kare Casimirs fork∈[1, m]\e(i);

(2) The matrixB = [{λ_k−, ϕ_l}]_m, wherek, lis indexed by(1, . . . , m)andk⁻is defined by Eq 4.7, is of the formB =DB⁰for

D= diag ((αi1, ωi1), . . . ,(αim, ωim)) = (1/di1, . . . ,1/dim),

andB⁰is an upper triangular positive integer matrix with diagonal elements being1;

(3) Lethw :B−→Hbe the highest weight map. Then the symplectic leaves ofPT(G^∗)are of the form

P_λ^∨ := hw^−t

R (λ^∨)×(S¹)^m, whereλ^∨ ∈X_∗⁺(H)⊗_Z+R+.

Proof. For item (1), letk∈[1, m]\e(i). By (7.16), we have{λ_k, ϕ_p} = 0forp 6k. All we need to show is{λ_k, ϕp}= 0forp > k. Sincek∈[1, m]\e(i), we knowp∈[1, m]\e(i)as well. Therefore, by (7.17),

{λ_k, ϕp}= (ωik, ωip)−(w0ωik, w0ωip) = 0, because the bilinear form isW-invariant.

For item (2), recall thats_iis the simple reflection generated byα_i. Note thats_iω_i =ω_i−α_i, andsjωi =ωiforj6=i. By (4.7), we haveωik =ωi_k−. Thus by (7.17): fork=l

{λ_k⁻, ϕk}= (ωi_k−, ωik)−(ωi_k−, si_k−+1· · ·sikωik)

= (ωik, ωik)−(ωik, sikωik) = (αik, ωik)>0.

For k > l, there are two possible cases. Fork > k⁻ > l, we have{λ_k−, ϕ_l} = 0by by (7.16). Fork > l > k⁻, we have:

{λ_k−, ϕ_l}= (ω_i

k−, ω_il)−(ω_i

k−, s_i

k−+1· · ·s_ilω_il) = (ω_ik, ω_il)−(ω_ik, ω_il) = 0.

The equalities follows from the facti_k⁻₊₁, . . . , i_lare all different fromi_kandω_ik =ω_i

k−. Fork < l, we have{λ_k−, ϕ_l}= ω_ik, ω_il−u⁻¹_k−u_lω_il

=c(ω_ik, α_ik)forc∈Z>0, because ωi−vωiis a non-negative integer linear combination ofαj’s and(αi, ωj) = 0fori6=j.

For item (3), recall that

(hw(g))^w0ωi = ∆_w0ωi,ωi(g), ∀g∈G^w0,e, i∈I.

The tropicalization ofhwwith respect to the chartσ(i)can then be written as a linear combina-tion of the cocharactersw0α^∨_i ∈X∗(H), with the tropical functions∆w0ωi,ωias coefficients:

hw^t_R=X

i

∆^t_w0ωi,ωi·(w0α_i^∨).

Together with the nondegeneracy of the matrixB in previous item, this proves the claim. ♦

8 | Poisson-Lie duality vs Langlands duality

8.1 Overview

LetKbe a compact connected semisimple Lie group. There are two very interesting duality constructions which involve K. First, one can associate to it the Langlands dual group K^∨ corresponding to the root system dual to the one ofK. Second, the groupKcarries the standard Poisson-Lie structureπK. As a Poisson-Lie group, it admits the dual Poisson-Lie groupK^∗.

The groupK^∨is a compact connected semisimple Lie group while the groupK^∗is solvable.

Despite this fact, they share some common features. LetT ⊂Kbe a maximal torus ofKand t = Lie(T)be its Lie algebra. The Lie algebrat^∨ = Lie(T^∨)of the maximal torusT^∨ ⊂K^∨ is in a natural duality witht:

t^∨∼= Hom(S¹, T)^∗⊗_ZR∼= Hom(T, S¹)⊗_ZR∼=t^∗.

The Lie algebra it ∼= t^∗ plays a role analogous to the one of Cartan subalgebra for the group K^∗. The isomorphism above is induced by the invariant scalar product onk = Lie(K)used to define the standard Poisson structures onKandK^∗.

Furthermore, both the Langlands dual group and the Poisson-Lie dual group can be used to parametrize representations of K (or finite dimensional representations of G = K^C). On one hand, by the Borel-Weil-Bott Theorem, geometric quantization of coadjoint orbits passing through dominant integral weights in t^∗ yields all irreducible representations ofK. By the Ginzburg-Weinstein Theorem [43], the Poisson spacesK^∗andk^∗ are isomorphic to each other and we can extend the Borel-Weil-Bott result to K^∗, where for a dominant integral weight λ∈t^∗ ∼=itwe consider theK-dressing orbit inK^∗passing throughexp(λ).

On the other hand, as we discussed in Chapter 3, fromB₋^∨ ⊂ G^∨ = (K^∨)^C, Berenstein-Kazhdan [16] constructed an integral polyhedral cone(B^∨₋,Φ^∨_BK)^tby tropicalization, together with a highest weight map

(hw^∨)^t: (B₋^∨,Φ^∨_BK)^t→X∗(H^∨) =X^∗(H).

The fiber(hw^∨)^−t(λ)carries a Kashiwara crystal structure, which isomorphisc to the the one of irreducible representations ofGwith highest weightλ.

This chapter is based on a joint work [6] with A. Alekseev, A. Berenstein and B. Hoffman.

81

82 CHAPTER 8. POISSON-LIE DUALITY VS LANGLANDS DUALITY It is the goal of this paper to establish a relation between the two duality constructions described above.

There are several tools that we are using to this effect. First, note that that of the double Bruhat cellsG^w0,e ⊂ B−andG^∨;w0,e ⊂ B₋^∨ is a pair of cluster varieties dual to each other.

In this case, the relationship between tropicalizations can be further improved: We show that our comparison mapψmaps one of these cones into the other, and preserves their Kashiwara crystal structure (up to some scaling). This gives a new perspective on a result of Kashiwara [58] and Frenkel-Hernandez [39].

For the discussion of the Poisson-Lie dualK^∗, we use the tool of partial tropicalization that we introduced in Chapter 7. Recall thatPT(K^∗)comes equipped with a constant Poisson struc-ture which induces integral affine strucstruc-tures on symplectic leaves. Together with the strucstruc-ture of the weight lattice ofK, they define a natural Bohr-Sommerfeld latticeΛ⊂(B−,ΦBK)^t.

We show that

ψ(Λ) = (B₋^∨,Φ^∨_BK)^t.

That is, the integral Bohr-Sommerfeld coneΛdefined by the Poisson-Lie data onK^∗is isomor-phic to the integral cone(B₋^∨,Φ^∨_BK)^tdefined by the potentialΦ^∨_BK. The isomorphism is given by the tropical duality map of the double cluster variety.

In more details, the cone(B₋^∨,Φ^∨_BK)^tparametrizes canonical bases of irreducibleG-modules.

For the representation with highest weightλ, the canonical basis inV_λ is parametrized by the points of (hw^∨)^−t(λ). The preimage of this set under the duality map ψ is exactly the set of points ofΛ which belongs to the integral symplectic leaf ofPT(K^∗)corresponding to the weightλ. The relations are depicted in Figure 8.1.

Integral coadjoint orbits IrreducibleG-modules Fibers of(hw^∨)^t

leaves ofPT(K^∗) Integral symplectic

Figure 8.1