We have seen in Proposition 4.4 that using the notion of virtual highest weight vector, one can replace the variables of the positive representationsPλ≃L2(R) ofUqqe(sl(2,R)) and recover the finite dimensional representations (up to a sign) of the compact quantum groupUq(sl2). This can obviously be generalized to higher rank split real quantum group Uqqe(gR) since the virtual highest weight vectors are also known.
In particular, the positive representation ofUqeq(gR) is given byP−→λ ≃L2(Rl(w0)) and parametrized by −→
λ ∈ Rrank(g)≥0 , where l(w0) is the length of the longest element w0 of the Weyl group. Following Proposition 4.4, what we get will be an expression of the irreducible finite dimensional representation V→−N of the corresponding compact quantum group Uq(gc), parametrized byrank(g) positive integers −→
N ∈Zrank(g)≥0 , and represented by a finite dimensionalC-vector subspace ofl2(Zl(w≥00)).
Example 6.1. Let us consider the positive representations of Uqeq(sl(3,R))generated by {Ei, Fi, Ki}i=1,2 onL2(R3, dudvdw)given by [16]:
Following the notion of virtual highest weight calculated in [21], we make a shift u −→
u−λ2, v−→v−λ1−λ2, w−→w−λ1, and using similar arguments of Proposition 4.4,
and obtain an expression for the finite dimensional representation VN1,N2 ofUq(sl3):
E1·vk,m,n= [k]qvk−1,m−1,n+1+ [m−n]vk,m−1,n, (6.7)
E2·vk,m,n= [n]qvk,m,n−1, (6.8)
F1·vk,m,n= [N1+k−m]qvk,m+1,n, (6.9) F2·vk,m,n= [N2−k]qvk+1,m,n+ [N2−2k+m−n]qvk,m,n+1, (6.10)
K1·vk,m,n=qk−2m+n+N1vk,m,n, (6.11)
K2·vk,m,n=q−2k+m−2n+N2vk,m,n, (6.12)
where we have removed the minus signs arising from the conversion. This seems to be a new explicit expression for the representation VN1,N2 that does not use a weight diagram.
We see that the highest weight vector is given by v0,0,0 and the lowest weight vector is given by vN2,N1+N2,N1. In general the basis of the representationVN1,N2 can be obtained for example by taking the action of the canonical basis [25]:
F1aF2bF1c·v0,0,0, a+c≤b, c≤N1, F2aF2bF1c·v0,0,0, a+c < b, c≤N2,
and it will be a vector subspace of the vector space V spanned by{vk,m,n} with0≤k≤ N2,0≤m≤N1+N2,0≤n≤N1.
Now the tensor product decomposition of two finite dimensional representations of Uq(gc) amounts to solving the functional equation for the coefficientsC−→−→n
k ,−→mfor v−→k =X
C−→−→k ,n−→mx→−m⊗y→−n.
Conjecture 6.2. The coefficientsC→−→−nk ,→−m can be represented explicitly by linear combina-tions of q-binomials.
This seems to be straightforward since the functional equations are just finite differ-ence equations expressed in terms of q-numbers. In particular it seems only necessary to calculate the expressions for the highest weight vectors, since they uniquely determined the rest of the basis. General methods to obtain the Clebsch-Gordan coefficients C→−→−n
k ,→−m
have been studied for the classical [1] and quantum groups [27] by solving similar func-tional equations. However, we are not aware of any explicit general formula in terms of q-binomials and the weight parameters −→
N in the quantum level except for the case of Uq(sl2) and at most the octet representations ofUq(sl3).
Applying the philosophy of Remark 2.5 by normalizing with the theory of canonical basis, and replacingq-binomials with quantum dilogarithms, the resultingintegral trans-formation is indeed the required intertwiners for the tensor product decomposition of the positive representations P−λ→
1 ⊗ P−λ→
2. Here one has to take into account the summation range of the Clebsch-Gordan equation which becomes nontrivial in higher rank, governed by the Littlewood-Richardson rule (see e.g. [14]). Also one has to show that the integral transformation is well-defined and unitary under some measure. In particular it should intertwine the action of thepositive Casimirscalculated in [21]. Investigating small cases of the Littlewood-Richardson rule, we made the following conjecture in [19]:
Conjecture 6.3. The positive representations forUqqe(gR)is closed under taking the tensor product, and it decomposes as
P−λ→
1⊗ P−λ→
2 ≃ Z ⊕
Rl(w+ 0)
P−→γdµ(−→γ), (6.13) where−→γ =P
α∈∆+γαωαsumming over all the positive roots∆+, whereγα∈R+ andωα
are the fundamental weights, with the abuse of notationωα:=ωα1+ωα2 ifα:=α1+α2
is not simple. The Plancherel measuredµ(−→γ)is a continuous measure given by dµ(−→γ) = Y
α∈∆+
sinh(2πbγα) sinh(2πb−1γα)dγα. (6.14)
Note that both sides of (6.13)is isomorphic toL2(R2l(w0)).
We believe that even establishing the conjecture for lower rank case ofUqqe(sl(3,R)) is enough to provide major breakthroughs in the theory of positive representations of split real quantum groups and its many applications as a completely new class of braided tensor categories.
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