For a connected simply connected compact semisimple Lie group K and a real parameter 0<q<1, the Drinfeld double G_q of the q-deformation K_q is a quantization of the complexification G=K_C. Arano (2017) classified the irreducible admissible representations of G_q by (P×X)/W and proved that for parameters (λ,ν) with ν almost real (small imaginary part relative to log q), the representation V_q(λ,ν) is unitary exactly when its classical counterpart for G is, yielding a full description of the unitary dual for type A (SL_q(n,C), modulo lattice shifts). The unresolved core is to complete the unitary dual of G_q for general K: decide unitarizability of every irreducible admissible V_q(λ,ν) without the almost-real restriction on ν, including parameters landing in singular blocks, where the irreducibles are already parametrized but their unitarizability remains open. As of the standard reference monograph of Voigt–Yuncken (2020) this is explicitly only an almost complete classification for general G_q, and no later work closes the gap.

Background

A compact quantum group is a Hopf *-algebra of functions carrying an analogue of the Pontryagin duality and harmonic analysis of compact groups; for 0<q<10<q<1 the q-deformation KqK_q of a compact Lie group KK is defined via the quantum coordinate algebra O(Kq)\mathcal{O}(K_q), the span of the matrix coefficients of the type-1 finite-dimensional modules of the quantized enveloping algebra Uq(g)U_q(\mathfrak{g}). The Drinfeld double D(Kq)D(K_q) (denoted GqG_q) is a Hopf *-algebra combining O(Kq)\mathcal{O}(K_q) and Uq(g)U_q(\mathfrak{g}) through their natural pairing; it should be thought of as a q-deformation of the complexification G=KCG=K_{\mathbb{C}} of KK, and its unitary representation theory is the quantum analogue of the representation theory of a complex semisimple group such as SL(n,C)SL(n,\mathbb{C}). This analogy was made precise through the theory of quantum Harish-Chandra modules — algebraic models of representations in which each irreducible representation of the maximal compact part appears with finite multiplicity — initiated by Joseph and Letzter. A representation of GqG_q is called admissible if, restricted to O(Kq)\mathcal{O}(K_q), it decomposes into finite-dimensional KqK_q-types with finite multiplicities; Arano previously proved that every irreducible unitary representation of GqG_q is admissible, so the unitary dual is realized inside the admissible dual. Classically the irreducible admissible dual of a complex semisimple group is parametrized by K-types and infinitesimal characters; in the quantum case Arano's Main Theorem parametrizes irreducible admissible GqG_q-modules by pairs (λ,ν)∈P×h∗(\lambda,\nu)\in P\times\mathfrak{h}^*, where PP is the weight lattice, h∗\mathfrak{h}^* is the dual of the Cartan subalgebra, and ν\nu is taken modulo the period lattice X=h∗/(2πilog⁡(q)−1Q∨)X=\mathfrak{h}^*/(2\pi i\log(q)^{-1}Q^{\vee}) built from the coroot lattice Q∨Q^{\vee}, with the Weyl group WW (the reflection group of the root system) acting diagonally. A parameter ν∈h∗\nu\in\mathfrak{h}^* is called almost real (with respect to qq) if log⁡(q) Im(ν)/(2π)\log(q)\,\mathrm{Im}(\nu)/(2\pi) is small, that is, its pairing with every root α\alpha has absolute value less than 1 — an open WW-invariant strip around the real part of h∗\mathfrak{h}^* in which the quantum and classical worlds are provably close. Arano's Theorem 4.9 shows that for almost real ν\nu the irreducible admissible representation Vq(λ,ν)V_q(\lambda,\nu) is unitary if and only if the corresponding classical representation V1(λ,ν)V_1(\lambda,\nu) of GG is, via a continuity-in-qq signature argument; since the unitary dual of complex semisimple groups is classically known, this transfers the classification to GqG_q throughout the almost-real strip. Moreover, unitarity of Vq(λ,ν)V_q(\lambda,\nu) is invariant under the coarser shifts ν↦ν−2πilog⁡(q)−1γ\nu\mapsto\nu-2\pi i\log(q)^{-1}\gamma with γ\gamma in the coweight lattice P∨P^{\vee}; for K=SU(n)K=SU(n) every parameter class admits an almost-real representative up to these shifts, because there the coweight lattice is the integer-coordinate lattice Zn/Z(1,…,1)\mathbb{Z}^n/\mathbb{Z}(1,\dots,1) and rounding each coordinate into [0,1)[0,1) moves any real ν\nu into the strip (Arano's Lemma 2.5), giving Corollary 4.11: a complete unitary-dual classification for SLq(n,C)SL_q(n,\mathbb{C}). For other simple types the shift lattice is too coarse to force ν\nu into the almost-real strip, and beyond the strip the comparison mechanism collapses. Two structural facts frame the remaining problem: the admissible category decomposes into blocks Adm(Gq)=⨁Λ∈12X/WAdm(Gq)Λ\mathrm{Adm}(G_q)=\bigoplus_{\Lambda\in\frac{1}{2}X/W}\mathrm{Adm}(G_q)_{\Lambda} indexed by central characters Λ∈12X/W\Lambda\in\frac{1}{2}X/W, where 12X=h∗/(πilog⁡(q)−1Q∨)\frac{1}{2}X=\mathfrak{h}^*/(\pi i\log(q)^{-1}Q^{\vee}), and the principal series module in the block Λ\Lambda induced from Λ′\Lambda' has parameter (λ,ν)=(Λ−Λ′,−Λ−Λ′−2ρ)(\lambda,\nu)=(\Lambda-\Lambda',-\Lambda-\Lambda'-2\rho) (Arano's Lemma 4.3); its irreducible quotient is nonzero exactly when no positive root α\alpha satisfies both (Λ+ρ,α∨)=0(\Lambda+\rho,\alpha^{\vee})=0 and (Λ′+ρ,α∨)∈Z≥0(\Lambda'+\rho,\alpha^{\vee})\in\mathbb{Z}_{\geq 0} (Arano's Corollary 4.5(1)), and the irreducibles so obtained exhaust all irreducible admissible modules at every central character, regular or singular (Arano's Theorem 4.2(iv), Corollary 4.5(2); Voigt–Yuncken Lemma 5.34, Theorems 5.47–5.48). Theorem 4.2(iii) establishes the categorical equivalence between admissible blocks and blocks of the quantum BGG category O\mathcal{O} — the analogue, for quantized enveloping algebras, of the Bernstein–Gelfand–Gelfand category of highest-weight modules — only at regular central characters (the central character is the character by which the center-like adjoint-finite part of Uq(g)U_q(\mathfrak{g}) acts; regularity means Λ\Lambda is fixed by no reflection under the dot action w.Λ=w(Λ+ρ)−ρw.\Lambda=w(\Lambda+\rho)-\rho), and the whole framework fixes 0<q<10<q<1 real, so roots of unity are never considered.

Problem Statement

Fix a connected simply connected compact semisimple Lie group KK with complexification G=KCG=K_{\mathbb{C}} and a real parameter 0<q<10<q<1, let Gq=D(Kq)G_q=D(K_q) be the Drinfeld double of the q-deformation of KK, and let Vq(λ,ν)V_q(\lambda,\nu) denote the irreducible admissible GqG_q-module of parameter (λ,ν)∈P×h∗(\lambda,\nu)\in P\times\mathfrak{h}^*, considered up to the diagonal Weyl-group action and the period shifts ν↦ν−2πilog⁡(q)−1γ\nu\mapsto\nu-2\pi i\log(q)^{-1}\gamma with γ∈Q∨\gamma\in Q^{\vee} (the coroot lattice). Determine completely the unitary dual of GqG_q: decide, for every (λ,ν)(\lambda,\nu) with no almost-reality restriction on ν\nu — that is, allowing the pairings (log⁡(q) Im(ν)/(2π),α)(\log(q)\,\mathrm{Im}(\nu)/(2\pi),\alpha) with the roots α\alpha to be arbitrarily large — whether Vq(λ,ν)V_q(\lambda,\nu) admits a positive-definite invariant Hermitian form, including every parameter class attached to a singular central character Λ\Lambda (one fixed by at least one reflection under the dot action), where the categorical equivalence of Arano's Theorem 4.2(iii) is not available. The classification must be given for every simple type beyond type A (where the unitarity-preserving shifts ν↦ν−2πilog⁡(q)−1γ\nu\mapsto\nu-2\pi i\log(q)^{-1}\gamma, γ∈P∨\gamma\in P^{\vee}, already reduce every parameter class to the known almost-real case) and must be consistent with the established anchors: the SLq(2,C)SL_q(2,\mathbb{C}) dual, the type-A classification, Arano's almost-real comparison theorem, and the spherical unitary dual.

References

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