The modular double - Representation of the quantum plane, its quantum double, and harmonic anal

#Q 2b− i

b(t2−λ)

e⁻²^π^bp^t2,

F = i

q−q⁻¹(q⁻¹²e^π^b⁽^t²^+λ)+q¹²e^−π^b⁽^t²^+λ))e²^π^bp^t2

#Q 2b+ i

b(t2+λ)

e²^π^bp^t2

which is precisely the representation denoted byPλ,−s₂/2. Combining Theorem8.11and8.14, and note that all the transformations used are unitary, we conclude the main theorem:

Theorem 8.15 As a representation of Uq(gl(2,R))L⊗Uq(gl(2,R))R,

L²(G L⁺_q(2,R)) ⊕ R

⊕ R⁺

Pλ,s⊗Pλ,−s|Sb(Q+2iλ)|²dλds (8.21)

and the equivalence is unitary.

In particular, we proved Ponsot–Teschner’s claim in [31] for Uq(sl(2,R))by setting s2=0 which corresponds to the determinant element N =AA, as well as the action of the central element K0.

8.6 The modular double

It is known that the representation spaceP_λ,t is actually a nontrivial representation for the modular double Uq(gl(2,R))⊗Uq(gl(2,R)). In this section, we describe its action using the same multiplicative unitary Wm.

The modular double counterpart Uq(gl(2,R))is defined in the same way with q replaced byq throughout. In particular, we can define the pairing betweenE,F,K,%K0

with A,B,A,B as before. Recall that A = A¹^/^b² and similarly for the other quan-tum plane variables. Hence, by the b ←→ b⁻¹duality of Wm, we can pair it with Uq(gl(2,R))and obtain:

Proposition 8.16 The left regular action of Uq(gl(2,R))on L²(G L⁺_q(2,R))is given by replacing b with b⁻¹, i.e.,

%K0=e^πb⁻

1 2 s2, K=e^−π^b⁻¹^s¹,

E = i

q−q⁻¹(q¹^/²e^π^b⁻¹⁽^s¹^−λ)+q⁻¹^/²e^−π^b⁻¹⁽^s¹^−λ))e⁻²^π^b⁻¹^p^s1, F = i

q−q⁻¹(q⁻¹^/²e^π^b⁻¹⁽^s¹^+λ)+q¹^/²e^−π^b⁻¹⁽^s¹^+λ))e²^π^b⁻¹^p^s1.

We note thatE andF are positive operators only when _2n¹₊₁ <b²< _2n¹ for n ∈N, and they are not necessarily essentially self-adjoint on the natural domainW [36].

To obtain essential self-adjointness, we have to apply Proposition4.1and define the operators on the respective transformed domain gb·W.

Ignoring the factors, we note from Theorem 8.11 that the summand of E and F are q-commuting; hence, we immediately obtain using Corollary3.14 (see also [2, Cor 1]).

Proposition 8.17 As positive self-adjoint operators, we have

K₀¹^/^b² =%K0, K¹^/^b² =K,

2 sin(πb⁻²)E=(2 sin(πb²)E)¹^/^b², 2 sin(πb⁻²)F=(2 sin(πb²)F)¹^/^b².

Similar analysis works as well for the right regular representation; hence, Theorem 8.15is actually an equivalence as a representation of the modular double

Uqq(gl(2,R))L⊗Uqq(gl(2,R))R,

where Uqq(gl(2,R)):=Uq(gl(2,R))⊗Uq(gl(2,R))denotes the modular double.

9 Representation theoretic meaning for certain integral transforms

In this section, we state without proof certain integral transformations of Gbthat arise in the calculations of certain representation relations.

Proposition 9.1 The pentagon equation W23W12 =W12W13W23is equivalent to the 4–5 relation (Lemma3.10)

Gb(α)Gb(β)Gb(γ ) Gb(α+γ )Gb(β+γ ) =

dτe⁻²^{πγ τ} Gb(α+iτ)Gb(β+iτ)

Gb(α+β+γ+iτ)Gb(Q+iτ). (9.1) By a change of variables and using the reflection formula, it can be rewritten as

Gb(α+γ )Gb(β+γ )Gb(α)Gb(β) Gb(α+β+γ )

dτe⁻²^πⁱ^(β+ⁱ^τ)(α+ⁱ^τ)Gb(α+iτ)Gb(β+iτ)Gb(γ −iτ)Gb(−iτ).

(9.2)

Scaling all the variables by b and taking the limit b−→0 by applying Theorem3.5, we recover precisely Barnes’ first lemma:

(a+c)(b+c)(a)(b)

(a+b+c) = 1

2π

(a+iτ)(b+iτ)(c−iτ)(−iτ)dτ (9.3) in the special case for d =0.

Next, as noted in Sect.8.4, by comparing Wm =_⊕

T^λ,^t^/²dtdμ(λ)as corepre-sentation, we obtain the following relation that is first observed by Volkov [44]:

Proposition 9.2 The 6–9 relation for Gb(x)can be written as

e⁻²^πτ(δ−ⁱ^τ)Gb(α+iτ)Gb(β+iτ)Gb(γ +iτ)Gb(δ−iτ)Gb(−iτ) Gb(α+β+γ+δ+iτ) dτ

= Gb(α)Gb(β)Gb(γ )Gb(α+δ)Gb(β+δ)Gb(γ +δ)

Gb(α+β+δ)Gb(α+γ+δ)Gb(β+γ+δ) , (9.4) where the contour goes alongRand separates the increasing and decreasing sequence of poles. By the asymptotic properties of Gb, the integral converges for any choice of parameters.

Again by scaling all the variables by b and applying Theorem 3.5, we recover Barnes’ second lemma:

(a+iτ)(b+iτ)(c+iτ)(d−iτ)(−iτ)

(a+b+c+d+iτ) dτ

= (a)(b)(c)(a+d)(b+d)(c+d)

(a+b+d)(a+c+d)(b+c+d) , (9.5) which in turn is a generalization of Pfaff–Saalschütz’s sum

j≥0

(n+m+l+k− j)!

(m−j)!(l− j)!(k− j)!(n+ j)!j! = (n+m+l)!(n+m+k)!(n+l+k)!

m!l!k!(n+m)!(n+l)!(n+k)! . (9.6) Finally, in [46], an alternative description of the multiplicative unitary lying instead inA⊗Ais defined to be (slightly modified to fit our definition):

V =gb(B⁻¹⊗q⁻¹B A⁻¹)e²^πⁱ^b2^log⁽^{q A B}⁻¹^)⊗^{log A}⁻¹, (9.7)

so that

V(x⊗1)V^∗=(x). (9.8)

On comparing

W^∗(1⊗x)W =(x)=V(x⊗1)V^∗ (9.9) as operators onH⊗H, we obtain a new relation involving Gb(x), which is essentially the same as the relation in [42, Theorem 5.6.7] in the case n=1:

Proposition 9.3 The 3–2 relation is given by

Gb(α+iτ)Gb(β−iτ)Gb(γ−iτ)e⁻²^πⁱ^(β⁻ⁱ^τ)(γ−ⁱ^τ)dτ=Gb(α+γ )Gb(α+β), (9.10) where the contour goes along R and separates the poles for iτ and −iτ. By the asymptotic properties for Gb, the integral converges for Re(α−β−γ ) < ^Q₂.

Acknowledgments I would like to thank my advisor Professor Igor Frenkel for suggesting the current project and providing useful insights into the problems. I would also like to thank L. Faddeev for pointing out several useful references, and to J. Teschner and Hyun Kyu Kim for helpful discussions. This work was partially supported by World Premier International Research Center Initiative (WPI Initiative), MEXT, Japan.

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