Invariant quantum measure on q -deformed twisted adjoint orbits
aa r X i v : . [ m a t h . QA ] J a n INVARIANT QUANTUM MEASURE ON q -DEFORMED TWISTED ADJOINTORBITS KENNY DE COMMER
Abstract.
Let U be a compact semisimple Lie group with complexification G and associated Cartaninvolution Θ. Let ν be an involutive complex Lie group automorphism of G commuting with Θ, andconsider the associated semisimple real Lie group G ν = { g ∈ G | ν ( g ) = Θ( g ) } . We consider q -deformedanalogues of the U -orbits of the quotient space G ν \ G , and determine for these the associated von Neumannalgebra and invariant state. Introduction
Let U be a connected, simply connected compact Lie group with complexification G = U C . Let Θ be theCartan involution of G with fixed points U , and put g ∗ = Θ( g ) − . Let ν be an involutive complex Lie groupautomorphism of G commuting with Θ. It gives rise to the real form G ν = { g ∈ G | ν ( g ) ∗ = g − } ⊆ G. Consider the quotient map π : G ν \ G → G ν \ G/U partitioning G ν \ G into U -orbits O νx = π − ( x ) , x ∈ G ν \ G/U with associated U -invariant probability measures µ x . Note that these can also be seen as twisted adjoint U -orbits on the space Z ν = { g ∈ G | ν ( g ) ∗ = g } , Ad νu ( g ) = ν ( u ) ∗ gu through the equivariant embedding G ν \ G ֒ → Z ν , G ν g ν ( g ) ∗ g. (0.1)The main goal of this paper is to construct quantum analogues of the ( O νx , µ x ), as well as the degenerate limitsof these orbits obtained by contraction, within the setting of von Neumann algebraic quantum homogeneousspaces. Note that any simply connected symmetric space of compact type can be realized as O ν [ e ] for some( U, ν ), where [ e ] = G ν U is the class of the unit e ∈ G within G ν \ G/U . Similarly, any flag manifold for U can be realized as a U -orbit at a point at infinity of U \ G . We mention that the degenerate limits can alsobe interpreted as orbits of U within a wonderful compactification of G ν \ G , cf. [EL01].Algebraic aspects of the quantum analogues of the above spaces were studied in [DCM18], to which wealso refer for further intuition and motivation. In this paper, we will take into account some finer spectralconditions to handle also analytic aspects. For the sake of the introduction, we explain some of the relevantconsiderations in an easy special case.Let G = SL ( N, C ) ⊆ M N ( C ) and let X X ∗ be the usual hermitian adjoint with associated Cartaninvolution Θ( g ) = ( g ∗ ) − . For 1 ≤ m, n with m + n = N define E = (cid:18) I m − I n (cid:19) . The work of K. De Commer was supported by the FWO grants G025115N and G032919N, and the grant H2020-MSCA-RISE-2015-691246-QUANTUM DYNAMICS. The author thanks the Fields Institute for the nice working environment. e obtain on SL ( N, C ) the involution ν ( g ) = Ad E ( g ) = E g E , and associated to ν we have the closed real Lie subgroup G ν = { g ∈ G | ν ( g ) ∗ = g − } = { g ∈ SL ( N, C ) | g ∗ E g = E } = SU ( m, n ) ⊆ SL ( N, C ) . Consider the ∗ -algebra O ( G R ) of regular functions on G as a real algebraic group, so O ( G R ) is generatedby the coordinate functions x ij ( g ) = g ij as well as their ∗ -conjugates x ij ( g ) = g ij . Inside we have the ∗ -subalgebra O ( G R ) G ν = { f ∈ O ( G R ) | ∀ g ∈ G ν , h ∈ G : f ( gh ) = f ( h ) } ⊆ O ( G R ) , with associated maximal real spectrum G ν \\ G = mSpec ∗ (cid:0) O ( G R ) G ν (cid:1) = {∗ -homomorphisms O ( G R ) G ν → C } , where we borrow notation from geometric invariant theory. We have a natural embedding G ν \ G ֒ → G ν \\ G, (0.2)but it is not bijective. One has however a concrete model for G ν \\ G : consider the real affine variety Z ν introduced above, Z ν = { h ∈ G | ν ( h ) ∗ = h } = { h ∈ SL ( N, C ) | ( E h ) ∗ = E h } , Ad νg ( h ) = h · g = ν ( g ) ∗ hg, g ∈ G, h ∈ Z ν . It comes equipped with an associated unital ∗ -algebra of regular functions O ( Z ν ) generated by the coordinatefunctions z ij ( h ) = ( E h ) ij , z ∗ ij = z ji . We can G -equivariantly identify G ν \\ G ∼ = Z ν by O ( Z ν ) ∼ = O ( G R ) G ν , z ij X k x ki E kk x kj , and under this isomorphism (0.2) becomes the injective G -equivariant map (0.1).In the following we simply view G ν \ G ⊆ Z ν by this map. Then G ν \ G is easily seen to be the connectedcomponent of the identity I N ∈ Z ν , consisting of those h with E h selfadjoint and of the same signature as E . However, this is not sufficient to have an algebraic description of G ν \ G inside Z ν as a semialgebraic setdetermined by polynomial equalities and polynomial inequalities. Instead, we break up G = U AN = AN U where U = G id = SU ( N ) = { g ∈ G | g ∗ = g − } is the associated maximal compact subgroup of SL ( N, C )and AN ⊆ SL ( N, C ) is the subgroup of upper triangular matrices with positive diagonal. Consider the AN -orbit Z + , reg ν = I N · AN ⊆ G ν \ G with Euclidian closure Z + ν . With a k ( h ) = det(( h ij ) i,j :1 → k )the leading minors of h , we can realize Z + , reg ν as the semialgebraic set Z + , reg ν = { h ∈ Z ν | ∀ k : a k ( h ) = a k ( E h ) /a k ( E ) > } , and G ν \ G becomes the union of the U -orbits through points of Z + , reg ν , G ν \ G = Z + , reg ν · U. (0.3)There is also a natural cell decomposition of G ν \ G : consider the natural implementation W = S N ⊆ U ( N )of the Weyl group of SU ( N ), and let W + ν = Stab W ( E ) = S m × S n . Using the natural extension of the right G -action to GL ( N, C ), put Z [ w ] , reg ν = I N · w − AN, [ w ] = wW + ν ∈ W/W + ν , and let Z [ w ] ν be its Euclidian closure. Consider Z reg ν = { h ∈ Z ν | ∀ k : a k ( h ) = 0 } , ( G ν \ G ) reg = Z reg ν ∩ G ν \ G. e have that ( G ν \ G ) reg is Euclidian dense in G ν \ G , with G ν \ G = [ [ w ] ∈ W/W + ν Z [ w ] ν , ( G ν \ G ) reg = G [ w ] ∈ W/W + ν Z [ w ] , reg ν . Consider now the U -orbits O νx ⊆ G ν \ G, x ∈ G ν \ G/U.
For example, O ν [ e ] is the symmetric space K ν \ U with K ν = U ∩ G ν = S ( U ( m ) × U ( n )). The cell decompositionof G ν \ G passes to each of these orbits, O νx = [ [ w ] ∈ W/W + ν Z [ w ] ν ∩ O νx , O ν, reg x := ( G ν \ G ) reg ∩ O νx = G [ w ] ∈ W/W + ν Z [ w ] , reg ν ∩ O νx , (0.4)with O ν, reg x dense in O νx .The above constructions can be performed for any compact semisimple Lie group U , and can then bequantized when ν is chosen in such a way that it preserves the Borel subgroup B of G = U C along which thequantization of U is performed [DCM18]. That is, for any 0 < q < ∗ -algebra O q ( Z ν ) can be constructed,along with a coaction by O q ( U ), with O q ( U ) the standard deformation of the Hopf ∗ -algebra of regularfunctions on U , in such a way that the classical setting is obtained by a suitable limiting procedure q → O q ( G ν \ G ) can be obtained by endowing O q ( Z ν ) with suitable spectral conditions .It turns out that the ∗ -algebra O q ( G ν \ G/U ) = O q ( G ν \ G ) O q ( U ) of O q ( U )-coinvariants in O q ( G ν \ G ) iscentral, so also quantized algebras O q ( O νx ) of associated orbit spaces can be obtained, parametrized bysuitable characters x on O q ( G ν \ G/U ). Such a ∗ -algebra comes equipped with a canonical von Neumannalgebraic completion L ∞ q ( O νx ), together with an invariant state R O νx,q . Our main result can now be statedqualitatively as follows. We denote T for the chosen maximal torus of U , and T ν for the subgroup of ν -fixedpoints. We endow T, T ν and T /T ν with their normalized Lebesgue measure, denoted generically by λ . Theorem. (Theorem 6.1 and Theorem 6.5) Let W be the Weyl group of U . Then there exist subgroups W + ν ⊆ W ν ⊆ W and Hilbert spaces H [ w ] for [ w ] ∈ W ν /W + ν such that L ∞ q ( O νx ) ∼ = (cid:16) ⊕ [ w ] ∈ W ν /W + ν B ( H [ w ] ) (cid:17) ⊗ L ∞ ( T /T ν ) . Moreover, the invariant state R O νx,q is given by Z O νx,q = 1 P [ w ] ∈ W ν /W + ν Tr( A [ w ] ) (cid:16) ⊕ [ w ] ∈ W ν /W + ν Tr( A [ w ] − ) (cid:17) ⊗ Z T/T ν − d λ ( θ ) where the A [ w ] are explicitly given as absolute values of a particular element a ρ ∈ O q ( G ν \ G ) in the associatedrepresentation. Remark 0.1. (1)
Our actual theorems allow more general input than an involution ν , so that alsodegenerate cases associated to general flag varieties can be treated. (2) In case of the compact Lie group U × U with involution ν ( u, v ) = ( v, u ) , we have an isomorphism O q ( O ν [ e ] ) ∼ = O q ( U ) . For this case, the above formula for the invariant state was obtained in [RY01] .We will deal with this specific example in Section 7.3. (3) In the case of quantized Hermitian symmetric spaces, closely related formulas were obtained in apurely algebraic setting in [Vak90] . However, the situation is slightly different there: only one ofthe cells appear, only the infinitesimal action of the quantized enveloping algebra is considered, andmoreover the relevant ∗ -structure of the acting quantized enveloping algebra corresponds to a non-compact real form. This is consistent with the fact that classically, one can indeed realize a non-compact hermitian Riemannian symmetric space as a cell within its compact Cartan dual, and thusas a bounded symmetric domain. The above cell structure on symmetric spaces is closely related to the decomposition into maximalleafs for a natural Poisson structure on symmetric spaces [FL04] . Although the Poisson structurethere is constructed by choosing the Borel in Iwasawa-Borel position, so that the base point at theunit is itself a one-point symplectic leaf, one can also construct the Poisson structure directly froma Borel preserved by the involution. Note that in turn, the symplectic leafs for this Poisson struc-ture are closely related to projections of B -orbits in G ν \ G . The latter are of course in one-to-onecorrespondence with the G ν -orbits on the flag space G/B [RS93] , whose structure can be examinedby considering restrictions of the action to rank one subgroups. The same idea is applied here in thequantum setting, similar to what was done for the quantization of U itself in [LS91] . The precise contents of this paper are as follows. In the first section , we prepare some general theory onspectral conditions for a ∗ -algebra in the presence of a compact quantum group action. In the second section ,we present some preliminaries on twisting data for semisimple Lie algebras and their associated Weyl groups.In the third section , we recall some of the theory of [DCM18], mainly to introduce notation. In the fourthsection , we obtain a version of the Harish-Chandra isomorphism for quantized enveloping algebras twistedby a twisting datum. In the fifth section we develop general results on the representation theory of O q ( Z ν ),which is then used in the sixth section to obtain our main theorem. In the seventh section some generalitieson the representation theory of O q ( Z ν ) are stated, and some concrete examples are considered such as thediagonal action of U × U on U as well as the quantization of the example mentioned in the introduction. Inan appendix , some calculations in low rank are gathered.1. Coactions on spectral ∗ -algebras Spectral ∗ -algebras. In the following, all algebras are assumed unital.
Definition 1.1. A spectral ∗ -algebra consists of a ∗ -algebra A together with a family P of bounded ∗ -representations of A on Hilbert spaces, called the spectral structure . In the following, we write a spectral ∗ -algebra as O ( X ). We write O ( X C ) for O ( X ) as an algebra, forgettingthe ∗ -structure and the spectral structure. We write O ∗ ( X ) = O ( X ∗ ) = O ( X ) /I ∗ I ∗ = ∩ π ∈ P Ker( π ) . We call I ∗ the ideal of non-admissibility . Note that we can have I ∗ = O ( X ), in which case we write X ∗ = ∅ .If A is a ∗ -algebra, the associated full spectral ∗ -algebra O ( X A ) is A together with the family of all itsbounded ∗ -representations on Hilbert spaces (possibly empty). For O ( X ) a spectral ∗ -algebra we write O ( X full ) for the associated full spectral ∗ -algebra. Remark 1.2.
This notion of spectral ∗ -algebra is very weak, and will in practice have little value if we do notimpose certain properties on P , such as being closed under arbitrary bounded direct sums. A more refinednotion of ‘closed spectral conditions’ was introduced in [DCF19, Definition 1.1] . The weak version above willhowever suffice for our purposes. If A is a ∗ -algebra and π a bounded ∗ -representation of A , we call π irreducible if the representation space H π has no non-trivial closed A -invariant subspaces. For π a general bounded ∗ -representation of A , wedenote by P Irr π the collection of all irreducible ∗ -representations π ′ which are weakly contained in π , meaningthat π ′ lifts to a ∗ -representation of the C ∗ -algebra π ( A ) k−k , the norm-closure of π ( A ). In this case wewrite π ′ π , so that P Irr π = { irreducible π ′ | π ′ π } . Definition 1.3.
Let O ( X ) = ( O ( X ) , P ) be a spectral ∗ -algebra. We denote P Irr = ∪ π ∈ P P Irr π . We call a bounded ∗ -representation π of O ( X ) admissible if P Irr π ⊆ P Irr , and we write P X = h P i = { π | π admissible bounded ∗ -representation } . he following lemma collects some easily verified properties. Lemma 1.4.
Let O ( X ) = ( O ( X ) , P ) be a spectral ∗ -algebra. Then (1) P ⊆ P X and P Irr ⊆ P X . (2) ( P X ) Irr = P Irr . (3) P X = h P X i = h P Irr i . (4) The ideal of non-admissibility does not change upon passing from P to P Irr or P X . General theory of compact quantum groups.
We recall some general theory on compact quantumgroups, see e.g. [NT13].Let H = ( H, ∆ , ε, S ) be a Hopf ∗ -algebra with a (necessarily unique) invariant state. We view H = O ( U ) asa full spectral ∗ -algebra, and interpret O ( U ) as the ∗ -algebra of regular functions on a ‘compact quantumgroup’ U .We denote the unique invariant state on O ( U ) as R U , and refer to it as the Haar integral . We write L ( U )for the GNS-completion of O ( U ) with respect to R U , C r ( U ) ⊆ B ( L ( U )) for the associated reduced C ∗ -algebra and C u ( U ) for the associated universal C ∗ -algebra. When U is coamenable, i.e. when the natural ∗ -homomorphism C u ( U ) → C r ( U ) is an isomorphism, we simply write C ( U ) for these C ∗ -algebras. We write L ∞ ( U ) = C r ( U ) ′′ ⊆ B ( L ( U )) for the associated von Neumann algebra.Recall that O ( U ) is spanned linearly by the matrix entries of its unitary corepresentations U ∈ B ( H ) ⊗ O ( U )on finite-dimensional Hilbert spaces H . Endowing the linear dual O ( U ) ′ with its convolution algebra structureand the ∗ -operation ω ∗ ( u ) = ω ( S ( u ) ∗ ) , we have a one-to-one correspondence between weak ∗ -continuous ∗ -representations π of O ( U ) ′ on finite-dimensional Hilbert spaces and unitary corepresentations U π of O ( U ) via π ( ω ) = (id ⊗ ω ) U π . We denote by ˆ ε the trivial representation of O ( U ) ′ with U ˆ ε = 1.For U π a unitary corepresentation on H π and ξ, η ∈ H π we write the associated matrix coefficient as U π ( ξ, η ) = ( ω ξ,η ⊗ id) U π , ω ξ,η ( X ) = h ξ, Xη i for X ∈ B ( H π ) . Then ω ( U π ( ξ, η )) = h ξ, π ( ω ) η i , ω ∈ O ( U ) ′ . The Haar integral on O ( U ) is determined by Z U , Z U U π ( ξ, η ) = 0 , π irreducible and π ≇ ˆ ε. More generally, if π, π ′ are inequivalent irreducible unitary representations of U we have Z U U π ( ξ, η ) ∗ U π ′ ( ξ ′ , η ′ ) = 0 , and there exists a unique character ˆ δ / ∈ O q ( U ) ′ with π (ˆ δ / ) strictly positive for all π and Z U U π ( ξ, η ) ∗ U π ( ξ ′ , η ′ ) = h η, η ′ ih ξ ′ , π (ˆ δ / ) ξ i Tr( π (ˆ δ / )) , π irreducible . (1.1)We call the associated family of functionalsˆ δ iz ∈ O ( U ) ′ , ˆ δ iz ( U π ( ξ, η )) = h ξ, π (ˆ δ / ) iz/ η i he family of Woronowicz characters . They are pointwise analytic in z , are characters of O ( U ) which are ∗ -preserving for z ∈ R , form a complex one-parametergroupˆ δ iz ˆ δ iw = ˆ δ i ( w + z ) with respect to convolution, and satisfy (ˆ δ iz ) ∗ = ˆ δ − iz . One can express the antipode squared of O ( U ) by means of the Woronowicz characters as S ( u ) = (ˆ δ − / ⊗ id ⊗ ˆ δ / )∆ (2) ( u ) , ∆ (2) = (∆ ⊗ id)∆ . We further define the modular group σ z as the complex one-parameter group of (non- ∗ -preserving) algebraautomorphisms of O ( U ) determined by σ z ( u ) = (ˆ δ − iz/ ⊗ id ⊗ ˆ δ − iz/ )∆ (2) ( u ) . Then σ = σ − i satisfies the modularity condition Z U ab = Z U bσ ( a ) . We extend in the following the convolution product to ∗ -representations: if π, π ′ ∈ P U , we write π ∗ π ′ = ( π ⊗ π ′ )∆ , H π ∗ π ′ = H π ⊗ H π ′ . Actions of compact quantum groups on spectral ∗ -algebras.Definition 1.5. Let U be a compact quantum group, and let O ( X ) be a spectral ∗ -algebra. A spectral rightcoaction of O ( U ) on O ( X ) is given by a ∗ -preserving right coaction α by O ( U ) , α : O ( X ) → O ( X ) ⊗ O ( U ) , satisfying the spectrality condition π ∈ P X , π ′ ∈ P U ⇒ π ∗ π ′ ∈ P X , where π ∗ π ′ = ( π ⊗ π ′ ) α. We then call X a quantum U -space . Note that α automatically descends to a coaction on O ∗ ( X ). Also note that if O ( X ) is a full spectral ∗ -algebra,any right ∗ -coaction is automatically spectral.If α does not satisfy the spectrality condition, we can enlarge P X to the set P XU = h{ π ∗ π ′ | π ∈ P X , π ′ ∈ P U }i ⊇ P X , and we denote the associated spectral ∗ -algebra as O ( XU ). In particular, O ( XU ) has the same underlying ∗ -algebra as O ( X ), and we can interpret X ⊆ XU ⊆ X full . We note the following properties. The secondproperty states in a way that any U -orbit of XU meets X and consists of the U -translates of any of its pointsin X . This nice behaviour comes however at the price of a strong condition on U , which will be satisfied forthe particular U we are interested in. Lemma 1.6.
Let U be a compact quantum group with C ( U ) a type I C ∗ -algebra, and let O ( X ) be a spectral ∗ -algebra with right coaction by O ( U ) . The following holds: (1) The coaction α is spectral on O ( XU ) . (2) If P X = h P i , then P XU = h{ π ∗ π ′ | π ∈ P , π ′ ∈ P U }i = h{ π ∗ π reg | π ∈ P }i , where π reg : O ( U ) → B ( L ( U )) is the regular representation. Note that, under the condition of the previous lemma, U is automatically coamenable by [CS19, Proposition2.5], upon noting that a type I C ∗ -algebra always admits a finite-dimensional ∗ -representation, cf. [BMT03]. roof. The first property in the lemma follows from the second, since then P ( XU ) U = h{ π ∗ π ′ ∗ π ′′ | π ∈ P X , π ′ , π ′′ ∈ P U } = h{ π ∗ π ′ | π ∈ P X , π ′ ∈ P U }i = P XU . In the second property the inclusions ⊇ are obvious. Let us show that P XU ⊆ h{ π ∗ π reg | π ∈ P }i . Write B π = π ( O ( X )) k−k and C π = π ( O ( U )) k−k for resp. π ∈ P X and π ∈ P U . Pick an irreducible π with π π ′ ∗ π ′′ for π ′ ∈ P X and π ′′ ∈ P U . We are to show that there exists e π ∈ P with π e π ∗ π reg .Let ω be a pure state on B π ′ ∗ π ′′ associated to π . Then ω can be lifted to a pure state on π ′ ( O ( X )) ⊗ π ′′ ( O ( U )) k−k = B π ′ ⊗ min C π ′′ . Since C π ′′ is type I , this lift must be a vector state associated to a tensor product π ⊗ π of irreduciblerepresentations of respectively B π ′ and C π ′′ . It follows that π π ∗ π . Since π π ′ , we can pick e π ∈ P with π e π . Since U is coamenable, we also have π π reg . But since we have the descent B e π ∗ π reg ⊆ B e π ⊗ min C π reg π ⊗ π → B π ⊗ min C π ⊇ B π ∗ π , we must also have π e π ∗ π reg . (cid:3) Quantum homogeneous spaces. If X is a quantum U -space, we denote the ∗ -algebra of coinvariantsas O ( X ) α = { f ∈ O ( X ) | α ( f ) = f ⊗ } . We write O ( X // U ) = O ( X ) α as a full spectral ∗ -algebra, and (1.2) O ( X / U ) = O ( X ) α as a spectral ∗ -algebra for P X / U = h{ π |O ( X ) α | π ∈ P X }i . (1.3) Definition 1.7. [Boc95]
We call X a quantum homogeneous space , or the action of U on X ergodic , if O ( X ) α = C . If X is a general quantum U -space, note that any f ∈ O ( X ) lies in the linear span of a finite set of elements f i ∈ O ( X ) which transform according to a unitary corepresentation U = ( u ij ) ij for O ( U ). It follows that P i f ∗ i f i ∈ O ( X ) α . Assuming that X is a quantum homogeneous space, each P i f ∗ i f i as above is scalar. Inparticular, any ∗ -representation π of O ( X ) on a (pre-)Hilbert space H is bounded, and there exists for each f ∈ O ( X ) a π -independent constant C f such that k π ( f ) k ≤ C f for all such π . Hence we can associate to O ( X ) the universal C ∗ -algebra C u ( X ) obtained from the seminorm k x k = sup {k π ( x ) k | π ∈ P X } . We have O ( X ) ։ O ∗ ( X ) ֒ → C u ( X ) . and the inclusion above has dense image.For X a quantum homogeneous space, we write R X for the associated normalized invariant functional on O ( X ), determined by Z X f = (cid:18) id ⊗ Z U (cid:19) α ( f ) . This functional descends to a faithful state on O ∗ ( X ). We write L ( X ) for the GNS-space of O ∗ ( X ) withrespect to R X , C r ( X ) for the reduced C ∗ -algebra, and L ∞ ( X ) for the associated von Neumann algebra. Thesecome equipped with respective coactions of C r ( U ) and L ∞ ( U ). When U is coamenable, the natural projectionmap C u ( X ) → C r ( X ) will be an isomorphism, and we simply write C ( X ) in this case.When U is coamenable, the spectral structure of O ( X ) will necessarily be the full one. Lemma 1.8.
Let α : O ( X ) → O ( X ) ⊗ O ( U ) be a spectral ergodic coaction, with U coamenable. Then O ( X ) is a full spectral ∗ -algebra. roof. By coamenability, any irreducible ∗ -representation of O ( X full ) is weakly contained in the regularrepresentation π X , reg of O ( X ) on L ( X ). It hence suffices to show that the latter is admissible for O ( X ). Ifhowever π ∈ P X is arbitrary, we have that π X , reg ⊆ π ∗ π U , reg , and the latter is admissible by spectrality ofthe coaction. (cid:3) Note that, for X homogeneous, O ∗ ( X ) also admits a unique modular automorphism σ : O ∗ ( X ) → O ∗ ( X )such that Z X xy = Z X yσ ( x ) , x, y ∈ O ∗ ( X ) , see [BDRV06, Proposition 2.10]. The automorphism σ extends uniquely to a complex analytic one-parametergroup of diagonalizable automorphisms σ z : O ∗ ( X ) → O ∗ ( X ) with σ = σ − i . We have α ◦ σ = ( σ ⊗ S ) ◦ α. (1.4)1.5. Centrally coinvariant coactions and quantum orbits.
In this paper, we will be interested in aslightly more general class of actions than just the ergodic ones. We use in the following definition thenotation introduced in (1.2) and (1.3).
Definition 1.9.
Let O ( X ) be a spectral ∗ -algebra with a spectral coaction by O ( U ) . We say that O ( X ) has central coinvariants if O ( X ) α ⊆ Z ( O ( X )) , the center of O ( X ) .We write in this case X // U = mSpec ∗ ( O ( X ) α ) = {∗ -characters of O ( X ) α } , X / U = { admissible ∗ -characters of O ( X / U ) } ⊆ X // U . We view X // U and X / U as topological spaces by the topology of pointwise convergence on O ( X ) α . Definition 1.10.
Let x ∈ X // U . We call a bounded ∗ -representation π of O ( X full ) an x -representation if π restricts to x on O ( X // U ) . By definition we have x ∈ X / U if and only if there exists an admissible x -representation π of O ( X ). Byelementary C ∗ -theory, we can also always choose this π to be irreducible.For x ∈ X // U we write e I x for the 2-sided ∗ -ideal e I x = O ( X ) { f − x ( f ) | f ∈ O ( X ) α } ⊆ O ( X ) . By central coinvariance α descends to an ergodic spectral coaction on O ( X ( x )) = O ( X ) / e I x , and we call X ( x ) the fiber of X at x . Here we interpret O ( X ( x )) as a spectral ∗ -algebra by P X ( x ) = { π ∈ P X | π is an x -representation } . We know from Lemma 1.8 that this is simply the full spectral ∗ -algebra structure if U is coamenable. Itfollows that in this case X / U consists of those x ∈ X // U for which the ideal of non-admissibility for O ( X ( x ))is not the whole algebra. We then write for x ∈ X / U O ( X ∗ ( x )) = O ∗ ( X ( x )) , and we call X ∗ ( x ) the fiber of X ∗ at x .Given an arbitrary coaction α , let E α be the projection map E α : O ( X ) → O ( X ) α , f (cid:18) id ⊗ Z U (cid:19) α ( f ) . The following lemma gives an easy characterisation of X full / U . emma 1.11. Let O ( X ) be a full spectral ∗ -algebra with a centrally coinvariant ∗ -coaction by O ( U ) . Let x ∈ X // U . Then x ∈ X / U if and only if the invariant functional Z X ( x ) = x ◦ E α on O ( X ) is a positive functional, i.e. Z X ( x ) f ∗ f ≥ for all f ∈ O ( X ) . Proof. If x ∈ X full / U , take any x -representation π of O ( X ), and take any non-zero vector ξ ∈ H π withassociated vector state ω ξ . Then we have Z X ( x ) f ∗ f = (cid:18) ( ω ξ ◦ π ) ⊗ Z U (cid:19) α ( f ∗ f ) ≥ , f ∈ O ( X ) . Conversely, if R X ( x ) is positive, let π x be the associated GNS-representation. The same argument whichshowed the existence of the universal C ∗ -envelope shows that π x is bounded, and thus an x -representationof O ( X ). (cid:3) In particular, we see that X / U is a closed subset of X // U . Note that if we start with a general spectralcoaction with central coinvariants, we then have the (possibly all strict) inclusions X / U ⊆ X full / U ⊆ X // U . Two examples.
Most of the following material can be found, as far as the algebraic constructions areconcerned, in standard reference works on quantum groups, see e.g. [Maj95, KS97].Consider for 0 < q < R -matrix R = q − / ( e ⊗ e + e ⊗ e ) + q / ( e ⊗ e + e ⊗ e ) + q / ( q − − q ) e ⊗ e ∈ M ( C ) ⊗ M ( C ) . Let O q ( M ( C )) be the algebra generated by a matrix M = (cid:18) a bc d (cid:19) ∈ M ( C ) ⊗ O q ( M ( C )) of indeterminatessuch that R M M = M M R . By direct verification one sees that O q ( M ( C )) has universal relations ac = qca, ab = qba, bc = cb, cd = qdc, bd = qdb, ad − da = ( q − q − ) bc. It is a bialgebra in a natural way such that M becomes a corepresentation. It becomes a ∗ -bialgebra by the ∗ -structure a ∗ = d, b ∗ = − qc. With V = C interpreted as row vectors, we write this ∗ -algebra as O q ( V R ) = O ( V R ,q ), as we view it as aquantization of the ∗ -algebra of regular functions on V as a real affine variety, with a, b corresponding to the(holomorphic) coordinate functions and d, c to their (anti-holomorphic) complex conjugates. We view here O q ( V R ) as a full spectral ∗ -algebra.Let D = ad − qbc = da − q − bc. Then D is a selfadjoint grouplike central element, and we write O q ( SU (2)) = O ( SU q (2)) for the Hopf ∗ -algebra obtained by imposing the extra relation D = 1. We write in this case the fundamental corepresen-tation M as U .Clearly the comultiplication gives rise to a coaction α : O q ( V R ) → O q ( V R ) ⊗ O q ( SU (2)) , (id ⊗ α ) M = M U . The following proposition is easily verified. roposition 1.12. The coaction on O q ( V R ) is centrally coinvariant, with O q ( V R ) α = C [ D ] . Identifying mSpec ∗ ( C [ D ]) ∼ = R via χ χ ( D ) , we have V R ,q //SU q (2) = R , V R ,q /SU q (2) = R ≥ . For λ = 0 we have O ∗ q ( V R (0)) = C with the trivial coaction.For λ > we have O ∗ q ( V R ( λ )) ∼ = O q ( SU (2)) equivariantly via M λ / U . Also the following theorem can be easily derived by hand, or deduced from the results in [LS91, Koo91].
Theorem 1.13.
A full list of inequivalent irreducible ∗ -representations of O ∗ q ( V R ( λ )) for λ > is given bythe ∗ -characters χ λ,θ (cid:18) a bc d (cid:19) = (cid:18) λ / e iθ λ / e − iθ (cid:19) , θ ∈ T = R / π Z , and the infinite-dimensional ∗ -representations S λ,θ = l ( N ) , ae n = λ / (1 − q n ) / e n − , be n = λ / e − iθ q n e n , θ ∈ T . When θ = 0 we drop the symbol θ , and when interpreting O q ( V R (1)) ∼ = O q ( SU (2)) we drop the symbol λ .We then have χ λ ∗ χ θ = χ λ,θ , S λ ∗ χ θ = S λ,θ , χ λ ∗ S = S λ and S λ ∗ S ∼ = Z T S λ,θ d θ. We also note that the Haar integral of SU q (2) can be realized with respect to the ∗ -representation Z T S ∗ χ θ d θ = l ( N ) ⊗ l ( Z ) , ae n,m = (1 − q n ) / e n − ,m , be n,m = q n e n,m − , since then Z SU q (2) x = (1 − q ) X n h be n , xbe n i , see [Wor87, Appendix A.1]. It follows easily from this that L ∞ ( SU q (2)) ∼ = B ( l ( N )) ⊗ L ∞ ( T ) . Let us now consider a second example. Let O br q ( M ( C )) be the algebra of braided Z = (cid:18) z wv u (cid:19) with relations R Z R Z = Z R Z R . It has the ∗ -structure Z ∗ = Z . As a full spectral ∗ -algebra, we interpret it as O q ( H ) = O ( H ,q ) with H the space of hermitian 2-by-2 matrices. It carries a right coaction by O q ( SU (2)) given as Z U ∗ Z U . (1.5)Write T = q − u + qz, D = uz − q − vw, which are called respectively the quantum trace and quantum determinant . We easily find by direct com-putation that the universal relations of O q ( H ) are centrality and selfadjointness of T, D together with therelations z ∗ = z, w ∗ = v, zw = q wz, vz = q zv, and q − vw = − D + qT z − q z , q − wv = − D + q − T z − q − z . he main results in the following proposition can be obtained from [Pod87, MN90, MNW91]. The computa-tion of the ∗ -products of representations can be done by spectral analysis along the lines of e.g. [DeC11, Ap-pendix B]. Proposition 1.14.
The coaction on O q ( H ) is centrally coinvariant with O q ( H ) α = C [ D, T ] . Identifying mSpec ∗ ( C [ D, T ]) ∼ = R via χ ( χ ( D ) , χ ( T )) = ( d, t ) , we have that H ,q //SU q (2) = R , H ,q /SU q (2) = S + ∪ S ∪ S − where, writing s n = q − n − + q n +1 for n ∈ N , S + = { ( c , cs n ) | c ∈ R × , n ∈ N } , S = { } × R , S − = R < × R . Moreover, we have the following fusion rules, denoting by H a Hilbert space of countably infinite dimensionencoding countable multiplicity: (1) For ( c , cs n ) ∈ S + we have O ∗ q ( H ( c , cs n )) ∼ = M n +1 ( C ) with unique irreducible ∗ -representation S c ,n ∼ = C n +1 , ze k = cq − n +2 k e k , ve = 0 . Convolution with O q ( SU (2)) -representations is given by S c ,n ∗ χ θ ∼ = S c ,n , S c ,n ∗ S ∼ = H ⊗ S c ,n . The invariant state on O ∗ q ( H ( c , cs n )) ∼ = M n +1 ( C ) is given by Z H ,q ( c ,cs n ) X = Tr( zX )Tr( z ) . (2) For (0 , t ) ∈ S we have the ∗ -character χ ( z ) = χ ( v ) = χ ( w ) = 0 . When t = 0 this is the onlyirreducible ∗ -representation, while if t = 0 the only other irreducible ∗ -representation is given by S ,t ∼ = l ( N ) , ze n = q n +1 te n , ve = 0 . Convolution with O q ( SU (2)) -representations is given by χ ∗ χ θ ∼ = χ χ ∗ S ∼ = S ,t , S ,t ∗ χ θ ∼ = S ,t , S ,t ∗ S ∼ = H ⊗ S ,t . For t = 0 the associated von Neumann algebra can be identified as L ∞ q ( H (0 , t )) = B ( S ,t ) , and under this isomorphism the invariant state on O ∗ q ( H (0 , t )) is given as Z H ,q (0 ,t ) X = Tr( zX )Tr( z ) . (3) For ( − c , t ) ∈ S − with c > and t = ( a − a − ) c for a > we have as complete list of non-equivalentirreducibles the ∗ -characters χ θ with χ θ ( z ) = 0 and χ θ ( v ) = qce iθ and the ∗ -representations S ±− c ,a ∼ = l ( N ) , ze n = ± ca ± q n +1 e n , ve = 0 . Writing S − c ,a = S + − c ,a ⊕ S −− c ,a , one has convolution with O q ( SU (2)) -representations given by χ θ ∗ χ θ ′ = χ θ +2 θ ′ , χ θ ∗ S ∼ = S − c ,a , S ±− c ,a ∗ χ θ ∼ = S ±− c ,a , S ±− c ,a ∗ S ∼ = H ⊗ S − c ,a . The associated von Neumann algebra can be identified as L ∞ q ( H ( − c , t )) = B ( S + − c ,a ) ⊕ B ( S −− c ,a ) , and under this isomorphism the invariant state on O ∗ q ( H ( − c , t )) is given (in obvious notation) as Z H ,q ( − c ,t ) X = Tr + ( zX ) − Tr − ( zX )Tr + ( z ) − Tr − ( z ) . The main goal of this paper will be to find higher rank analogues of the above formulas for the ‘quantuminvariant measures’ on the V R ,q ( λ ) and H ,q ( d, t ). . Twisting data
In this section we fix an index set I of cardinality | I | . We further fix a Euclidian space V ∼ = R | I | with innerproduct ( − , − ) and a reduced finite root system ∆ ⊆ V which spans V . We choose a system of positiveroots ∆ + and associated simple roots Φ = { α r | r ∈ I } ⊆ ∆ + with indexation by I . We denote by Γ theassociated Dynkin diagram on I , with arrows pointing towards the shorter roots. We write α ∨ = α,α ) α forthe coroots, ̟ r ∈ V for the fundamental weights such that ( ̟ r , α ∨ s ) = δ rs , and ̟ ∨ r for the fundamentalcoweights such that ( ̟ ∨ r , α s ) = δ rs . We write the Cartan matrix A = ( a rs ) r,s = (( α ∨ r , α s )) r,s . We write Q + = N { α r | r ∈ I } and Q = Q + − Q + for resp. the cone of positive roots and the root lattice,and similarly P + = N { ̟ r | r ∈ I } and P = P + − P + for the cone of dominant integral weights and theweight lattice.For J ⊆ I a subset, we write Γ J for the subdiagram of Γ with vertex set J and associated Cartan matrix A J = ( a rs ) r,s ∈ J . We call Γ J a full Dynkin subdiagram . We write ∆ J = ∆ ∩ Z { α r | r ∈ J } for the associatedroot system.2.1. Twisting data.Notation 2.1.
For τ an involution of I , we write • I τ for the τ -fixed vertices, • I ∗ for a fixed fundamental domain of τ inside I \ I τ . Assume now moreover that τ is a Dynkin diagram involution, allowing τ = id. Then we can extend τ uniquely to an isometric involution on V such that τ ( α r ) = α τ ( r ) . This extension preserves the root system and its positive roots, τ (∆) = ∆ , τ (∆ + ) = ∆ + . Definition 2.2.
We call Γ connected if its root system is irreducible.We call Γ τ -connected if I = J ∪ τ ( J ) with Γ J connected.We call a subdiagram Γ ′ ⊆ Γ a τ -connected component if Γ ′ is a maximal τ -connected full subdiagram. Any Γ is obviously the disjoint union of its τ -connected components. Remark 2.3.
Note that Γ is τ -connected if and only if it is connected or it is the disjoint union of twoisomorphic Dynkin diagrams which are switched under τ . We call the latter (Γ , τ ) of direct product type. The following notion will play an important rle in the paper.
Definition 2.4.
We call a couple ν = ( τ, ǫ ) = ( τ ν , ǫ ν ) a twisting datum if τ is an involution of the Dynkindiagram Γ and ǫ is a C -valued function on I with ǫ τ ( r ) = ǫ r , for all r ∈ I. (2.1) We write H = { ν } ⊆ Inv(Γ) × C | I | for the space of all twisting data.We write J ν = J ǫ = Supp( ǫ ) = { r ∈ I | ǫ r = 0 } for the locus of non-degeneracy . Note that J ν is τ -invariant. We will need various extra conditions on twisting data, depending on the context. We gather these conditionsin the following definition. efinition 2.5. We call a twisting datum ν • regular if J ν = I , • positive if ν is regular and ǫ r > for all r ∈ I τ , • strongly positive if ǫ r > for all r ∈ I , • of symmetric pair type if | ǫ r | = 1 for all r ∈ I . • a gauge if ǫ r = 1 for r ∈ I τ and | ǫ r | = 1 for r ∈ I ∗ , • ungauged if ǫ r ≥ for r ∈ I ∗ , • reduced if ǫ r ∈ {− , , } for r ∈ I τ and ǫ r ∈ { , } for r ∈ I ∗ , • strongly reduced if ν is reduced and each τ -connected component of J ν has at most one r with τ ( r ) = r and ǫ r = − .We write the respective spaces as H × , H > , H ≫ , H s , H g , H ug , H r , H rr . (2.2) The trivial twisting datum + is given by τ = id and ǫ r = 1 for all r . Remark 2.6.
The notion of twisting datum is very similar to that of ‘extended τ -signature’ [EL01] , but weallow more flexibility. We identify the space of all twisting data with underlying involution τ as H τ = { ǫ } ⊆ C | I | , and we index in this case all notations in (2.2) with τ . Similarly, if we fix also a subset J ⊆ I as the locusof non-degeneracy, we index all notations in (2.2) with J .The space H τ forms a monoid under pointwise multiplication, with group of invertibles H × τ . Definition 2.7.
We call two twisting data ν, ν ′ multiplicatively equivalent , and write ν ∼ m ν ′ , if τ = τ ′ and if ǫ and ǫ ′ lie in the same orbit under multiplication with H >τ . We stress that under this equivalence the values at I τ can only be multiplied with strictly positive numbers,whereas the values at I ∗ ∪ τ ( I ∗ ) can be multiplied with non-zero conjugate complex numbers. Clearly eachtwisting datum is multiplicatively equivalent to a unique reduced twisting datum.If ν = ( τ, ǫ ) ∈ H J , we can extend ǫ to monoid homomorphisms ǫ Q : ( Q J + , +) → ( C , · ) , ǫ Q ( α r ) = ǫ r , (2.3) ǫ P : ( P J + , +) → ( C , · ) , ǫ P ( ̟ r ) = ǫ r , (2.4)where Q J + = Z { α r | r ∈ J } + N { α r | r / ∈ J } , P J + = Z { ̟ r | r ∈ J } + N { ̟ r | r / ∈ J } . We then have in general that ǫ Q ( τ ( α )) = ǫ Q ( α ) and ǫ P ( τ ( ̟ )) = ǫ P ( ̟ ) for α ∈ Q J + and ̟ ∈ P J + .2.2. Folding.
Let τ be an involution of the Dynkin diagram Γ. We write V → V τ , v v + = 12 ( v + τ ( v )) , V → V − τ , v v − = 12 ( v − τ ( v ))for the projection maps onto resp. the τ -fixed and the − τ -fixed points. Proposition 2.8. [Ste67, Theorem 32]
The subset ˆ∆ = { α + | α ∈ ∆ } ⊆ V τ is a (possibly non-reduced) root system with associated positive roots { α + | α ∈ ∆ + } and associated systemof simple roots ˆΦ = { α r, + | r ∈ I } . emark 2.9. When Γ is τ -connected, the case of non-reduced ˆ∆ only appears for the Dynkin diagramof type A n with non-trivial involution. In the following, we denote ˆ I = I/τ, ˆ r = { r, τ ( r ) } . The simple roots in ˆΦ can be labeled by ˆ I upon putting α ˆ r = α r, + = α τ ( r ) , + . We call a simple root α ˆ r a folded simple root if | ˆ r | = 2.We define the associated folded Dynkin diagram ˆΓ = Γ /τ via the folded Cartan matrixˆ A ˆ r ˆ s = 2 ( α ˆ r , α ˆ s )( α ˆ r , α ˆ r ) . We denote by ̟ ˆ r ∈ V τ the fundamental weights for ˆ∆. We have the general formula ̟ ˆ r = X s ∈ ˆ r λ s ̟ s , λ r = 12 + 14 ( α ∨ r , α τ ( r ) ) . (2.5)Hence ̟ ˆ r = ̟ r, + except for the middle two vertices in type A n with non-trivial τ , in which case ̟ ˆ r = ̟ r, + .We list below the connected Dynkin diagrams with non-trivial involution, together with their folded version.Type Dynkin diagram Folded Dynkin diagram Folded type A n n ˆ1 ˆ n BC n A n +1 n n ′ ˆ1 ˆ n n ′ C n +1 D n n -2 n -1 n n -2 ˆ n B n − E F Definition 2.10.
We define the folding operation on ungauged twisting data as H ug → ˆ H , ( τ, ǫ ) (id , ˆ ǫ ) where ˆ ǫ ˆ r = ǫ r . Note that the folding operation is well-defined since ǫ r = ǫ τ ( r ) for ungauged twisting data. It then satisfiesin general the rule ˆ ǫ ˆ Q ( α + ) = ǫ Q ( α ) , α ∈ ˆ∆ . Note also that a twisting datum (id , η ) in ˆ H appears as the folding of a deformation in H ug if and only if η ˆ r ≥ r . .3. Weyl group actions on twisting data.
Let W ⊆ End( V ) be the Weyl group of ∆, with associatedsimple reflections s r for r ∈ I . For J ⊆ I , we denote W J ⊆ W for the parabolic subgroup generated by the s r with r ∈ J .Let τ be an involution of Γ. Then τ defines a group automorphism of W in the obvious way, denoted again τ . We let W τ ⊆ W be the subgroup of τ -fixed elements. On the other hand, we let ˆ W ⊆ End( V τ ) be theWeyl group of ˆΓ. Theorem 2.11. [Ste67, Theorem 32](1)
The restriction W τ → End( V τ ) , w w | V τ induces an isomorphism W τ ∼ = ˆ W . (2) Under this correspondence ˆ s r s ˆ r with ˆ s r the longest element in the parabolic Weyl group W { r,τ ( r ) } . More concretely, we have • ˆ s r = s r for ( α ∨ r , α τ ( r ) ) = 2, so r ∈ I τ , • ˆ s r = s r s τ ( r ) for ( α ∨ r , α τ ( r ) ) = 0, • ˆ s r = s r s τ ( r ) s r for ( α ∨ r , α τ ( r ) ) = − Corollary 2.12.
For each τ -invariant ω ∈ P there exists w ∈ W τ with wω ∈ P + . In the following we will identify W τ ∼ = ˆ W . Definition 2.13.
Let ν = ( τ, ǫ ) ∈ H J be a twisting datum with locus of non-degeneracy J . We define theassociated Weyl group as W ν = W τJ := W τ ∩ W J ⊆ W. We define the
Weyl group action W ν × H τ,J → H τ,J , ( wǫ ) Q ( α ) = ǫ Q ( w − α ) . Note that this is well-defined since W J preserves Q J + . Note also that the action of W ν descends to each H τ,J ′ with J ′ ⊇ J . Definition 2.14.
We call two twisting data ν = ( τ, ǫ ) and ν ′ = ( τ ′ , ǫ ′ ) Weyl group equivalent , and write ν ∼ W ν ′ , if • τ = τ ′ , • ǫ and ǫ ′ have the same support, and • there exists w ∈ W ν = W ν ′ such that ǫ ′ = wǫ . Lemma 2.15.
The group H >τ is stable under W τ .Proof. Let ǫ ∈ H >τ . If r ∈ I τ , then s r multiplies each entry ǫ t with a power of ǫ r >
0. If r ∈ I ∗ and t ∈ I ,then ˆ s r α t = α t − m rt ( α r + α τ ( r ) ) for some integer m rt , hence ˆ s r multiplies ǫ t with a power of | ǫ r | > τ -conjugate symmetry of ǫ . (cid:3) It follows that we can consider the action of W ν ⋉ H >τ on H τ,J . Definition 2.16.
We call two twisting data ν, ν ′ ∈ H equivalent , and write ν ∼ ν ′ , if ν, ν ′ ∈ H τ,J for some τ, J and ν, ν ′ lie in the same orbit under the W ν ⋉ H >τ -action. Lemma 2.17.
Each twisting datum ν is equivalent to a strongly reduced twisting datum. roof. By the multiplicative action of H >τ , the twisting datum ν is equivalent to a reduced one. It is thenenough to prove the lemma when I is connected, ν is reduced and Supp( ǫ ) = I , so in particular W ν = W τ . Inthis case ν can be interpreted as a Vogan diagram. The result thus follows from the Borel and de Siebenthaltheorem [Kna02, Section 6.96]. (cid:3) In fact, the proof loc. cit. shows that one can achieve the last step using only W I τ . We will at a later pointrefine this further, see Corollary 2.29.2.4. ν -compact roots.Definition 2.18. Let ν = ( τ, ǫ ) ∈ H ug J be an ungauged twisting datum. We say that a root β ∈ ˆ∆ is ν -compact if β ∈ ˆ∆ ˆ J and at least one of the following two conditions is satisfied: • ˆ ǫ ˆ Q ( β ) > , or • there exists r ∈ I ∗ ∩ J with ( ̟ ∨ ˆ r , β ) odd.We write ˆ∆ c for the set of ν -compact roots in ˆ∆ . Remark 2.19.
Note that the second condition is in general not subsumed by the first. For example, consider , the Dynkin diagram A with non-trivial involution and ǫ = ǫ = 1 and ǫ = − . Put α = α + α .One sees that ˆ ǫ ˆ Q ( α + ) = − , but α + = α ˆ1 + α ˆ2 , so ˆ α is ν -compact. Proposition 2.20.
The subset ˆ∆ c ⊆ ˆ∆ is a root subsystem.Proof. We may assume that Γ is τ -connected with J = I . Since the proposition is automatic for Γ of directproduct type, by ungaugedness, we may assume in fact that Γ is connected. If τ = id, then ˆ∆ c = ∆ c consistsof all α ∈ ∆ with ǫ Q ( α ) >
0, which clearly form a root subsystem. If τ = id, we may again by ungaugednessassume that Γ is not of type A n . The proposition then follows from the following lemma. (cid:3) When ∆ is any irreducible reduced root system, we write resp. ∆ s and ∆ l for the root subsystems of shortand long roots. Lemma 2.21.
Assume that Γ is a connected Dynkin diagram, not of type A n , with non-trivial involution.Then (1) ˆ∆ s consists precisely of those β with ( ̟ ∨ ˆ r , β ) odd for at least one r ∈ I ∗ . (2) If ǫ ∈ H ug , × τ , then ˆ∆ c is a root subsystem of ˆ∆ .Proof. Necessarily Γ is simply laced, and the only ˆΓ which can arise are those of type B m , C m or F . Sincewe can only have ( α r , α τ ( r ) ) = 0 for τ ( r ) = r , as we have excluded the case A n , it is further clear that r ∈ I τ if and only if α ˆ r is long. The first statement of the lemma thus says that β ∈ ˆ∆ s if and only if β contains at least one short simple root with odd multiplicity in its decomposition β = P ˆ r k ˆ r α ˆ r . That thelatter property is true for β ∈ ˆ∆ s follows immediately by inspection [OV90, Appendix, §
2, Table 1]. Theconverse direction can either again be verified by inspection, or one can argue that any long root is Weylgroup conjugate to a simple long root, with the action of s ˆ r for ˆ r short adding even multiples of α ˆ r .For the second property, note that by the first partˆ∆ c = ˆ∆ l,c ∪ ˆ∆ s , where ˆ∆ l,c = { β ∈ ˆ∆ l | ˆ ǫ ˆ Q ( β ) > } . Clearly ˆ∆ l,c and ˆ∆ s are root subsystems. Since ( β, γ ∨ ) ∈ Z for all β ∈ ˆ∆ l and γ ∈ ˆ∆ s , we have s γ ( ˆ∆ l,c ) =ˆ∆ l,c for all γ ∈ ˆ∆ s . Since ˆ∆ s is automatically preserved by the whole Weyl group ˆ W , it follows that ˆ∆ c is aroot system. (cid:3) Definition 2.22.
We define W + ν ⊆ W ν to be the Weyl group of the root system ˆ∆ c . .5. Twisted dot-action.
We now come to the definition of the twisted dot-action, which will be importantfor a Harish-Chandra-type formula later on. Note that, in contrast to the classical dot-action, we work herein the multiplicative setting, and the dot-action is with respect to a shift by ǫ in stead of by the halfsum ofall positive roots. Definition 2.23.
Let ν = ( τ, ǫ ) ∈ H ug be an ungauged twisting datum. We define an action · ǫ : W ν × H × τ → H × τ , ( w · ǫ λ ) P ( ω ) = ǫ Q ( ω − w − ω ) λ P ( w − ω ) , ω ∈ P. Note that this indeed defines an action. The following lemma follows by a direct computation.
Lemma 2.24.
The value of ˆ s r · ǫ λ remains everywhere the same as λ except at the positions t ∈ { r, τ ( r ) } ,where the value λ t gets multiplied by • ǫ t λ P ( α t ) − when ( α ∨ r , α τ ( r ) ) ∈ { , } , • | ǫ t | | λ P ( α t ) | − when ( α ∨ r , α τ ( r ) ) = − . Definition 2.25.
Let ν = ( τ, ǫ ) ∈ H ug J . We define W − ν as the set of elements w = s r n . . . s r ∈ W ν with r , . . . , r n ∈ J τ such that for all ≤ k ≤ n − s r k +1 . . . s r ) · ǫ H ≫ τ = ( s r k . . . s r ) · ǫ H ≫ τ . (2.6)Clearly (2.6) is the same as asking ( s r k +1 . . . s r ) · ǫ + / ∈ ( s r k . . . s r ) · ǫ H ≫ τ , or that at each successive step asign change is introduced. Theorem 2.26. (1)
The map W − ν → W ν /W + ν , w wW + ν (2.7) is bijective. (2) If w , w ∈ W − ν and w · ǫ H ≫ τ = w · ǫ H ≫ τ , then w = w .Proof. Note that all λ ∈ W ν · ǫ H ≫ τ satisfy λ r > ǫ r = 0. Using Lemma 2.24, it is then easily seenthat we may restrict to the case where I is τ -connected with I = Supp( ǫ ). Since the theorem is trivial incase we are in the product type situation, we may in fact assume that I is connected.Assume now first that τ = id, so in particular W ν = W . Note that in this case the ν -compact roots inˆ∆ = ∆ are simply the α with ǫ Q ( α ) > Claim : W + is the global stabilizer of H ≫ id under the ǫ -twisted dot-action.Choosing a compact root α and λ ∈ H ≫ id , we have( s α · ǫ λ ) r = ǫ Q ( α ) ( ̟ r ,α ∨ ) λ P ( s α ( ̟ r )) > , hence W + leaves H ≫ id globally stable.For the converse direction, we first fix a connected and simply connected compact Lie group U with com-plexification G = U C , together with a maximal torus T ⊆ U and Borel subgroup T C ⊆ B ⊆ G realizing theroot system ∆ with its particular choice of positive roots. Let Θ be the Cartan involution of G with fixedpoints U , and put g ∗ = Θ( g ) − . We further make ν into a complex Lie group involution on G uniquelydetermined by ν ( X α ) = sgn( ǫ Q ( α )) X α for X α ∈ g a root vector at weight α in the Lie algebra g of G . Let G ν = { g ∈ G | ν ( g ) ∗ = g − } , so that G ν is the real semisimple Lie group associated to the Vogan diagram sgn( ǫ ). As is well-known, W + = W ( T : G ν ) = N G ν ( T ) /T, (2.8)see e.g. [Kna01, Chapter IX. § w ∈ W which leaves H ≫ id globally stable, and pick a representative e w ∈ N U ( T ). It is by (2.8) sufficientto show that also e w ∈ G ν . But let V ̟ be an irreducible unitary representation of U at highest weight ̟ , nd let E ̟ be the operator on V ̟ determined by E ̟ v = sgn( ǫ Q ( ̟ − wt( v ))) v for v a weight vector at weightwt( v ) ∈ P . Then π ̟ ( ν ( u )) = E ̟ π ̟ ( u ) E ̟ for u ∈ U . It follows that π ̟ ( ν ( e w ) ∗ e w ) v = E ̟ π ̟ ( e w ) − E ̟ π ̟ ( e w ) v = sgn( ǫ Q (wt( v ) − w wt( v ))) v = v by the assumption on w . This proves ν ( e w ) ∗ = e w − , hence e w ∈ G ν and so w ∈ W + . This finishes the proofof the claim.Let us show now that the map (2.7) is bijective.To see that the map (2.7) is surjective, write any w ∈ W as a product of s r and remove those s r whichact by multiplication with H ≫ id within the expression w · ǫ +. Then we obtain an element w − such that w − · ǫ H ≫ id = w · ǫ H ≫ id . By the previous claim, w = w − w + for w + ∈ W + , proving surjectivity.To see that the map (2.7) is injective, we again first make a claim. Claim : W + ∩ W − = { e } .Indeed, assume that w ∈ W − , and write w = s r n . . . s r as in (2.6). We may clearly take this to be a reducedexpression. Consider the roots γ k = s r . . . s r k − ( α r k ) . By the assumption on w , we have R < = ( s r k . . . s r · +) r k ( s r k − . . . s r · +) r k R > = ǫ Q ( α r k )( s r k − . . . s r · +) P ( α r k ) − R > = ǫ Q ( γ k ) R > , hence all the γ k are non-compact. Since w − ∆ + ∩ ∆ + = ∆ + \ { γ , . . . , γ n } , (2.9)we can not have w ∈ W + unless w is trivial. This proves the claim.Assume now that w , w ∈ W − with w W + = w W + , or equivalently w · ǫ H ≫ id = w · ǫ H ≫ id . Write w = s r n . . . s r and w = s t m . . . s t as in (2.6). Then w = s t . . . s t m s r n . . . s r is seen to be an expressionstill satisfying (2.6), hence w ∈ W − . On the other hand, since w stabilizes H ≫ id we have w ∈ W + . From theabove claim, it follows that w = e and hence w = w .This ends the proof that (2.7) is bijective, and it also follows at once that property (2) in the theorem holds.Let us turn now to the case τ = id. We may assume that Γ is not of type A n , since in this case W − ν istrivial and W + ν = W τ . Consider the Dynkin subdiagram Γ ′ = Γ I τ with the twisting datum ν ′ = (id , ǫ ′ )where ǫ ′ is the restriction of ǫ . Since any α r with r ∈ I τ has ( α r , α ∨ s ) = ( α r , α ∨ τ ( s ) ) for all s , it follows byLemma 2.24 and (2.1) that W − ν coincides with its counterpart W − ν ′ for Γ ′ . By the first part of the proof, wehave a bijection W − ν × W + ν ′ → W I τ . (2.10)The first part of the proof also shows already that property (2) in the theorem holds.Let us show now that (2.7) is injective. It is sufficient to show that W + ν ′ = W + ν ∩ W I τ . (2.11)Since W I τ is parabolic, it follows by Chevalley’s lemma that W + ν ∩ W I τ is generated by reflections s β acrossroots β ∈ ˆ∆ c ∩ ∆ I τ = (∆ I τ ) c , implying (2.11).To see that (2.7) is surjective, it is by (2.10) sufficient to show that W I τ × W + ν → W ν is surjective. However, we have by Lemma 2.21 that W + ν ⊇ ˆ W s , the Weyl group of the short roots ˆ∆ s . Sinceˆ W s is normal in ˆ W , we find W I τ W + ν ⊇ W I τ ˆ W s = W I τ ˆ W s W I τ . Hence W I τ ˆ W s is a group containing all s ˆ r , so W I τ W + ν = W I τ ˆ W s = W ν . (cid:3) e end this section by recording the following results. Fix ν an ungauged deformation datum. Lemma 2.27.
Let w ∈ W − ν and v ∈ W + ν \ { e } . Let ρ = 12 X α ∈ ∆ + α = X r ∈ I ̟ r . Then w − ρ − ( wv ) − ρ ∈ ˆ Q + c \ { } .Proof. If v = s β r . . . s β rn is a reduced decomposition of v , with β k the simple roots of ˆ∆ c , then w − ρ − ( wv ) − ρ = n X p =1 ( w − ρ, s β r . . . s β rp − β ∨ r p ) β r p . Since all s β rn . . . s β rp +1 β r p ∈ ˆ∆ + c , and since ( ρ, α ∨ + ) ∈ Z for all α ∈ ∆, it is hence sufficient to show that( w − ρ, β ) > β ∈ ˆ∆ + c . In turn, this will follow if we can show wβ ∈ ˆ∆ + . But assume the latterwere not the case. Writing w = ˆ s r m . . . ˆ s r as in (2.6), it follows that we must then have ˆ s r p . . . ˆ s r β = − α ˆ r p for some p . This however implies β = ˆ s r . . . ˆ s r p − α ˆ r p , which is non-compact by the proof of Theorem 2.26.This contradiction ends the proof. (cid:3) Lemma 2.28.
For all w ∈ W + ν and ω ∈ P one has ǫ Q ( ω − wω ) > . (*) Conversely, if β ∈ ˆ∆ is such that (*) holds for w = s β and all ω ∈ P τ , then β is ν -compact.Proof. It is clear that (*) will hold once it holds for all w = s β with β ∈ ˆ∆ c . In this case, (*) states thatˆ ǫ ˆ Q ( β ) ( ω,β ∨ ) > , ∀ ω ∈ P τ . (**)We may assume that Supp( ǫ ) = I and that Γ is connected and not of type A n with non-trivial involution.The condition is clearly satisfied for the first case in Definition 2.18. In the second case we have that β is ashort root in ˆ∆, and so ( ω, β ∨ ) ∈ Z .Assume now conversely that β ∈ ˆ∆ is such that (*) holds for s β , or equivalently (**) holds for all ω ∈ P τ .Then clearly β must be supported on ˆ J = Supp(ˆ ǫ ), so we may suppose that J = I . If then β contains α r with r ∈ I τ with odd multiplicity, it follows upon chosing ω = ̟ r that ˆ ǫ ˆ Q ( β ) >
0, hence β is ν -compact. Ifon the other hand β contains all α r with r ∈ I τ with even multiplicity, then necessarily β must contain afolded root with odd multiplicity. Since the latter are short, we are done. (cid:3) Corollary 2.29. If ν ∼ ν ′ are equivalent ungauged twisting data, there exists w ∈ W − ν and f ∈ H >τ suchthat ǫ ′ = f wǫ .Proof. Note first that if β ∈ ˆ∆ + c , then for α ∈ ∆ + we have( s β ǫ ) Q ( α ) = ǫ Q ( α )ˆ ǫ ˆ Q ( β ) ( α,β ∨ ) . Now except for type A n with non-trivial involution, we always have α + ∈ P for α ∈ ∆ + . Since ǫ is howeverpositive anyway for type A n by the ungaugedness assumption, we deduce from the previous lemma that s β ǫ = gǫ with g ∈ H ≫ τ .Assume now that ǫ ′ = f ( wǫ ) for w ∈ W ν and f ∈ H >τ . We can write w = uv with u ∈ W − ν and v ∈ W + ν . Bythe above we see that ǫ ′ = f ′ ( uǫ ) for some f ′ ∈ H >τ . (cid:3) . Quantized enveloping algebras and their deformations
Quantized enveloping algebras.
Let g be a complex semisimple Lie algebra of rank l . We fix a setof Chevalley generators S = { h r , e r , f r | r ∈ I } ⊆ g , indexed by a set I with | I | = l . We write b ± for the associated Borel subalgebras, h for the Cartan subalgebraand n ± for the associated nilpotent subalgebras. We write u ⊆ g R for the compact form of g determinedby the above choice of Chevalley generators, and we write t = h ∩ u for the maximal torus of u . We write a = i t ⊆ h R . When viewing g as a real Lie algebra by forgetting the complex structure, we write g R .We use the notation of the previous section for the associated root system ∆ on a ∼ = a ′ . We will also use theshorthand notation d r = 12 ( α r , α r ) . Fix in the following 0 < q <
1. Algebraically there is no issue allowing also q >
1, but this will influencespectral conditions later on.
Definition 3.1.
The quantized enveloping algebra U q ( b + ) is the algebra generated by K + ω , E r for r ∈ I and ω ∈ P , with K +0 = 1 and commutation relations K + ω K + χ = K + ω + χ , K + ω E r = q ( ω,α r ) E r K + ω and the quantum Serre relations whose precise form we will not need in what follows, see e.g. [KS97, Section6.1.2] . Similarly one defines U q ( b − ). We write its generators as K − ω and F − r , with in particular K − = 1 and K − ω K − χ = K − ω + χ , K − ω F r = q − ( ω,α r ) F r K − ω . The algebras U q ( b ± ) form Hopf algebras for the comultiplication K ± ω K ± ω ⊗ K ± ω , E r E r ⊗ K + r ⊗ E r , F r F r ⊗ ( K − r ) − + 1 ⊗ F r . (3.1)We let U q ( g ) be the associated quantized enveloping algebra of g , which is generated by a copy of the Hopfalgebras U q ( b + ) and U q ( b − ) with the relations K + ω = K − ω =: K ω and interchange relations[ E r , F s ] = δ r,s K r − K − r q r − q − r , where we use the shorthand K r = K α r and q r = q d r . Endowed with the ∗ -structure K ∗ ω = K ω , E ∗ r = F r K r we obtain a Hopf ∗ -algebra which we denote U q ( u ). Its antipode squared is given by S = Ad( K − ρ ) . (3.2)Occasionally, we will use U ′ q ( u ) as generated by the E r , F r and K r .We view U q ( u ) as a spectral ∗ -algebra with admissible finite-dimensional ∗ -representations those for whichall K ω have positive spectrum. Up to unitary equivalence the irreducible admissible ∗ -representations areparametrized by P + , and we write V ̟ for a ∗ -representative associated to ̟ ∈ P + . The correspondence isdetermined by the fact that V ̟ has a highest weight vector ξ ̟ , which we normalize by k ξ ̟ k = 1, with K ω ξ ̟ = q ( ω,̟ ) ξ ̟ , E r ξ ̟ = 0 for all r. When V π is a finite-dimensional admissible ∗ -representation and v ∈ V π is a joint eigenvector of the K ω , wewrite wt( v ) for the unique element of P such that K ω v = q ( ω, wt( v )) v. We write U ̟ ( ξ, η ) for the matrix coefficients of a highest weight ∗ -representation π ̟ of highest weight ̟ . efinition 3.2. We write O q ( U ) = O ( U q ) = { U π ( ξ, η ) } for the Hopf ∗ -algebra of matrix coefficients of finite-dimensional admissible ∗ -representations of U q ( u ) . Here the ∗ -structure is determined by( f ∗ , X ) = ( f, S ( X ) ∗ ) , f ∈ O q ( U ) , X ∈ U q ( u ) . The definition remains the same if we replace U q ( u ) by U ′ q ( u ).We view O q ( U ) as a q -deformation of the ∗ -algebra O ( U ) of regular functions on the simply connectedcompact Lie group U integrating u . The algebra O q ( U ) separates elements in U q ( u ), and we may identify U q ( u ) ⊆ O q ( U ) ′ , the linear dual of O q ( U ). Moreover, O q ( U ) has a positive Haar integral, and is henceassociated to a compact quantum group which we write U q . We have Z U q U ̟ ( ξ, η ) = δ ̟, h ξ, η i , and the Woronowicz character is given byˆ δ iz ( U ̟ ( ξ, η )) = q iz ( ρ, wt( η )) h ξ, η i , (3.3)where ρ = 12 X α ∈ ∆ + α = l X r =1 ̟ r . Formally, we can write ˆ δ iz = K izρ . We further note the following. Proposition 3.3 ([LS91, Section 4.6.7], [NT13, Theorem 2.7.14]) . The compact quantum group U q is coa-menable, and C q ( U ) = C ( U q ) is a type I C ∗ -algebra. We write L ∞ q ( U ) = L ∞ ( U q ) for the associated von Neumann algebra.When forgetting the ∗ -structure, we write O q ( U ) rather as O q ( G ) = O ( G q ) , with G q viewed as a q -deformation of the simply connected complex algebraic group G integrating g .Note that when l = 1, we indeed obtain that O q ( SU (2)), as defined in Section 1.6, is dual to U q ( su (2)).More precisely, the duality is determined by the 2-dimensional ∗ -representation π / with associated corep-resentation U such that ( U, K ̟ ) = (cid:18) q q − (cid:19) , ( U, E α ) = (cid:18) q / (cid:19) , where α is the simple root, normalized such that ( α, α ) = 2, and ̟ = α/ E = E α and K = K α . Then U ′ q ( su (2)) is generated by K, E, E ∗ .3.2. Deformations of quantized enveloping algebras from twisting data, and associated coac-tions.
We keep the setting of the previous section.Let µ, ν be twisting data. One can create a ( µ, ν )-twisted Drinfeld double of U q ( b + ) and U q ( b − ) as follows,see [DCNTY19, Section 3.4] and [DCM18, Section 2.1]. Definition 3.4.
For µ, ν ∈ H twisting data, the ( µ, ν )-deformed quantized enveloping algebra U µ,νq ( e g ) , with e g = g ⊕ h , is generated by U q ( b + ) and U q ( b − ) with interchange relations K + ω K − χ = q ( ω,τ ν ( χ ) − τ µ ( χ )) K − χ K + ω ,K − ω E r = q ( α r ,τ µ ( ω )) E r K − ω , K + ω F r = q − ( α r ,τ ν ( ω )) F r K + ω , (3.4) We allow somewhat more generality by allowing complex values for the parameters, but this extra generality is inessentialand will be removed in a moment. E r , F s ] = δ r,τ ν ( s ) ǫ ν,r K + r − δ r,τ µ ( s ) ǫ µ,s ( K − s ) − q r − q − r . It is easy to see that the multiplication maps U q ( b + ) ⊗ U q ( b − ) → U q ( e g ) , U q ( b − ) ⊗ U q ( b + ) → U q ( e g )are vector space isomorphisms. The above algebra becomes a ∗ -algebra for( K + ω ) ∗ = K − ω , E ∗ r = F r K − r , in which case we write it as U µ,νq ( e u ), where e u is interpreted as u ⊕ a . The U µ,νq ( e u ) form a ∗ -compatibleHopf-Galois system, also called co-groupoid [Bic14], for the maps∆ : U µ,νq ( e u ) → U µ,κq ( e u ) ⊗ U κ,νq ( e u ) , given by (3.1) on the components U q ( b ± ). In particular, the U ν,νq ( e u ) are Hopf ∗ -algebras. We write U q ( e u ) = U ++ q ( e u ). We see that U q ( u ) coincides with the quotient of U q ( e u ) by putting K + ω = K − ω .In the following, we will only be interested in the case where µ = + is the trivial twisting datum. We writein this case the associated deformed quantized enveloping ∗ -algebra U νq ( e u ) = U + ,νq ( e u )with ν = ( τ, ǫ ). It comes equipped with a left coaction by U ++ q ( e u ) and hence by U q ( u ).Fix now ν = ( τ, ǫ ). We use again the following shorthand notations for ω ∈ a ′ ∼ = Q ⊗ Z R and r ∈ I , ω ± = 12 (id ± τ ) ω, c ω = q ( ω − ,ω − ) , c r = c α r . (3.5)Write U νq ( u ) for the ∗ -subalgebra of U νq ( e u ) generated by the E r , E ∗ r and the T ω := c − ω K + − τ ( ω ) K −− ω . We have for U νq ( u ) the universal relations T = 1 and T ω + χ = T ω T χ , T ∗ ω = T τ ( ω ) ,T ω E r = q − ω + ,α r ) E r T ω , T ω E ∗ r = q ω + ,α r ) E ∗ r T ω (3.6) E r E ∗ s − q − ( α r ,α s ) E ∗ s E r = δ r,τ ( s ) ǫ r c r T − α s − δ r,s q r − q − r , together with the usual Serre relations between the E r . As one notices, the Cartan part of U νq ( u ) shouldrather be interpreted as U q ( h ν ), with h ν = t τ ⊕ a − τ . Note also that when ν = +, we obtain an embedding U + q ( u ) ֒ → U q ( u ), but it is not surjective as we onlyreach elements in the Cartan algebra with weights in 2 P .It is easily checked that the left coaction of U q ( e u ) on U νq ( e u ) gives rise to a ∗ -preserving left coaction γ : U νq ( u ) → U q ( u ) ⊗ U νq ( u ) ,T ω K − ω − τ ( ω ) ⊗ T ω , E r E r ⊗ K r ⊗ E r . (3.7)Moreover, the ∗ -algebra U νq ( e u ) is through its Galois object structure endowed with a left U q ( e u )-module ∗ -algebra structure, called the Miyashita-Ulbrich action , given in this concrete case by X ◮ Y = X (1) Y S ( X (2) ) , Y ∈ U νq ( e u ) , X ∈ U q ( b ) ∪ U q ( b − ) . (3.8)This is easily seen to descend to a left U q ( u )-module ∗ -algebra structure on U νq ( u ) (although the latteraction will not be inner anymore if τ = id). The triple ( U νq ( u ) , γ, ◮ ) forms a ∗ -compatible Yetter-Drinfeld U q ( u )-module algebra.The next lemma shows that all U νq ( u ), with ν ranging in an equivalence class of twisting data, are isomor-phic as Yetter-Drinfeld U q ( u )-module ∗ -algebras. The isomorphism can moreover be chosen such that it iscompatible with certain ‘spectral conditions’, which will be explored further in the next sections. emma 3.5. Let ν, ν ′ ∈ H τ be equivalent twisting data. Then there exists f ∈ H × τ such that χ f : U ν ′ q ( u ) → U νq ( u ) , T ω f P ( τ ( ω )) T ω , E r E r extends to an isomorphism of Yetter-Drinfeld U q ( u ) -module ∗ -algebras. Moreover, if ǫ ′ = λ ( wǫ ) for w ∈ W ν and λ ∈ H >τ , then f can be chosen so that the following condition with respect to ( w, ǫ ) holds: f P ( ω ) ǫ Q ( ω − w − ω ) > , ∀ ω ∈ P τ . (3.9) Proof.
It is clear that χ f will be a ∗ -isomorphism of Yetter-Drinfeld module ∗ -algebras as soon as f ∈ H × τ with ǫ ′ r = f P ( α r ) ǫ r for all r ∈ I .Assume now that ǫ ′ = λ ( wǫ ) for w ∈ W ν and λ ∈ H >τ . Choose N ∈ N such that P ⊆ N Q , and pick χ ∈ H >τ such that χ Nr = λ r for all r ∈ I . Define g ∈ H × τ by g P ( ω ) = Q χ k r r if ω = P r k r ( N − α r ) with k r ∈ N . Thenclearly g P ( α r ) = λ r , and g P ( ω ) > ω ∈ P τ . Define now f P ( ω ) = g P ( ω ) ǫ Q ( ω − w − ω ) − , ω ∈ P. Then f clearly satisfies the requirements. (cid:3) Remark that this allows us to restrict ourselves to deal with strongly reduced twisting data only. The generalcase is however convenient to keep around for using scaling arguments, though we will from now on alwaysassume that at least ν ∈ H ug when considering U νq ( u ).3.3. Interpretation of U νq ( u ) as quantized function algebra. We want to introduce certain spectralconditions on U νq ( u ). These are more easily justified for the quantized function algebra interpretation of U νq ( u ) which we now introduce. Definition 3.6.
Let ν = ( τ, ǫ ) be an ungauged twisting datum. We define O q ( Z reg ν ) as the U q ( u ) -module ∗ -algebra generated by elements a ω for ω ∈ P and x r , x ∗ r with r ∈ I such that the map ι ν : O q ( Z reg ν ) → U νq ( u ) , a ω T ω , x r E r is a U q ( u ) -equivariant ∗ -isomorphism. Remark 3.7. • Note that, with respect to the notation in [DCM18, (2.36)] , we have rescaled the a ω so that ω a ω is a homomorphism. • As we are working in the concrete setting where q is a definite number, the above renaming processis of course purely formal. Within the deformation setting where q is a parameter, the choice ofgenerators for O q ( Z reg ν ) has to be made more judiciously as to obtain the correct limit as a functionalgebra when q is set equal to . The following theorem collects some of the results in [DCM18, Section 2]. Note first that τ can be interpretedas a Hopf ∗ -algebra isomorphism of U q ( u ) by τ ( K ω ) = K τ ( ω ) , τ ( E r ) = E τ ( r ) , τ ( F r ) = F τ ( r ) . (3.10)By duality, we also have that τ determines a Hopf ∗ -algebra automorphism of O q ( U ). Theorem 3.8.
Endow O q ( U ) with the τ -twisted adjoint U q ( u ) -module structure X ◮ U π ( ξ, η ) = U π ( τ ( S ( X (1) )) ∗ ξ, X (2) η ) . (3.11) Then there exists a unique injective U q ( u ) -module map i ν : O q ( U ) → O q ( Z reg ν ) (3.12) such that i ν ( U ̟ ( ξ ̟ , ξ ̟ )) = c ̟ a ̟ . (3.13) Definition 3.9.
We define O q ( Z ν ) = i ν ( O q ( U )) ⊆ O q ( Z reg ν ) to be the image of i ν . heorem 3.10. The following properties hold for O q ( Z ν ) . • O q ( Z ν ) is a ∗ -subalgebra of O q ( Z reg ν ) , • O q ( Z ν ) ∩ C [ a ω | ω ∈ P ] = C [ a ̟ | ̟ ∈ P + ] , • O q ( Z reg ν ) is generated as an algebra by O q ( Z ν ) and a − ρ , and • O q ( Z ν ) is generated as a U q ( u ) -module by the a ̟ . It follows in particular from the above theorem that the ◮ -action on O q ( Z reg ν ) restricts to a module ∗ -algebrastructure on O q ( Z ν ) which integrates to an O q ( U )-coaction ρ ν : O q ( Z ν ) → O q ( Z ν ) ⊗ O q ( U ) , f ρ ν ( f )where (id ⊗ X ) ρ ν ( f ) = X ◮ f, X ∈ U q ( u ) . In the following, we will also write Z π ( ξ, η ) = i ν ( U π ( ξ, η )) , and we pair O q ( Z ν ) with U q ( u ) along i ν . Then(id ⊗ ρ ν ) Z π = (id ⊗ τ )( U π ) ∗ Z π, U π, . (3.14)We can also transport the coaction γ of (3.7) through the isomorphism ι ν into a coaction e γ by U q ( u ) on O q ( Z reg ν ), so e γ : O q ( Z reg ν ) → U q ( u ) ⊗ O q ( Z reg ν ) , a ω K − ω − τ ( ω ) ⊗ a ω , x r E r ⊗ K r ⊗ x r . (3.15) Lemma 3.11.
The coaction e γ restricts to a coaction of U q ( u ) on O q ( Z ν ) .Proof. Let us write e γ ( f ) = f ( − ⊗ f (0) , f ∈ O q ( Z reg ν ) . Using the Yetter-Drinfeld module structure, we have e γ ( X ◮ f ) = X (1) f ( − S ( X (3) ) ⊗ ( X (2) ◮ f (0) ) , f ∈ O q ( Z reg ν ) , X ∈ U q ( u ) . (3.16)In particular, e γ ( X ◮ a ̟ ) = X (1) K − ̟ − τ ( ̟ ) S ( X (3) ) ⊗ ( X (2) ◮ a ̟ ) , ̟ ∈ P + , X ∈ U q ( u ) . Since the a ̟ generate O q ( Z ν ) as a U q ( u )-module, this proves the lemma. (cid:3) On the subalgebra O q ( Z ν ) the map ι ν can also be given by a global formula. Let R ∈ U q ( b + ) ˆ ⊗ U q ( b − )be the universal R -matrix of U q ( g ) in some completed tensor product, normalized such that R ( ξ ⊗ η ) = q − (wt( ξ ) , wt( η )) ξ ⊗ η for ξ a highest weight and η a lowest weight vector. Viewing O q ( Z ν ) again as functionals on U q ( g ), we havethe following. Let E be the unique element in the completion O q ( U ) ′ ∼ = Q ̟ B ( V ̟ ) of U q ( g ) such that inthe highest weight ∗ -representation V ̟ we have E v = ǫ Q ( ̟ − wt( v )) v. (3.17)Let R τ = ( τ ⊗ id) R = (id ⊗ τ ) R . Proposition 3.12. [DCM18, Proposition 2.30]
For all f ∈ O q ( Z ν ) one has ι ν ( f ) = ( f ⊗ id)( R τ, ( E ⊗ R ) . (3.18) Note that the normalization of the R -matrix in Section 1.6 was chosen compatibly with this convention. ere the right hand side is easily seen to be a well-defined element in U q ( b + ) U q ( b − ) ⊆ U νq ( e g ).We record for the computations in the following section that, writing κ − ( f ) = ( f ⊗ id) R , κ + ( f ) = (id ⊗ f ) R − , f ∈ O q ( G ) , (3.19)we have the following general commutation rule κ + ( g ) κ − ( f ) = r ν ( f (1) , g (1) ) κ − ( f (2) ) κ + ( g (2) ) r ( f (3) , S − ( g (3) )) (3.20)inside U q ( e u ), where r ( f, g ) = ( f ⊗ g, R ) and r ν ( f, g ) = ( f ⊗ g, R ν ).4. Centrally coinvariant coactions from twisting data and the Harish-Chandra morphism
Centrally coinvariant coactions.
Fix an ungauged twisting datum ν = ( τ, ǫ ) ∈ H ug , and recall theintegrable U q ( u )-module ∗ -algebra O q ( Z ν ) introduced in Definition 3.9, with associated O q ( U )-coaction ρ ν .Let O q ( Z ν //U ) = O q ( Z ν ) ρ ν be the ρ ν -coinvariants, or equivalently the U q ( u )-invariant part of O q ( Z ν ) under ◮ . In using this notation,we interpret O q ( Z ν ) as a full spectral ∗ -algebra. Lemma 4.1.
We have O q ( Z ν //U ) ⊆ Z ( O q ( Z ν )) , where the right hand side denotes the center of O q ( Z ν ) . Hence the coaction ρ ν is centrally coinvariant.Proof. By means of the equivariant embedding ι ν it is sufficient to show that the U q ( u )-invariant elements U νq ( u ) inv in U νq ( u ) for the action (3.8) are central. But for the U q ( e u )-module ∗ -algebra U νq ( e u ) we have that U νq ( e u ) inv = Z ( U νq ( e u )) since the associated U q ( e u )-action is the Miyashita-Ulbrich action for a Galois object,see e.g. [CGW06, Theorem 3.2.7]. One can also check this property directly in the case at hand . (cid:3) Consider the projection map E = E ρ ν : O q ( Z ν ) → O q ( Z ν //U ) , E ( f ) = id ⊗ Z U q ! ρ ν ( f ) . (4.1)To have a more concrete expression for E , we first make the following observations.Recall that τ defines a Hopf ∗ -algebra involution of U q ( u ) by (3.10). Fix ̟ ∈ P + . We have that the Z ̟ ( ξ, η )span a U q ( u )-representation isomorphic to V ∗ τ ( ̟ ) ⊗ V ̟ . We hence see that this span contains no invariantelement unless τ ( ̟ ) = ̟ , in which case it contains a one-dimensional subspace of invariant elements. Letfor ̟ = τ ( ̟ ) the operator J ̟ : V ̟ → V ̟ be uniquely determined by the requirements J ̟ ξ ̟ = ξ ̟ , J ̟ Xv = τ ( X ) J ̟ v, X ∈ U q ( u ) , v ∈ V ̟ . One easily sees that J ̟ is a selfadjoint operator with J ̟ = 1. For ω ∈ P with τ ( ω ) = ω , we write j ̟ ( ω ) = Tr( J ̟ | V ̟ ( ω ) ) , (4.2)with V ̟ ( ω ) the weight ω subspace. Lemma 4.2. (1)
The functions ω j ̟ ( ω ) are independent of q . (2) Each function ω j ̟ ( ω ) is W τ -invariant.Proof. The first statement follows since j ̟ ( ω ) ∈ Z depends continuously on q . The second statement thenfollows (for q = 1) by [FSS96, (5.26)]. (cid:3) emma 4.3. With { e i } an orthonormal basis of V ̟ , we have E ( Z ̟ ( ξ, η )) = δ ̟,τ ( ̟ ) h ξ, J ̟ η i dim q ( V ̟ ) X i,j h e j , K ρ J ̟ e i i Z ̟ ( e i , e j ) , (4.3) where dim q ( V ̟ ) = Tr( K ρ ) .Proof. Using (3.14), we obtain E ( Z ̟ ( ξ, η )) = X i,j Z U q τ ( U ̟ ( e i , ξ )) ∗ U ̟ ( e j , η ) ! Z ̟ ( e i , e j ) . From the orthogonality relations (1.1) and (3.3) for O q ( U ), we obtain that this is 0 for τ ( ̟ ) = ̟ , while for τ ( ̟ ) = ̟ this reduces to X i,j Z U q U ̟ ( J ̟ e i , J ̟ ξ ) ∗ U ̟ ( e j , η ) ! Z ̟ ( e i , e j ) , from which (4.3) follows immediately. (cid:3) Using (3.13), we obtain
Corollary 4.4.
Let ̟ ∈ P + with τ ( ̟ ) = ̟ and define z ̟ = dim q ( V ̟ ) E ( a ̟ ) = X i,j h e j , K ρ J ̟ e i i Z ̟ ( e i , e j ) . Then the z ̟ form a linear basis for O q ( Z ν //U ) . We recall now from [DCM18, Lemma 2.33] that the a ̟ satisfy the commutation a ̟ Z ( ξ, η ) = q ̟ + , wt( ξ ) − wt( η )) Z ( ξ, η ) a ̟ . (4.4)This still holds with ̟ replaced by a general weight in P when interpreting the identity within O q ( Z reg ν ). Lemma 4.5.
Define σ z : O q ( Z ν ) → O q ( Z ν ) , σ z ( Z ( ξ, η )) = q iz (2 ρ, wt( ξ ) − wt( η )) Z ( ξ, η ) , (4.5) and write σ = σ − i . Then the σ z are algebra automorphisms with E ( f g ) = E ( gσ ( f )) , ∀ f, g ∈ O q ( Z ν ) . Proof.
Within O q ( Z reg ν ) we can write σ = Ad( a ρ ) (4.6)It follows from this that all σ z are algebra automorphisms.By (4.4), Lemma 4.3 and τ -invariance of the ̟ + , we have that E ( a ̟ z ) = E ( za ̟ ) , ∀ ̟ ∈ P + , ∀ z ∈ O q ( Z ν ) . (4.7)Note further that we can write σ ( Z ( ξ, η )) = Z ( K ρ ξ, K − ρ η ) . Using (3.11) and (3.2) it follows that σ ( X ◮ Y ) = S ( X ) ◮ σ ( Y ) , X ∈ U q ( u ) , Y ∈ O q ( Z ν ) . (4.8)From invariance of E , we also have E (( X ◮ Y ) Z ) = E ( Y ( S ( X ) ◮ Z )) . (4.9) ence E (( X ◮ a ̟ ) Y ) = (4.9) E ( a ̟ ( S ( X ) ◮ Y )) = (4.7) E (( S ( X ) ◮ Y ) a ̟ )= (4.9) E ( Y ( S ( X ) ◮ a ̟ )) = (4.8) E ( Y σ ( X ◮ a ̟ )) . Since the X ◮ a ̟ span O q ( Z ν ) linearly, we obtain the lemma. (cid:3) Harish-Chandra homomorphism.
Let U q ( h ν ) ⊆ U νq ( u ) be the algebra linearly spanned by the T ω .Let U q ( n ) be the unital algebra generated by the E r , and U q ( n − ) the algebra generated by the E ∗ r . Lemma 4.6.
An element in U q ( n ) commutes with the T ω if and only if it is a scalar.Proof. If X ∈ U q ( n ) commutes with the T ω , we may assume it is a sum of elements of the form E α i . . . E α in ,each of which commutes with T ω . Putting α = P α i k ∈ Q + , we then have by (3.6) that ( ω + , α ) = 0 for all ω ∈ P , hence α + = 0. But then α = − τ ( α ). Since α ∈ Q + , this entails α = 0. Hence X is scalar. (cid:3) Let P : U νq ( g ) → U q ( h ν ) , XY Z → ε ( X ) Y ε ( Z ) , X ∈ U q ( n − ) , Y ∈ U q ( h ν ) , Z ∈ U q ( n ) (4.10)be the projection map on the Cartan part with respect to this particular triangular decomposition. UsingLemma 4.6, the standard argument gives that P is a ∗ -homomorphism on the degree zero-part of U νq ( u ) withrespect to the natural P -grading. Hence we have in particular the Harish-Chandra ∗ -homomorphism χ HC : Z ( U νq ( u )) → U q ( h ν ) , X P ( X ) . From the proof of Lemma 4.1, we see that ι ν maps O q ( Z ν //U ) into Z ( U νq ( u )). Hence we can consider the ∗ -homomorphism χ HC ◦ ι ν : O q ( Z ν //U ) → U q ( h ν ) . Recall the elements z ̟ introduced in Corollary 4.4. Theorem 4.7 (Harish-Chandra formula) . For ̟ ∈ P + with τ ( ̟ ) = ̟ we have χ HC ( ι ν ( z ̟ )) = X ω ∈ P + ̟ − ω ∈ Q + τ ( ω )= ω j ̟ ( ω ) | Stab W ν ( ω ) | − ǫ Q ( ̟ − ω ) X w ∈ W ν ǫ Q ( ω − wω ) q − ρ,wω ) T wω ! . (4.11) Proof.
We would like to use formula (3.18), but unfortunately the expression in that formula is with respectto the opposite triangular decomposition of the one for the Harish-Chandra projection (4.10). We hencehave to first bring (3.18) in the right form.Endow O q ( Z ν ) with the coalgebra structure inherited from O q ( U ) through the map i ν in (3.12). If f ∈O q ( Z ν //U ) we have that f (2) ⊗ S ( τ ( f (1) )) f (3) = f ⊗ , from which one obtains ∆( f ) = (id ⊗ S τ )∆ cop ( f )and hence f ( XY ) = f ( τ ( S ( Y )) X ) , X, Y ∈ U q ( g ) . (4.12) sing the shorthand notation R = R ⊗ R for the universal R -matrix, and using the maps κ ± from (3.19),we find using (3.18) that for f ∈ O q ( Z ν //U ) ι ν ( f ) = f (1) ( τ ( R )) f (2) ( E ) f (3) ( R ′ )) R R ′ = f (2) ( E ) κ + ( S ( τ ( f (1) ))) κ − ( f (3) )= (3.20) f (4) ( E ) r ν ( f (5) , S ( τ ( f (3) ))) κ − ( f (6) ) κ + ( S ( τ ( f (2) ))) r ( f (7) , τ ( f (1) ))= f ( τ ( R R ′ S ( ν ( R ′′ ))) E R ′′ R ′′′ R ) R ′′′ R ′ = (4.12) f ( τ ( R ′ S ( ν ( R ′′ ))) E R ′′ R ′′′ R S − ( R )) R ′′′ R ′ = f ( τ ( R ′ ) E S ( R ′′ ) R ′′ R ′′′ R S − ( R )) R ′′′ R ′ , where in the last line we used τ ( ν ( Y )) E = E Y for Y ∈ U q ( b − ).Let us now write u = S ( R ) R , v = S ( u ) = R S ( R ) , so that also v − = R S − ( R ) , u − = R S ( R ) . We have that Ad( u ) implements S and uv − = K − ρ , the grouplike implementing S , see e.g. [Maj95,Proposition 2.1.8 and Corollary 2.1.9].Since we can write ι ν ( f ) = f ( τ ( R ) E u R ′ v − ) R ′ R , we thus obtain ι ν ( f ) = f ( τ ( R ) E S ( R ′ ) K − ρ ) R ′ R . Since R = e RQ with e R = P α ∈ Q + e R α , where e R = 1 ⊗ e R α ∈ U q ( n ) α ⊗ U q ( n − ) − α , and where Q ( ξ ⊗ η ) = q − (wt( ξ ) , wt( η )) ξ ⊗ η, we see that χ HC ( ι ν ( z ̟ )) = X i,j h e j , K ρ J ̟ e i i P ( ι ν ( Z ̟ ( e i , e j )))= X i,j h e j , K ρ J ̟ e i ih e i , τ ( R ) E S ( R ′ ) K − ρ e j i P ( R ′ R )= X i,j h e j , K ρ J ̟ e i ih e i , τ ( Q ) E S ( Q ′ ) K − ρ e j i Q − ′ Q +1 = X i h e i , J ̟ e i i ǫ Q ( ̟ − wt( e i )) q − ρ, wt( e i )) K −− wt( e i ) K + − τ (wt( e i )) = X ω ∈ P̟ − ω ∈ Q + τ ( ω )= ω j ̟ ( ω ) ǫ Q ( ̟ − ω ) q − ρ,ω ) T ω . (4.13)Now if ω ∈ P with τ ( ω ) = ω , ̟ − ω ∈ Q + and ǫ Q ( ̟ − ω ) = 0, it follows that ̟ − ω ∈ Q + J , where J = Supp( ǫ ).From Corollary 2.12, we can find w ∈ W ν such that wω is dominant integral on Q J . Since however we alsomust have ̟ − wω ∈ Q + J and ̟ ∈ P + , we see that in fact wω ∈ P + . It follows that χ HC ( ι ν ( z ̟ )) = X ω ∈ P + ̟ − ω ∈ Q + τ ( ω )= ω | Stab W ν ( ω ) | − ǫ Q ( ̟ − ω ) X w ∈ W ν j ̟ ( wω ) ǫ Q ( ω − wω ) q − ρ,wω ) T wω . We can now deduce (4.11) from Lemma 4.2.(2). (cid:3) et us now transport the above triangular decomposition and Harish-Chandra homomorphism to O q ( Z reg ν ).Write N + for the unital algebra generated by the x r , N − for the unital algebra generated by the x ∗ r , and A for the algebra generated by the a ω . Consider the associated triangular decomposition N − ⊗ A ⊗ N + ∼ = O q ( Z reg ν ) (4.14)where the isomorphism is a vector space isomorphism induced by multiplication. The projection to the A -component gives the ∗ -homomorphism e χ HC : O q ( Z ν //U ) → A , z ̟ X ω ∈ P + ̟ − ω ∈ Q + τ ( ω )= ω j ̟ ( ω ) | Stab W ν ( ω ) | − ǫ Q ( ̟ − ω ) X w ∈ W ν ǫ Q ( ω − wω ) q − ρ,wω ) a wω ! . (4.15)View A as functions on H × τ by a ω ( µ ) = µ P ( ω ) , (4.16)using the notation of (2.4). We will use a q -deformation of the ǫ -twisted dot action introduced in Definition2.23. Definition 4.8.
We define the q -deformed ǫ -twisted dot action of W ν on H × τ by · ǫ,q = · q ρ ǫ , so ( w · ǫ,q λ ) P ( ω ) = ǫ Q ( ω − w − ω ) q (2 ρ,ω − w − ω ) λ P ( w − ω ) Corollary 4.9.
Let µ, µ ′ ∈ H × τ , and assume that e χ HC ( z ̟ )( µ ) = e χ HC ( z ̟ )( µ ′ ) , for all ̟ ∈ P + with τ ( ̟ ) = ̟. Then there exists w ∈ W ν and a gauge γ ∈ H g τ such that µ ′ = γ ( w · ǫ,q µ ) . Proof.
For ̟ ∈ ( P + ) τ we write ˆ a ̟ = X w ∈ W ν ǫ Q ( ̟ − w̟ ) q − ρ,w̟ − ̟ ) a w̟ , and we let A W ν τ be the subspace of A spanned by the ˆ a ̟ . By triangularity of the map( P + ) τ → A W ν τ , ̟ e χ HC ( z ̟ ) = X ω ∈ P + ̟ − ω ∈ Q + τ ( ω )= ω j ̟ ( ω ) | Stab W ν ( ω ) | − ǫ Q ( ̟ − ω ) q − ρ,ω ) ˆ a ω with respect to the natural partial order on ( P + ) τ given by ̟ ≥ ̟ ′ if and only if ̟ − ̟ ′ ∈ Q + , it followsthat e χ HC is surjective onto A W ν τ .Note now that the functions ˆ a ̟ can be written as µ ˆ a ̟ ( µ ) = X w ∈ W ν a ̟ ( w · ǫ,q µ ) . Consider the action of H g τ ∼ = T | I ∗ | on H × τ by multiplication. Then the associated action on A satisfies Z H g τ a ̟ ( χ − ) d χ = δ ̟,τ ( ̟ ) a ̟ . Since the a ̟ for ̟ ∈ P + separate points in H × τ , we must thus have that the range of e χ HC separates the(compact) H g τ × W ν -orbits of H × τ , from which the corollary follows. (cid:3) . Highest weight ∗ -representations for O q ( Z ν ) and O q ( G ν \ G R )5.1. The spectral ∗ -algebra O q ( G ν \ G R ) . Let ν = ( τ, ǫ ) ∈ H ug be an ungauged twisting datum. We recallthat we view O q ( Z ν ) as a full spectral ∗ -algebra. In the following we also write this as O q ( Z ν ) = O ( Z ν,q ) = O q ( G ν \\ G R ) = O ( G ν,q \\ G R ,q ) . Here G is again the connected, simply connected complex Lie group integrating g , written as G R whenviewing it specifically as a real Lie group in stead of a complex one. The symbol G ν evokes a closed realLie subgroup of G R determined by ν , see [DCM18] for more information on this interpretation. When ν isa reduced symmetric twisting datum, G ν is the real form of G determined by ν as its Vogan diagram.Recall from Lemma 4.1 that the coaction ρ ν is centrally coinvariant. We can hence consider the topologicalspaces G ν,q \\ G R ,q /U q = Z ν,q /U q ⊆ Z ν,q //U q = G ν,q \\ G R ,q //U q . However, as one can glean from this notation, there is also a more refined spectral condition one can impose.
Definition 5.1. If π is a bounded ∗ -representation of O q ( Z ν ) on a Hilbert space H π , we call a joint eigenspaceof the π ( a ̟ ) a weight space . We call π a weight ∗ -representation if H π is the closure of the sum of its weightspaces.If ξ lies in a weight space, we call λ ∈ H τ with a ̟ ξ = λ P ( ̟ ) ξ, ∀ ̟ ∈ P + the associated weight. Note that ∗ -compatibility ensures that λ is indeed a τ -twisting datum. Lemma 5.2.
Any irreducible ∗ -representation π of O q ( Z ν ) is a weight ∗ -representation, and all its weightslie in the same H ≫ τ -orbit for the multiplication action.Proof. It follows from (4.4) that the absolute values | π ( a ̟ ) | obey q -commutation relations with π ( O q ( Z ν )).By irreducibility, we then either have | π ( a ̟ ) | = 0 or Ker( | π ( a ̟ ) | ) = 0 with the spectrum of | π ( a ̟ ) | a setwith 0 as its only possible accumulation point. It also follows from (4.4) and irreducibility that the polarparts of the π ( a ̟ ) are central, hence scalar. It is now immediate that π is a weight ∗ -representation. (cid:3) Definition 5.3.
We call an irreducible ∗ -representation π of O q ( Z ν ) • of regular type if π ( a ρ ) has zero kernel, • positive if π ( a ̟ r ) ≥ for all r ∈ I τ . Note that if π is an irreducible ∗ -representation of regular type, it follows by Lemma 5.2 that there existsa dense subspace V π ⊆ H π , namely the algebraic direct sum of the weight spaces, on which π extends to a(non-bounded) ∗ -representation of O q ( Z reg ν ). Definition 5.4.
We write O q ( Z + ν ) for O q ( Z ν ) as a spectral ∗ -algebra with respect to the family P + of allirreducible, positive representations of regular type. Borrowing notation from the introduction, the irreducible positive ∗ -representations of regular type of O q ( Z + ν ) correspond to the ‘open cell’ Z + , reg ν,q .The coaction ρ ν will in general not be spectral on O q ( Z + ν ). We therefore apply the construction describedin Lemma 1.6, noting that the hypotheses are satisfied by Proposition 3.3. Definition 5.5.
We define O q ( G ν \ G R ) as the spectral ∗ -algebra O q ( Z + ν U ) with spectral coaction ρ ν . We can then also consider Z + ν,q U q /U q = G ν,q \ G R ,q /U q . It is not true of course that any x -representation for x ∈ G ν,q \ G R ,q /U q will be either positive or of regulartype. We do however have the following. emma 5.6. For any x ∈ G ν,q \ G R ,q /U q there exists a positive x -representation of regular type.Proof. Let x ∈ G ν,q \ G R ,q /U q , and let π be an irreducible admissible x -representation of O q ( G ν \ G R ). ByLemma 1.6 we can find π ′ ∈ P + and a ∗ -representation π ′′ of O q ( U ) such that π π ′ ∗ π ′′ . Clearly π ′ mustbe an x -representation. (cid:3) For x ∈ G ν,q \ G R ,q /U q we let O q ( O νx ) be the associated fiber of O q ( G ν \ G R ), which by Lemma 1.8 may beinterpreted as a full spectral ∗ -algebra. We let C q ( O νx ) be the associated C ∗ -envelope. We denote by L ∞ q ( O νx )the von Neumann algebraic envelope of O q ( O νx ).Our goal is to understand the L ∞ q ( O νx ) for x ∈ G ν,q \ G R ,q /U q , together with their invariant states R O νx,q . Thefinal result will be presented in the next section. For now, we will develop some of the representation theoryof O q ( Z ν ) and O q ( Z reg ν ) as preparation.5.2. Admissible representations.
Recall the triangular decomposition O q ( Z reg ν ) = N − A N + from (4.14).Recall also the correspondence (4.16). Definition 5.7.
Let λ ∈ H × τ . We call highest weight ∗ -representation of O q ( Z reg ν ) at weight λ any non-zero O q ( Z reg ν ) -module with ∗ -compatible pre-Hilbert space structure for which there exists a cyclic weight vector ξ λ vanishing under N + and such that a ω ξ λ = a ω ( λ ) ξ λ , ω ∈ P. Lemma 5.8.
For each λ ∈ H × τ there can exist up to isomorphism at most one highest weight ∗ -representationat weight λ .Proof. The argument is standard, see e.g. [JL92]. By the triangular decomposition (4.14), one can form thehighest weight Verma module M λ ∼ = N − at weight λ , say with highest weight vector ξ λ ∼ = 1. There thenexists a unique invariant Hermitian form on M λ , determined by h xξ λ , yξ λ i = e P ( x ∗ y )( λ ) , x, y ∈ N − , with e P = ι − ν ◦ P ◦ ι ν and P the projection map (4.10). The subspace N λ = { v ∈ M λ | h v, −i = 0 } ⊆ M λ must be a maximal proper submodule by cyclicity of ξ λ and invariance of the hermitian form. The hermitianform must then descend to a non-degenerate hermitian form on V λ = M λ /N λ .If now W λ is any highest weight ∗ -representation at λ , we obtain a unique-up-to-scalar non-zero morphism π : M λ → W λ by universality. Since the positive-definite form on W λ must retract to a multiple of theunique invariant hermitian form on M λ , it follows that N λ = Ker( π ), and π must descend to an injectivemap V λ → W λ , which is then bijective by cyclicity of the highest weight vector. In particular, the hermitianform on V λ must be positive-definite, and the map V λ ∼ = W λ a multiple of a unitary. (cid:3) In the following we will denote by V λ the highest weight ∗ -representation at λ , if it exists. Lemma 5.9.
Let V λ be a highest weight ∗ -representation of O q ( Z reg ν ) at weight λ ∈ H × τ . Then V λ canbe completed to an irreducible O q ( Z ν ) -representation ( π λ , H λ ) of regular type, and any irreducible O q ( Z ν ) -representation of regular type arises in this way for a unique λ .Proof. The fact that V λ completes to a bounded ∗ -representation of O q ( Z ν ) follows by the same argument asin Section 1.3. Clearly H λ is then irreducible and of regular type. Conversely, assume ( π, H π ) is an irreducible O q ( Z ν )-representation of regular type. As in the proof of Lemma 5.2, a ρ must necessarily have a boundedand discrete spectrum in any irreducible ∗ -representation. Since a ̟ r x r is a multiple of E r ◮ a ̟ r ∈ O q ( Z ν ),and a ρ a ̟ r x r = q − ρ,α r ) a ̟ r x r a ρ , we must have that a ̟ r x r ξ = 0 for all r for some non-zero weight vector ξ , say at weight λ . If π is of regular type, we must also have x r ξ = 0 for all r . Hence ξ is a highest weightvector. By irreducibility, ξ must be cyclic for H π and hence also for V π , so V π ∼ = V λ . (cid:3) ow clearly any V λ will factor over a character of the center of O q ( Z ν ), and in particular over a character x of O q ( Z ν /U ). We call x = x λ the associated central character . Definition 5.10.
We call λ ∈ H × τ a highest weight (associated to the twisting datum ν ) if there existsa highest weight ∗ -representation V λ of O q ( Z reg ν ) at weight λ . We call λ an admissible highest weight ifmoreover the associated central character x λ lies in G ν,q \ G R ,q /U q . We then also say that V λ is an admissiblehighest weight ∗ -representation.We denote e Λ ν = { λ a highest weight associated to ν } ⊆ H × τ , Λ ν = { admissible highest weights } ⊆ e Λ ν , and Λ >ν = Λ ν ∩ H >τ , Λ ≫ ν = Λ ν ∩ H ≫ τ . For x ∈ G ν,q \\ G R ,q /U q we further define Λ ν ( x ) = { λ ∈ e Λ ν | x = x λ } . The following proposition shows that there is no clash of terminology.
Proposition 5.11.
Let λ ∈ e Λ ν . Then the following are equivalent: (1) H λ is admissible as a ∗ -representation of O q ( G ν \ G R )(2) λ ∈ Λ ν (3) x λ ∈ G ν,q \ G R ,q /U q .Proof. The equivalence between (2) and (3) is simply the definition of Λ ν . To see that (1) ⇒ (3), we can useLemma 1.6 together with the fact that tensoring with an O q ( U )-representation does not change the centralcharacter by Lemma 4.1. Together with Lemma 1.8, the latter lemma also gives the reverse implication (3) ⇒ (1). (cid:3) Properties of the set Λ ν of admissible highest weights. We are not yet able to give a completedescription of the sets Λ ν for general u and general ν , but see Section 7 for some partial results. For now,we derive some structural properties of these sets. Lemma 5.12.
For any x ∈ G ν,q \\ G R ,q /U q the set Λ ν ( x ) is closed under multiplication by H g τ .Proof. By the descent of the coaction ρ ν to O q ( O νx ), we have an action of the maximal torus T on O q ( O νx ).More precisely, the character e iH : U ̟ ( ξ, η ) e i ( H, wt( η )) , H ∈ a acts on a ̟ by multiplication with e i ( H − τ ( H ) ,̟ ) , from which the lemma follows. (cid:3) From Corollary 4.9, Lemma 5.6 and the fact that e χ HC ( z ̟ )( λ ) ξ λ = e χ HC ( z ̟ ) ξ λ = x ( z ̟ ) ξ λ , ̟ ∈ P τ, + , we obtain the following. Corollary 5.13. (1)
For x ∈ G ν,q \\ G R ,q /U q the set Λ ν ( x ) consists of finitely many H g τ -orbits. (2) Λ ν is contained in the W ν -orbit of Λ >ν under the · ǫ,q -action. Corollary 5.14.
Let H be a bounded representation of O q ( O νx ) , and assume that all a ∗ ̟ r a ̟ r have purelydiscrete spectrum not containing zero. Then there exists a measurable field of multiplicity Hilbert spaces λ
7→ G λ over Λ ν ( x ) such that H ∼ = Z ⊕ Λ ν ( x ) d λ H λ ⊗ G λ . ere we view Λ ν ( x ) as a closed subset of C | I | . Proof.
Let u s be the polar part of π H ( a ̟ s ) for s ∈ I ∗ . Then the u s commute amongst themselves and withthe operators in π H ( O q ( O νx )). It follows that H ∼ = Z ⊕ T | I ∗| d θ H θ . where each H θ is a representation of O q ( O νx ) still satisfying the given assumption. However, as now the polarpart of a ̟ s for s ∈ I ∗ is a fixed scalar, it follows from Corollary 5.13 that each H θ must be a direct sumwith components from a finite family of highest weight representations H λ . The corollary is now clear. (cid:3) In the upcoming Theorem 5.26, we aim to provide a more refined version of Corollary 5.13.(2). We first needto examine some low-rank cases.Recall O q ( H ) with its O q ( SU (2))-coaction introduced in Section 1.6. Lemma 5.15.
Consider the localisation O q ( H reg2 ) := O q ( H )[ z − ] . Then O q ( H ) ⊆ O q ( H reg2 ) , and O q ( H reg2 ) is the universal ∗ -algebra generated by elements a ± , x, D with a, D selfadjoint, with D central and with ax = q − xa and xx ∗ − q − x ∗ x = Da − − q − q − . The precise correspondence is given by z = a, v = q − / ( q − − q ) xa. Moreover, the infinitesimal action of U ′ q ( su (2)) on O q ( H ) extends to an inner action on O q ( H reg2 ) , deter-mined by K ◮ X = a − Xa, E ◮ X = xX − a − Xax, E ∗ ◮ X = x ∗ X − a − Xax ∗ . (5.1) Proof.
The fact that O q ( H ) ⊆ O q ( H reg2 ) is straightforward, as is the fact that O q ( H reg2 ) is presented by theabove generators and relations under the given correspondence.Consider now O q ( Z ν ) in rank 1 for ν = (id , ǫ ) with ǫ a formal (central) variable. The fact that ǫ is herea central abstract element, and not a scalar, is inessential to apply the theory from the previous sections.In particular, we know from [DCM18, Lemma 2.20] that the generating matrix Z ̟ of O q ( Z ν ) satisfies thereflection equation by [DCM18, Lemma 2.20]. We hence obtain a U ′ q ( su (2))-linear ∗ -homomorphism π : O q ( H ) → O q ( Z ν ) , Z Z ̟ . (5.2)We claim that π ( a ) = a ̟ , π ( x ) = x α , π ( D ) = ǫ, (5.3)Then from the universal relations of source and target it is clear that π must be an isomorphism. Moreover, itwill then follow from the U ′ q ( su (2))-modularity of the inclusion (3.12) that indeed the infinitesimal U ′ q ( su (2))-module ∗ -algebra structure on O q ( H ) extends to O q ( H reg2 ) by means of the given inner actions.To see that (5.3) holds, note that automatically π ( a ) = π ( z ) = a ̟ by definition of a ̟ . We then also find π ( v ) = q − / ( q − − q ) x α a ̟ upon writing out π ( E ◮ z ) = E ◮ π ( z ). Hence π ( x ) = x α . Now writing out π ( E ◮ w ) = E ◮ π ( w ), we also find that π ( u ) = π ( z ) − q / π ( E ◮ w ) = a ̟ − ( q − − q )( x α a ̟ x ∗ α − x ∗ α a ̟ x α ) = ǫa − ̟ + q ( q − − q ) x α x ∗ α a ̟ . (5.4)Hence π ( D ) = π ( uz − q − vw ) = ǫ + q ( q − − q ) x α x ∗ α a ̟ − q − ( q − − q ) x α a ̟ x ∗ α = ǫ (cid:3) orollary 5.16. Consider u = su (2) with simple root α and fundamental weight ̟ = α . Let ν = (id , ǫ ) be a twisting datum with ǫ Q ( α ) = ǫ ∈ R . Then there exists a unique O q ( SU (2)) -colinear ∗ -homomorphism π : O q ( H ) → O q ( Z ν ) such that z a ̟ . The kernel of this map is the ideal generated by D − ǫ . From Proposition 1.14 we now obtain the following.
Corollary 5.17.
Consider u = su (2) with twisting datum ǫ ∈ R . Let W ǫ be the Weyl group, so W is trivialand W ǫ = W = { , s } if ǫ = 0 . Identify H × id ∼ = R × . • If ǫ > , then Λ ν = Λ >ν = q − N ǫ / while e Λ ν = ± Λ ν . Moreover, ( s · ǫ,q e Λ ν ) ∩ e Λ ν = ∅ . • If ǫ = 0 , then Λ ν = Λ >ν = R > while e Λ ν = R × . • If ǫ < , then Λ ν = e Λ ν = R × with Λ >ν = R > and s · ǫ,q Λ >ν = − Λ >ν . Lemma 5.18.
Consider u = su (2) ⊕ su (2) with roots α , α . Let ν = ( τ, ǫ ) be the ungauged twisting datumgiven by τ (1) = 2 and ǫ = ǫ = ǫ ≥ . (1) If ǫ > , then λ ∈ Λ ν if and only if λ P ( ̟ ) = q − n ǫ / e iθ for n ∈ N and θ ∈ R . (2) If ǫ = 0 , then Λ ν = H × τ ∼ = C × .Proof. See Appendix, Theorem A.6. (cid:3)
Corollary 5.19.
In the situation of Lemma 5.18 with ǫ = 0 , one has W ν = { , ˆ s } with ˆ s = s s and (ˆ s · ǫ,q Λ ν ) ∩ Λ ν = ∅ . Lemma 5.20.
Consider u = su (3) with roots α , α . Assume that τ (1) = 2 , and let ǫ = ǫ = ǫ ≥ be atwisting datum. (1) If ǫ > , then λ ∈ Λ ν if and only if λ P ( ̟ ) = q − n ǫe iθ for n ∈ N and θ ∈ R . (2) If ǫ = 0 , then Λ ν = H × τ ∼ = C × .Proof. See Appendix, Theorem A.11. (cid:3)
Corollary 5.21.
In the situation of Lemma 5.20 with ǫ = 0 , one has W ν = { , ˆ s } with ˆ s = s s s and (ˆ s · ǫ,q Λ ν ) ∩ Λ ν = ∅ . We now consider u of general rank. Recall again the O q ( SU (2))-coactions introduced in Section 1.6. Lemma 5.22.
For r = τ ( r ) , there exists a unique U q r ( su (2)) -equivariant ∗ -homomorphism κ r : O q r ( H ) → O q ( Z ν ) , (cid:18) z wv u (cid:19) (cid:18) z r w r v r u r (cid:19) such that z r = a ̟ r . Then v r = q − / r ( q − r − q r ) x r a ̟ r , and furthermore κ r ( D ) = D r := ǫ r a ̟ r − α r and κ r ( T ) = T r := q − r ( q r − q − r ) x ∗ r x r a ̟ r + q − r a ̟ r + ǫ r q r a ̟ r − α r . Proof.
The existence of a ∗ -homomorphism κ r : O q r ( H reg2 ) → O q ( Z reg ν ) , z a ̟ r , v q − / r ( q − r − q r ) x r a ̟ r , D ǫ r a ̟ r − α r follows immediately from the defining relations in Lemma 5.15. This map is then seen to be U ′ q ( su (2))-linear,and hence U q ( su (2))-linear, by the explicit formula (5.1). Since z generates O q ( H ) as an U q ( su (2))-module ∗ -algebra, we obtain that κ r maps O q ( H (2)) equivariantly into O q ( Z ν ), and that moreover this is the uniquesuch map sending z to a ̟ r . The formula for κ r ( T ) now follows by an easy explicit computation, recyclingfor example the formula (5.4) (upon replacing ǫ by ǫ r a ̟ r − α r ). (cid:3) emma 5.23. For r = τ ( r ) , there exists a unique U q r ( su (2)) -equivariant ∗ -homomorphism κ r : O q r ( V R ) → O q ( Z ν ) , (cid:18) a bc d (cid:19) (cid:18) a r b r c r d r (cid:19) such that c r = a ̟ r . Then d r = q − / r ( q − r − q r ) a ̟ r x ∗ r and κ r ( D ) = D r with D r := (1 + q − r ( q − r − q r ) x ∗ r x r ) a ̟ r + ̟ τ ( r ) . (5.5) Proof.
It is immediate from the defining relations of O q r ( V R ) and O q ( Z reg ν ) that there exists a unique ∗ -homomorphism κ r : O q r ( V R ) → O q ( Z ν ) , c a ̟ r , d q − / r ( q − r − q r ) a ̟ r x ∗ r . It is then also immediate that κ r ( D ) is given by (5.5).To see that κ r is equivariant (and is then uniquely determined by its value on c ), we note that K ◮ (cid:0) c d (cid:1) = (cid:0) q r c q − r d (cid:1) , E ◮ (cid:0) c d (cid:1) = (cid:16) q / r c (cid:17) , E ∗ ◮ (cid:0) c d (cid:1) = (cid:16) q / r d (cid:17) . It is straightforwardly verified that κ r is U q r ( su (2))-equivariant on the two-dimensional module spanned by c, d . Hence κ r is U q r ( su (2))-equivariant as c, d generate O q r ( V R ) as an U q r ( su (2))-module ∗ -algebra. (cid:3) We introduce the following ∗ -subalgebras on O q ( Z ν ). Definition 5.24.
We let A r be the ∗ -algebra generated by the range of κ r in Lemma 5.22 and Lemma 5.23.We let A loc r be the ∗ -algebra generated by A r and the a ± α r , a ± ̟ r . For r = τ ( r ) , we let A loc r,τ ( r ) be the ∗ -algebragenerated by A loc r and A loc τ ( r ) . Note that for r ∈ I τ , A loc r is simply the unital ∗ -algebra generated by x r , a ± α r and a ± ̟ r , corresponding upto the Cartan part with the rank one case. When r / ∈ I τ , A loc r,τ ( r ) is the unital ∗ -algebra generated by the a ± ̟ r , a ± α r , x r and x τ ( r ) , corresponding up to the Cartan part with a rank two case with non-trivial involution.Consider now the ‘big cell’ ∗ -representation ( s , S ) of the ∗ -algebra O q ( SU (2)), given by S = l ( N ) and s ( a ) e n = (1 − q n ) / e n − , s ( b ) e n = q n e n , as well as the ∗ -characters χ θ : a e iθ , b θ ∈ R / π Z , cf. Section 1.6.Let π r : O q ( U ) → O q r ( SU (2) r )be the natural restriction map obtained by restricting O q ( U ) to functionals on the isomorphic copy of U q r ( su (2)) generated by E r , F r , K α r ∈ U q ( u ). By these factorisations, the above ∗ -representations can beinterpreted as ∗ -representations ( s r , S r ) and χ ( r ) θ of O q ( U ).For H λ an irreducible ∗ -representation of regular type of O q ( G ν \ G R ), the following theorem describes itsfusion rules with the ∗ -representations s r and χ ( r ) θ . We will use the following notation: for r ∈ I ∗ ∪ τ ( I ∗ )and θ ∈ R we define η r ( θ ) ∈ H g τ by η r ( θ ) s = 1 for s / ∈ { r, τ ( r ) } and η r ( θ ) r = e iθ , η r ( θ ) τ ( r ) = e − iθ . Theorem 5.25.
Let x ∈ G ν,q \ G R ,q /U q , and let λ ∈ Λ ν ( x ) . We denote by H some generic multiplicityHilbert space. (1) If r ∈ I τ and λ P ( α r ) ǫ r ≥ , then π λ ∗ s r ∼ = H ⊗ π λ , π λ ∗ χ ( r ) θ ∼ = π λ . If r ∈ I τ and λ P ( α r ) ǫ r < , then also s r · ǫ,q λ ∈ Λ ν ( x ) and π λ ∗ s r ∼ = H ⊗ ( π λ ⊕ π s r · ǫ,q λ ) , π λ ∗ χ ( r ) θ ∼ = π λ . (3) If r ∈ I ∗ ∪ τ ( I ∗ ) , then π λ ∗ s r ∼ = H ⊗ Z T π λη r ( θ ) d θ, π λ ∗ χ ( r ) θ ∼ = π λη r ( θ ) . Proof.
The fusion rules with the χ ( r ) θ are immediate, so we concentrate on the fusion rules with the s r .In all cases, we have that π λ ∗ s r ( a ̟ s ) = a ̟ s ⊗ , s / ∈ { r, τ ( r ) } by (3.11) and the definition of a ̟ s . On the other hand, by Lemma 5.22 and Lemma 5.23 we see that π λ ◦ κ r is a ∗ -representation of either O q r ( H ) or O q r ( V R ) under which respectively z and c get sent to operatorswith discrete spectrum not containing zero. It then follows from the computations in Section 1.6 that π λ ◦ κ r is a direct sum of irreducible ∗ -representations satisfying the same property. From the fusion rules computedthere, we moreover see that π λ ∗ s r ( a ̟ r a ∗ ̟ r ) must have discrete spectrum not containing 0.Since π λ ∗ s r still has the same central character x , it follows from Corollary 5.14 that π λ ∗ s r is a directintegral over amplifications of the π λ ′ with λ ′ ranging inside the W ν ⋉ H g τ -orbit of λ .Now if r ∈ I τ the spectrum of the a ̟ s for s ∈ I ∗ ∪ τ ( I ∗ ) remains the same, so we must actually have adirect sum (possibly with repetition) over the elements in the W ν -orbit of λ under the · ǫ,q -action. For each λ ′ which appears, we have a surjective O q ( Z ν )-intertwiner H λ ⊗ S r ։ H λ ′ . Since H λ = N − ξ λ , and since0 < q <
1, it follows that | λ ′ s | ≤ | λ s | for s = r . On the other hand, if this is a strict inequality for some s = r , the above surjection must be zero on ξ λ ⊗ S r , and must hence be zero everywhere by using the generalidentity ρ ν ( O q ( Z ν ))(1 ⊗ O q ( U )) = O q ( Z ν ) ⊗ O q ( U ) . It follows that λ ′ s = λ s for s = r , and that a highest weight vector ξ λ ′ of H λ ⊗ S r lies in A r ξ λ ⊗ S r , thelatter space vanishing under all x s with s = r . Since λ P ( α r ) ǫ r ≥ λ P (2 ̟ r − α r ) ǫ r ≥
0, it noweasily follows from Lemma 5.22 and Proposition 1.14 that we must have λ ′ ∈ { λ, s α r · ǫ,q λ } , with λ ′ = λ if λ P ( α r ) ǫ r ≥ λ and s r · ǫ,q λ appearing as highest weights if λ P ( α r ) ǫ r < r = τ ( r ). Then again we can conclude as above that any vector in H λ ⊗ S r vanishing underall x s with s / ∈ { r, τ ( r ) } must lie in A loc r,τ ( r ) ξ λ ⊗ S r . If now H λ ′ appears in the direct integral decompositionof H λ ⊗ S r , we must have | λ ′ r | ∈ {| λ r | , | (ˆ s r · ǫ,q λ ) r |} by applying Corollary 5.13 to the rank 2 case withnon-trivial involution. By Corollary 5.19 and Corollary 5.21, we see however that only the value | λ r | canappear.Finally, to see that we must have a direct integral over the full circle, we simply note that H λ must decomposeas a direct sum of A r -representations, each of which has full spectrum with uniform multiplicity for theunitary part of a ̟ r inside its tensor product with S r by Lemma 5.23 and the fusion rules described inTheorem 1.13. (cid:3) Theorem 5.26.
Let x ∈ G ν,q \ G R ,q /U q . The map H g τ × W − ν × Λ ≫ ν ( x ) → Λ ν ( x ) , ( f, w, λ ) f ( w · ǫ,q λ ) is a well-defined bijection.Proof. Recall that W − ν was defined in Definition 2.25. Using then Lemma 2.24, we see from Lemma 5.12and Theorem 5.25 that the above map is well-defined.To see that it is surjective, pick λ ∈ Λ ν with associated central character x = x λ . By definition of admissibleweight and Lemma 1.6, there exists a λ ′ ∈ Λ ≫ ν ( x ) such that π λ is weakly contained in π λ ′ ∗ π reg , where π reg36 s the regular representation of O q ( U ) on L q ( U ). However, with w = s r . . . s r N a longest word in W , onehas by [LS91] π reg ∼ = H ⊗ s r ∗ . . . ∗ s r N ∗ Z T χ θ d θ with H a multiplicity Hilbert space and with χ θ ( U ̟ ( ξ, η )) = e πi (wt( η ) ,θ ) h ξ, η i , θ ∈ T ∼ = a /Q ∨ . Hence π λ must be weakly contained in some π λ ′ ∗ s r ∗ . . . ∗ s r N ∗ χ θ with λ ′ ∈ Λ ≫ ν . By Theorem 5.25, theremust exist a finite subsequence r i , . . . , r i K in J τν such that π λ = π s riK ...s ri · ǫ,q λ ′ ∗ χ θ , where in the finitesubsequence we may assume that at each step, there is a place in J τν where the sign is changed. Hence wemay assume s r iK . . . s r i ∈ W − ν , and so our map is surjective.To see that the map is injective, note that the sign patterns of λ and f ( w · ǫ,q λ ) determine w and f byTheorem 2.26. (cid:3) From Theorem 2.26 we obtain also the following.
Corollary 5.27.
For each x ∈ G ν,q \ G R ,q /U q the set Λ ≫ ν ( x ) consists of at most one point. We will in the following write Λ ≫ ν ( x ) = { λ x } , and call λ x the positive highest weight associated to x .The above results on Λ ν ( x ) were obtained using the coaction by O q ( U ). For the next results, we will obtaininformation about the total set Λ ν using the coaction e γ by U q ( u ) on O q ( Z ν ) defined in (3.15). Lemma 5.28. If λ ∈ Λ ≫ ν , then also λq − ( ̟ + τ ( ̟ )) ∈ Λ ≫ ν for all ̟ ∈ P + .Proof. Choosing π λ with λ ∈ Λ ≫ ν , we see that ( π ̟ ⊗ π λ ) ◦ e γ contains the highest weight vector ξ ̟ ⊗ ξ λ atweight λq − ( ̟ + τ ( ̟ )) . (cid:3) Also Λ ν is stable under the above operation, but we need one extra argument. Proposition 5.29. If λ ∈ Λ ν , then also λq − ( ̟ + τ ( ̟ )) ∈ Λ ν for all ̟ ∈ P + .Proof. We have to show that ( π ̟ ⊗ π λ ) ◦ e γ is again admissible as an O q ( G ν \ G R )-representation. By definitionof admissibility, it is actually sufficient to show that( π ̟ ⊗ π λ ⊗ π reg ) ◦ (id ⊗ ρ ν ) e γ is admissible for all λ ∈ Λ ≫ ν , where π reg is the regular representation of O q ( U ) on L q ( U ). However, by theYetter-Drinfeld compatibility (3.16) we have( π ̟ ⊗ id ⊗ id)(id ⊗ ρ ν ) e γ ( f ) = U ∗ ̟, (( π ̟ ⊗ id ⊗ id)( e γ ⊗ id) ρ ν ( f )) U ̟, , from which the result is obvious. (cid:3) Combining this with Theorem 5.26, we obtain the following useful corollary extending Lemma 5.28.
Corollary 5.30. If λ ∈ Λ ≫ ν , then also λq − w − ( ̟ + τ ( ̟ )) ∈ Λ ≫ ν for all ̟ ∈ P + and w ∈ W − ν . To end this section, we will show that a certain positivity assumption is unperturbed by equivalence oftwisting data. In the next section, we will in fact show that this positivity assumption is satisfied for alltwisting data, see Proposition 6.4. Let us write again λ = q γ for γ ∈ a τ with λ r = q (2 γ,̟ r ) . Lemma 5.31.
Consider the following property for a reduced twisting datum ν :For all γ ∈ a τ with λ = q γ in Λ ≫ ν one has ( ρ − γ, β ) > , ∀ β ∈ ˆ∆ + c . (*) Then Property (*) is preserved under equivalence of twisting data. roof. In the proof, let us write ˆ∆ c = ˆ∆ ν,c for emphasis on the ν -dependence.Assume that ν, ν ′ are equivalent, and assume that Property (*) holds for ν . By Corollary 2.29, we have ǫ ′ = f wǫ for w ∈ W − ν and f a gauge. By Lemma 3.5 and Theorem 5.26 we then have for γ ∈ a τ and λ = q γ ∈ Λ ≫ ν as above that | w · ǫ,q λ | = q w ( γ − ρ )+ ρ ) ∈ Λ ≫ ν ′ , establishing a bijection Λ ≫ ν → Λ ≫ ν ′ , λ
7→ | w · ǫ,q λ | . We are now to verify that ( wρ − wγ, β ) > β ∈ ˆ∆ + ν ′ ,c . But this is clear since ˆ∆ + ν ′ ,c = w ˆ∆ + ν,c , cf.(2.9). (cid:3) The von Neumann algebra L ∞ q ( O νx ) and its invariant integral R O νx,q Fix again an ungauged twisting datum ν ∈ H ug τ , and let x ∈ G ν,q \ G R ,q /U q . Recall from the end of Section5.1 that O q ( G ν \ G R ) → O q ( O νx ) denotes the associated fiber at x , with von Neumann algebraic completion L ∞ q ( O νx ) and associated invariant integral R O νx,q . Our goal is to present concrete models for the couple L ∞ q ( O νx ) , Z O νx,q ! . The structure of L ∞ q ( O νx ) is easily obtained from the results of the previous section. Theorem 6.1.
Let λ x ∈ Λ ≫ ν ( x ) be the positive highest weight associated to x . Then L ∞ q ( O νx ) ∼ = (cid:16) ⊕ w ∈ W − ν B ( H w · ǫ,q λ x ) (cid:17) ⊗ L ∞ ( T s ) , (6.1) where s = dim a − τ .Proof. For θ ∈ T s ∼ = a − τ / ( Q ∨ ) − τ , consider f θ ∈ H g τ given by ( f θ ) P ( ω ) = e πi ( ω,θ ) . From Theorem 5.26,it follows that all π f θ ( w · ǫ,q λ x ) are mutually inequivalent irreducible ∗ -representations of O q ( G ν \ G R ). Hencealso the direct integral H x = ⊕ w ∈ W − ν Z T s H f θ ( w · ǫ,q λ x ) d θ is an admissible representation of O q ( G ν \ G R ). On the other hand, it follows from Theorem 5.25 that theoperations ∗ s r and ∗ χ θ leave H ⊗ H x stable (with H some multiplicity Hilbert space of countable dimension).Since the regular representation π reg of O q ( U ) is a product of the various s r and χ θ up to multiplicity, andsince L ∞ q ( O νx ) is the σ -weak closure of ( π λ x ∗ π reg )( O q ( O νx )), it follows that L ∞ q ( O νx ) is the σ -weak closure of O q ( O νx ) in its representation on H x . Since however the π f θ ( w · ǫ,q λ x ) are mutually inequivalent, the latter vonNeumann algebra is simply L ∞ q ( O νx ) ∼ = ⊕ w ∈ W − ν Z T s B ( H f θ ( w · ǫ,q λ x ) ) dθ ∼ = (cid:16) ⊕ w ∈ W − ν B ( H w · ǫ,q λ x ) (cid:17) ⊗ L ∞ ( T s ) . (cid:3) Let us now describe in more detail the associated invariant integral R O νx,q where we will use the isomorphism(6.1) implicitly. We will also abbreviate π w = π w · ǫ,q λ x , w ∈ W − ν , and use the notation wt( Z ̟ ( ξ, η )) = wt( η ) − τ (wt( ξ )) . Then (id ⊗ χ θ ) ρ ν ( f ) = e πi (wt( f ) ,θ ) f, θ ∈ T ∼ = a /Q ∨ . roposition 6.2. There exist c w = c w ( λ x ) > for w ∈ W − ν such that Z O νx,q f = X w ∈ W − ν c w Tr( π w ( f ) A w ) Z T s e πi (wt( f ) ,θ ) d θ, (6.2) where A w = | π w ( a ρ ) | and where we identify once more T s ∼ = a − τ / ( Q ∨ ) − τ .Proof. From (4.5), it follows that the modular automorphism of O q ( O νx ) is implemented as Ad( ⊕ w ∈ W − ν A itw ⊗ ⊕ w ∈ W − ν H w ⊗ L ( T s ). Since R O νx,q is faithful, this already implies that Z O νx,q f = X w ∈ W − ν c w Tr( π w ( f ) A w ) Z T s e πi (wt( f ) ,θ ) g w ( θ ) d θ, for c w > g w ∈ L ( T s , d θ ). Since however R O νx,q ∗ χ θ = R O νx,q for all θ ∈ a /Q ∨ ,it follows that g w is a constant, which can then be absorbed into c w . (cid:3) In the remainder of this section, we will determine the c w more explicitly. To simplify some of the compu-tations in what follows, we will from now on make the standing assumption that ǫ is reduced. Lemma 6.3.
Assume ̟ ∈ P + with τ ( ̟ ) = ̟ . Let λ = λ x = q γ ∈ H ≫ τ , where γ ∈ a τ . Then Z O νx,q a ̟ = 1dim q ( V ̟ ) P w ∈ W ν d sgn( w ) ǫ Q ( ̟ + ρ − w ( ̟ + ρ )) q γ − ρ,w ( ̟ + ρ )) P w ∈ W ν d sgn( w ) ǫ Q ( ρ − wρ ) q γ − ρ,wρ ) . (6.3)Note that the sign function d sgn on ˆ W is to be taken with respect to the simple reflections s ˆ r for ˆ r ∈ I/τ . Alsonote that the proof will show that the right hand side, up to the factor dim q ( V ̟ ) − , is actually a polynomialin q , so that there is no issue evaluating the quotient. Proof.
Recall first the projection map E from (4.1), the Harish-Chandra homomorphism e χ HC from (4.15)and the functions j ̟ introduced in (4.2). Then since zξ λ = e χ HC ( z ) ξ λ for z ∈ Z ( O q ( Z ν )), we have Z O νx,q a ̟ = (( e χ HC ◦ E )( a ̟ ))( λ ) , which by (4.13) becomes Z O νx,q a ̟ = 1dim q ( V ̟ ) X α ∈ ( Q + ) τ j ̟ ( ̟ − α ) ǫ Q ( α ) q γ − ρ,̟ − α ) . It now suffices to show that X α ∈ ( Q + ) τ j ̟ ( ω − α ) ǫ Q ( α ) q γ − ρ,̟ − α ) = P w ∈ W ν d sgn( w ) ǫ Q ( ̟ + ρ − w ( ̟ + ρ )) q γ − ρ,w ( ̟ + ρ )) P w ∈ W ν d sgn( w ) ǫ Q ( ρ − wρ ) q γ − ρ,wρ ) . (6.4)Assume first that Supp( ǫ ) = I . Then we can take e ǫ ∈ H × τ such that e ǫ P ( α ) = ǫ Q ( α ) for α ∈ Q + . We canhence write X α ∈ ( Q + ) τ j ̟ ( ω − α ) ǫ Q ( α ) q γ − ρ,̟ − α ) = e ǫ P ( ̟ ) X α ∈ ( Q + ) τ j ̟ ( ̟ − α ) e ǫ P ( ̟ − α ) − q γ − ρ,̟ − α ) . Using now the ‘twining Weyl character formula’ [Jan73], where we use the form as given in [FSS96, (5.56)],we find X α ∈ ( Q + ) τ j ̟ ( ̟ − α ) e ǫ P ( ̟ − α ) − q γ − ρ,̟ − α ) = P w ∈ W τ d sgn( w ) e ǫ P ( w ( ̟ + ρ )) − q γ − ρ,w ( ̟ + ρ )) P w ∈ W τ d sgn( w ) e ǫ P ( wρ ) − q γ − ρ,wρ ) , which after multiplication with e ǫ P ( ̟ ) leads to X α ∈ ( Q + ) τ j ̟ ( ω − α ) ǫ Q ( α ) q γ − ρ,̟ − α ) = P w ∈ W τ d sgn( w ) ǫ Q ( ̟ + ρ − w ( ̟ + ρ )) q γ − ρ,w ( ̟ + ρ )) P w ∈ W τ d sgn( w ) ǫ Q ( ρ − wρ ) q γ − ρ,wρ ) . ince all sums concerned are finite, this formula is by continuity still valid for arbitrary ǫ ∈ H ug τ . Sincemoreover any ̟ + ρ − w ( ̟ + ρ ) contains α r with a strictly positive coefficient whenever w contains s r in itsreduced expression, it follows that we can take the above sums over W ν , obtaining (6.4). (cid:3) Let now V w · ǫ,q q γ ( α + ) be the weight space inside V w · ǫ,q q γ at weight ( w · ǫ,q q γ ) q α + for α ∈ Q + . Comparing(6.3) with (6.2), we find for ̟ ∈ P + with ̟ = τ ( ̟ ) and λ x = q γ ∈ Λ ≫ ν , by using the usual Weyl characterformula and noting that | ǫ Q ( ρ − w − ρ ) | = 1 for w ∈ W ν by the assumption that ǫ is reduced, Z O νx,q a ̟ = X w ∈ W − ν X α + ∈ ˆ Q + c w ǫ Q ( ̟ − w − ̟ ) q ̟ + ρ,α + + ρ + w ( γ − ρ )) dim( V w · ǫ,q q γ ( α + ))= 1dim q ( V ̟ ) P w ∈ W ν d sgn( w ) ǫ Q ( ̟ + ρ − w ( ̟ + ρ )) q γ − ρ,w ( ̟ + ρ )) P w ∈ W ν d sgn( w ) ǫ Q ( ρ − wρ ) q γ − ρ,wρ ) = P w ∈ W sgn( w ) q − ρ,wρ ) P w ∈ W ν d sgn( w ) ǫ Q ( ρ − wρ ) q γ − ρ,wρ ) P w ∈ W ν d sgn( w ) ǫ Q ( ̟ + ρ − w ( ̟ + ρ )) q γ − ρ,w ( ̟ + ρ )) P w ∈ W sgn( w ) q − ρ,w ( ̟ + ρ )) = e γ G ( ̟ ) H ( γ ) w ( ̟ )where G ( ̟ ) = X w ∈ W sgn( w ) q − ρ,w ( ̟ + ρ )) , e γ = P w ∈ W sgn( w ) q − ρ,wρ ) P w ∈ W ν d sgn( w ) ǫ Q ( ρ − wρ ) q γ − ρ,wρ ) and H ( γ ) w ( ̟ ) = X w ∈ W ν d sgn( w ) ǫ Q ( ̟ + ρ − w ( ̟ + ρ )) q γ − ρ,w ( ̟ + ρ )) . On the other hand, by Theorem 2.26 we can write H ( γ ) w ( ̟ ) = X w ∈ W − ν ǫ Q ( ̟ − w − ̟ ) ǫ Q ( ρ − w − ρ ) d sgn( w ) F ( γ ) w ( ̟ )where F ( γ ) w ( ̟ ) = X v ∈ W + ν d sgn( v ) q γ − ρ,v − w − ( ̟ + ρ )) . Note that in the formula for F ( γ ) w we could remove a factor ǫ Q ( w − ( ̟ + ρ ) − v − w − ( ̟ + ρ )) by reducednessof ǫ and the positivity result in Lemma 2.28.We have obtained so far the identity X w ∈ W − ν X α + ∈ ˆ Q + c w ǫ Q ( ̟ − w − ̟ ) q ̟ + ρ,α + + ρ + w ( γ − ρ )) dim( V w · ǫ,q q γ ( α + ))= e γ G ( ̟ ) X w ∈ W − ν ǫ Q ( ̟ − w − ̟ ) ǫ Q ( ρ − w − ρ ) d sgn( w ) F ( γ ) w ( ̟ ) . Consider now for a function g : W − ν → {±} the set P + g = { ̟ ∈ ( P + ) τ | ∀ w ∈ W − ν : g ( w ) ǫ Q ( ̟ − w − ̟ ) > } . Then P + g is either empty or contains a translate of 2( P + ) τ in a τ . As all expressions involved in the aboveidentity consist of finite linear combinations of analytic functions on the open unit ball of exp( a τ ) multipliedwith functies of the form q ω q ( ω,κ ) for κ ∈ a τ , it then follows by the above property of each non-empty P + g that in fact X w ∈ W − ν X α + ∈ ˆ Q + c w ǫ Q ( ̟ − w − ̟ ) q ω + ρ,α + + ρ + w ( γ − ρ )) dim( V w · ǫ,q q γ ( α + ))= e γ G ( ω ) X w ∈ W − ν ǫ Q ( ̟ − w − ̟ ) ǫ Q ( ρ − w − ρ ) d sgn( w ) F ( λ ) w ( ω ) or all ̟ ∈ ( P + ) τ and all dominant ω ∈ a τ . In particular, since the characters ̟ ǫ Q ( ̟ − w − ̟ ) are alldistinct by Theorem 2.26, they must be linearly independent, and hence c w X α + ∈ ˆ Q + q ω + ρ,α + + ρ + w ( γ − ρ )) dim( V w · ǫ,q q γ ( α + )) = e γ G ( ω ) ǫ Q ( ρ − w − ρ ) d sgn( w ) F ( λ ) w ( ω )for all dominant ω ∈ a τ and all w ∈ W − ν . Rearranging, and using that we already know the c w to be positive,this becomes c w X α + ∈ ˆ Q + q ω + ρ,α + ) dim( V w · ǫ,q q γ ( α + )) = | e γ | (cid:12)(cid:12)(cid:12)P v ∈ W + ν d sgn( v ) q ρ − γ,w − ( ω + ρ ) − v − w − ( ω + ρ )) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)P w ∈ W sgn( w ) q ω + ρ,ρ − w − ρ ) (cid:12)(cid:12) (6.5)We now want to take the limit of both sides, putting ω = nρ and letting n tending to infinity. However, wefirst want to exclude the possibility of having singularities on the right hand side which cancel each otherout. We therefore state first the following proposition. Proposition 6.4.
Let ν be a reduced twisting datum, and let λ = q γ ∈ Λ ≫ ν . Then ( ρ − γ, β ) > for all β ∈ ˆ∆ + c . (6.6)Before we prove this proposition, let us first show how it can be used to prove the following theorem. Theorem 6.5.
Assume that ǫ ∈ H r τ is a reduced twisting datum. Then Z O νx,q f = P w ∈ W − ν Tr( π w ( f ) A w ) R T s e πi (wt( f ) ,θ ) d θ P w ∈ W − ν Tr( A w ) , ∀ f ∈ O q ( Z ν ) . Proof.
Consider in (6.5) the case where ω = ( n − ρ for n ∈ Z > , c w X α + ∈ ˆ Q + q n ( ρ,α + ) dim( V w · ǫ,q q γ ( α + )) = | e γ | (cid:12)(cid:12)(cid:12)P v ∈ W + ν d sgn( v ) q n ( ρ − γ,w − ρ − v − w − ρ ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)P w ∈ W sgn( w ) q n ( ρ,ρ − w − ρ ) (cid:12)(cid:12) (6.7)Since dim( V w · ǫ,q q γ ( α + )) ≤ X α ′ + = α + P ( α ′ ) , with P Kostant’s partition function, we have that x X α + ∈ ˆ Q + x ( ρ,α + ) dim( V w · ǫ,q q γ ( α + ))is uniformly convergent on ( − , n tend to infinity, the left hand side of (6.7)tends to c w . On the other hand, by Lemma 2.27 and Proposition 6.4, we obtain upon letting n tend toinfinity that the right hand side tends to | e γ | . We thus obtain c w = | e γ | , from which the theorem immediately follows. (cid:3) Let us now come back to the proof of Proposition 6.4.
Proof (of Proposition 6.4).
For Γ ′ ⊆ Γ a τ -stable Dynkin subdiagram we let ν ′ be the twisting datum on Γ ′ obtained by restriction. For λ ∈ H ≫ τ we let λ ′ ∈ H ≫ τ ′ be the unique element such that λ P ′ ( α r ) = λ P ( α r ) forall r ∈ I ′ . Then clearly λ ∈ Λ ≫ ν ⇒ λ ′ ∈ Λ ≫ ν ′ . It is also clear that if β ∈ ˆ∆ + ν,c with β in the integer span of the α r, + with r ∈ I ′ , then β ∈ ˆ∆ + ν ′ ,c . Since also( ρ, β ) = ( ρ ′ , β ), it follows that it suffices to prove the proposition in the case where β contains all α r in itsdecomposition. In particular, we may restrict to the case where I is τ -connected and ǫ is regular. Finally,by Lemma 5.31 we may replace ǫ by any of its representatives within its equivalence class. We will however ot always consider the strongly reduced versions, but will take ‘preferred choices’ allowing for our inductionprocess to go through.Fix now ν and λ = q γ ∈ Λ ≫ ν . It is enough to show that (6.6) holds for β a simple root in ˆ∆ + c .Assume first that τ = id and u ν has non-trivial center (the hermitian symmetric case). Taking ν in itsstrongly reduced form, we have for this case that any simple compact root in ˆ∆ + c is actually simple in ∆ + .Hence (6.6) holds by the considerations in the beginning of the proof.Assume now that τ = id but u ν is semisimple. We recall that we use the Vogan diagram formalism for reducedsymmetric deformation data, coloring the roots with negative value black. For specific implementations ofroot systems, we will use the conventions as in [OV90, Appendix, §
2, Table 1].We will prove this case by induction. We start with the cases which will serve as the induction basis. • If our Vogan diagram is given by , with short root α , then the positive compact roots are α , β = α + 2 α . Clearly ( ρ − γ, α ) >
0. As α , β are orthogonal, it further easily follows that for λ = q γ ∈ Λ ≫ ν we have X v ∈ W + ν d sgn( v ) q n ( ρ − γ,ρ − v − ρ ) = A n + q n ( ρ − γ,β )( ρ,β ∨ ) B n with A n convergent and B n →
1. Since the left hand side of (6.7) converges (for w = e ), it followsthat we must have ( ρ − γ, β ) > • If our Vogan diagram is given by , we note that the simple compact roots are α , β = α +2 α and δ = α + α + α , which are again mutually orthogonal. Using the case proven above, we thensee that ( ρ − γ, α ) >
0, ( ρ − γ, β ) > X v ∈ W + ν d sgn( v ) q n ( ρ − γ,ρ − v − ρ ) = A n + q n ( ρ − γ,δ )( ρ,δ ∨ ) B n with A n convergent and B n →
1. We can conclude again that also ( ρ − γ, δ ) > • Finally, if our Vogan diagram is given by the type G diagram , our compact positive roots areeasily seen to be α and β = 3 α + 2 α . As these are again orthogonal, the same method as in theprevious cases allows to conclude that (6.6) holds also for this case.Let us now prove the remaining cases by induction on the rank. We will follow the list in [OV90, Appendix, §
2, Table 7, type I]. • Assume we are in the case of the Vogan diagram encoding the semisimple Lie algebra so p, l − p )+1 , p ℓ , where we may assume p ≥ + is given as β = α p − + 2 α p + . . . + 2 α l . We may hence restrictto the case p = 2. In this case, we have ℓ ∼ = ℓ , which apart from the trivial compact simple roots has also the compact roots α + α + α and α + 2 α + . . . + 2 α l . Hence we can chop our diagram into two pieces of lower rank to deal withthese two components. The first component gives a hermitian symmetric pair, while by inductionthe second pair can be reduced to the case ∼ = , This table rather contains the Kac diagrams, but one easily passes to the associated Vogan diagram by chopping of thesimple root α . hich was already dealt with before. • Assume that we are dealing with the Vogan diagram of sp p,l − p , given by p ℓ with 1 ≤ p ≤ ⌊ l/ ⌋ . Now the compact simple root which is not a simple root in ∆ + is given as2 α p + 2 α p +1 + . . . + 2 α l − + α l . We may hence assume that p = 1. Now ℓ ∼ = ℓ , which has non-trivial compact simple roots given by α + α and 2 α + . . . + 2 α l − + α l . Choppingup the diagram, we are by induction reduced to the case , which was already dealt with in theinduction basis. • Assume that we are dealing with the Vogan diagram of so p, l − p ) for 2 ≤ p ≤ ⌊ l/ ⌋ , p ℓ − ℓ . The compact simple root which is not simple in ∆ + is given by α p − + 2 α p + . . . + 2 α l − + α l − + α l ,so we can restrict to the case p = 2. In this case ℓ − ℓ ∼ = ℓ − ℓ ,where the non-trivial compact simple roots of the latter are given by α + α + α and α + 2 α + . . . + 2 α l − + α l − + α l . Chopping up the diagram, we can by induction reduce to the case ∼ = , where the simple compact roots of the latter are α , α + α + α , α + α + α , α + α + α , all ofwhich live on components of hermitian symmetric type (that have already been dealt with).This already takes care of all Vogan diagrams of classical type with τ = id. We now deal with the exceptionalcases. • For the diagram we note that the root α + 2 α + 2 α + α + α is compact, hence we can reduce to a diagram oftype D . • For the diagram we have the compact root α + 2 α + 3 α + 2 α + α + 2 α , so that we can reduce to the case E . For the diagram we have the compact root 2 α + 2 α + 3 α + 2 α + α + 2 α , so that we can reduce to the case E . • For the diagram , we have the compact root α + 2 α + 3 α + 4 α + 3 α + 2 α + 2 α , so that we can reduce to thecase E . • Using the equivalence ∼ = , we have for the latter diagram the compact roots α + α + α + α and α + 2 α + 3 α + 4 α +3 α + α + 2 α , so that we can chop the diagram up in the pieces A and E , which have alreadybeen dealt with. • Using the equivalence ∼ = , we have for the latter diagram the compact roots α + α + α and 2 α + 2 α + α , so that we canreduce to the cases B and C , which have already been dealt with. • Using the equivalence ∼ = , we see that the latter has the compact root 2 α + α , so that we can reduce to the case B alreadydealt with.This deals with all the cases where τ = id.Assume now that τ = id. If ǫ = +, then we need to prove the result for all β = α r, + . Since we only needto consider the case where the support of β is the whole of I , this means that we are in the case of either A × A or A with non-trivial involution. The theorem follows in this case by the concrete computationsin Theorems A.6 and A.11.Finally, assume that τ = id and ǫ = +. We take the following representative of ǫ within its equivalence class,listing in the final column the simple ν -compact roots for this preferred representative.Diagram Strong reduction Preferred representative Simple ν -compact roots A n +1 n n ′ α ˆ1 , . . . , α ˆ n , α ˆ n + α n ′ D n , p even ⌊ n − p ⌋ n − n p n − n α ,...,α p − ,α p + α p +1 ,...,α n − + α n − ,α ˆ n E α ˆ1 , α ˆ2 , α , α ˆ2 + α s one sees, the simple ν -compact roots are all supported by twisting data which have been treated already,except for the case . By Theorem A.6, we have γ = − m ( ̟ + ̟ )+ t̟ for m ∈ N and t ∈ R . Sinceˆ∆ + c = { α ˆ1 , α ˆ1 + α } and W + ν = { e, ˆ s , s ˆ s s , ˆ s s ˆ s s } , we easily compute for ω = a ( ̟ + ̟ )+ b̟ ∈ ( P + ) τ that, with z = q a +2 and z = q b +2 , X v ∈ W + ν d sgn( v ) q ρ − γ, ( ω + ρ ) − v − ( ω + ρ )) = (1 − z m +11 )(1 − ( z z ) m +1+2(1 − t ) ) . By (6.7) for w = e , we see that we must have2( ρ − γ, α ˆ1 + α ) = m + 1 + 2(1 − t ) > , proving the last remaining case. (cid:3) Elements in G ν,q \ G R ,q /U q In this section, we will give some results, complete or partial, concerning the elements x ∈ G ν,q \ G R ,q /U q and the associated Λ ν ( x ).7.1. Correspondence between G ν,q \ G R ,q /U q and G ν ′ ,q \ G R ,q /U q when ν and ν ′ are equivalent. Let ν, ν ′ ∈ H ug τ be equivalent ungauged twisting data, say ǫ ′ = g ( wǫ ) for w ∈ W ν and g ∈ H >τ . In Lemma 3.5, weconstructed an isomorphism χ f : U ν ′ q ( u ) → U νq ( u ) using an f ∈ H × τ such that (3.9) holds. We can transport χ f to a ∗ -isomorphism e χ f : O q ( Z reg ν ′ ) → O q ( Z reg ν ) , a ω f P ( τ ( ω )) T ω , x r x r . Since χ f moreover respected the U q ( u )-action, it follows that also O q ( Z ν ′ ) ∼ = → e χ f O q ( Z ν ) , Z ( O q ( Z ν ′ )) ∼ = → e χ f Z ( O q ( Z ν )) . We hence obtain a bijectionΞ f : G ν,q \\ G R ,q //U q → G ν ′ ,q \\ G R ,q //U q , x x ′ = x ◦ e χ f . Proposition 7.1.
The map Ξ f determines a bijection Ξ f : G ν,q \ G R ,q /U q → G ν ′ ,q \ G R ,q /U q . Proof.
It is sufficient to prove that if x ∈ G ν,q \ G R ,q /U q , then x ′ ∈ G ν ′ ,q \ G R ,q /U q .Pick λ x ∈ Λ ≫ ν ( x ). Further write w = uv with u ∈ W − ν and v ∈ W + ν . By Theorem 5.26, we have that also u · ǫ λ x ∈ Λ ν ( x ), and then π u · ǫ λ x ◦ e χ f is a highest weight representation of O q ( Z ν ′ ) with central character x ′ at highest weight λ x ′ with( λ x ′ ) P ( ω ) = f P ( ω )( u · ǫ λ x ) P ( ω ) = f P ( ω ) ǫ Q ( ω − u − ω ) q ( ω − u − ω, ρ ) λ P ( u − ω ) , ω ∈ P τ . However, the condition (3.9) on f implies that f P ( ω ) ǫ Q ( ω − w − ω ) = f P ( ω ) ǫ Q ( ω − u − ω ) ǫ Q ( u − ω − v − u − ω ) > , so by Lemma 2.28 also f P ( ω ) ǫ Q ( ω − u − ω ) > , ω ∈ P τ . Hence λ x ′ ∈ Λ ≫ ν ( x ′ ), and x ′ ∈ G ν ′ ,q \ G R ,q /U q . (cid:3) .2. Characters of O q ( Z ν ) . In [BK15b] the authors constructed universal solutions (= universal K -matrices)for the τ -modified reflection equation for U q ( g ), which by the philosophy in [KoSt09] can be considered asgiving characters on O q ( Z ν ) - see also [DCM18] where it was shown how these solutions can be slightlymodified to produce ∗ -characters. One then has the following general result. Proposition 7.2.
Let χ ∈ Char ∗ ( O q ( Z ν )) be a ∗ -character, and let x = x χ = χ | Z ( O q ( Z ν )) . Let π reg be theregular representation of O q ( U ) . Then the ∗ -representation ρ χν := ( χ ⊗ π reg ) ρ ν of O q ( Z ν ) on L q ( U ) is an x -representation of regular type.Proof. Let us first show that ρ χν ( a ρ ) is not the zero operator. Assume to the contrary that ρ χν ( a ρ ) = 0.It is well-known that O q ( U ) is a domain [Jos95, Lemma 9.1.9], so then all a ̟ with ̟ ∈ P + \ { } act aszero. Since ρ χν is an O q ( U )-comodule map, and the a ̟ generate O q ( Z ν ) as an U q ( u )-module, it followsthat ρ χν ( Z ̟ ( ξ, η )) = 0 for all ̟ = 0, i.e. the Haar integral is a character on O q ( Z ν ). However, from[DCM18, Lemma 2.26] it would then follow that the Haar integral is also a character on O q ( U ), which isabsurd.It follows that ρ χν contains a non-trivial subrepresentation of regular type. If λ ∈ H × τ is a highest weightappearing in the decomposition of this subrepresentation, write λ r = u r | λ r | . Then it follows that wehave produced an admissible (positive) highest weight for O q ( Z ν ′ ) where ǫ ′ r = u P ( α r ) ǫ r . Let x ′ be thecorresponding central character. We then have a ∗ -representation of O ∗ q ( O ν ′ x ′ ) on L q ( U ). Since however theHaar state of O q ( U ) ⊆ B ( L q ( U )) is given by a vector state, and since the Haar state of O q ( U ) restricts to theinvariant state on O ∗ q ( O ν ′ x ′ ), it follows that the above must extend to an embedding L ∞ q ( O ν ′ x ′ ) ⊆ B ( L q ( U )).From Theorem 6.1, we however know that the support projection of a ρ equals 1, so ρ χν must be regular. (cid:3) In general, determining if x χ ∈ G ν,q \ G R ,q /U q involves computing some spectral information of the operator ρ χν ( a ρ ), namely the existence of a positive eigenvalue. This can in principle be a difficult problem. We willshow however that this problem can be circumvened by a deformation argument, exploiting the fact that thesignature of a ρ in any L ∞ q ( O νx ) only depends on ν and not on q or x . We will only show how this phenomenonworks for the case of symmetric twisting data, as they are the ones that have been treated explicitly in[DCM18].Since we will now need to work with variable q , we will index all notations with either q as subscript orsuperscript. Recall then from [NT11, Lemma 1.1] that, for each ̟ ∈ P + , one can create ∗ -representations π q̟ of U q ( u ) for 0 < q ≤ V ̟ such that each π q̟ is a highestweight representation at weight ̟ , such that the weight space decomposition of V ̟ is independent of q , andsuch that the π q̟ ( E r ) depend continuously on q ∈ (0 ,
1] for each r ∈ I . In the following, we will fix for each ̟ ∈ P + such a continuously varying family of representations on the fixed Hilbert space V ̟ . One then hasthe following theorem. Theorem 7.3. [NT11, Theorem 1.2]
There exists a unique structure of continuous field of C ∗ -algebras on (0 , with fibers given by the C q ( U ) , such that the U q̟ ( ξ, η ) ∈ O q ( U ) form continuous sections. Note that for any f ∈ O ( U ), one has a unique corresponding f q ∈ O q ( U ) using that the U ̟ ( e i , e j ) form abasis of O q ( U ). The above theorem then states that the norm q
7→ k f q k varies continuously.To proceed, note that also O q ( Z ν ) can be meaningfully defined as an algebra for q = 1. Indeed, by [DCM18,Definition 2.24] one can view O q ( Z ν ) as a copy of O ( U ) with the new product f ∗ q g = ( g (2) ⊗ f (1) , Ω ǫ,q ) r q ( f (2) , g (3) ) f (3) · q g (4) r q ( f (4) , τ ( S q ( g (1) ))) , where · q denotes the product of O q ( U ) transported to O ( U ), where r q ( f, g ) = ( f ⊗ g ) R q and where Ω ǫ,q actson each V ̟ ⊗ V ̟ ′ by Ω ǫ,q ◦ ι = ι ◦ E , ∀ ι ∈ Hom U q ( u ) ( V ̟ ′′ , V ̟ ⊗ V ̟ ′ ) , Also the ‘flag case’ where ǫ = ǫ was dealt with in [DCM18], but the case of a general ungauged twisting datum, interpolatingbetween the flag case and the symmetric case, was not. ith E as in (3.17). Since R q ( ξ ⊗ η ) is continuous in q , with limit ξ ⊗ η at q = 1, this product then variescontinuously in q for the weak topology on O ( U ) induced by its duality with Q ̟ End( V ̟ ) (to see that theΩ q,ǫ also vary continuously, use that ξ ̟ ⊗ η w ̟ ′ is cyclic, for η w ̟ ′ a unit lowest weight vector in V ̟ ′ ).Hence O ( Z ν ) = O ( Z ν ) may be identfied with O ( U ) with the product f ∗ g = ( g (1) ⊗ f (1) , Ω ǫ ) f (2) g (2) . (7.1)We then denote by e f q the copy of f ∈ O ( U ) inside O q ( Z ν ).Our key lemma will be the following. Lemma 7.4.
Assume that χ q : O q ( Z ν ) → C are a family of ∗ -characters such that q χ q ( e f q ) is continuouson (0 , for each f ∈ O ( U ) . Assume that that there exists v ∈ U such that χ ( Z α ( τ ( v ) ξ α , vξ α )) > for all α ∈ Q + . Then Λ ≫ ν ( x q ) = ∅ for all < q < , where x q = ( χ q ) | Z ( O q ( Z ν )) .Proof. The hypotheses imply that the functions u χ ( Z α r ( τ ( u ) ξ α r , uξ α r )) have a point of R | I | > in theirjoint spectrum. By continuity, it follows that also the operators( χ q ⊗ π reg ,q ) ρ ν,q ( a α r ,q ) ∈ O q ( U ) ⊆ C q ( U )have a point of R | I | > in their joint spectrum for q close to 1. However, by Theorem 6.1 the joint spectrum of the a α r must hit the same connected components of ( R \{ } ) | I τ | × C | I ∗ | for all q . Hence the ( χ q ⊗ π reg ,q ) ρ ν,q ( a α r ,q )must have a point of R | I | > in their joint spectrum for all < q < O q ( Z reg ν ) in which all a α r ,q have positivespectrum. Since the commutation relations of O q ( Z reg ν ) only involve the a α r ,q non-trivially, it follows easilythat one can then construct a highest weight representation of O q ( Z reg ν ) at λ ∈ Λ ≫ ν ( x q ). (cid:3) We are now ready to prove the following theorem.
Theorem 7.5.
Let ν = ( τ, ǫ ) be a symmetric twisting datum. Then G ν,q \ G R ,q /U q = ∅ .Proof. We follow the terminology and notation of [DCM18].We may assume that ν is reduced. From the discussion in [DCM18, Appendix B] one sees that one canfind a concrete Satake diagram ( X, τ τ ) such that ǫ is an ( X, τ τ )-admissible sign function. We let θ be theassociated Satake involution.Let χ q be the character on O q ( Z ν ) given by χ q ( Z q̟ ( ξ, η )) = h ξ, K q η i , with K q determined by [DCM18, (4.31)]. As in the proof of [DCM18, Theorem 4.54], one has that the K q in B ( V ̟ ) depend continuously on q , with K = E e ǫ − m m X e z − ττ = E e ǫ − e zm m X , where for the second equality we have used [DCM18, Lemma 4.26] (note that we have interchanged thenotations τ and τ τ with respect to the convention in [DCM18]).Let now J ̟ : V ̟ → V τ ( ̟ ) be the unique (unitary) intertwiner between π ̟ ◦ τ and π τ ( ̟ ) sending ξ ̟ to ξ τ ( ̟ ) .Put v ̟ = J ̟ E e ǫ − e zm m X . From the definition of θ and the centrality of E e ǫ − , it follows that v ̟ π ̟ ( u ) = π τ ( ̟ ) ( θ ( u )) v ̟ , u ∈ U. Since by construction θ and ν are inner equivalent, we can find k ∈ U such that ν ( u ) = k − θ ( kuk − ) k, u ∈ U. In particular, since π τ ( ̟ ) ( ν ( u )) = Ad( J ̟ E )( π ̟ ( u )) , u ∈ U, t follows that E − J − ̟ k − J ̟ E e ǫ − e zm m X k = e ǫ − τ ( k ) − e zm m X k. is central. Hence for all α ∈ Q + , one has χ ( Z α ( τ ( k ) ξ α , kξ α )) = h τ ( k ) ξ α , K kξ α i = h ξ α , τ ( k ) − E e ǫ − e zm m X kξ α i = h ξ α , τ ( k ) − E e ǫ − e zm m X kξ α i = h ξ α , E e ǫ − τ ( k ) − e zm m X kξ α i = h ξ α , E ξ α i = k ξ α k > . From Lemma 7.4 it now follows that Λ ≫ ν ( x χ q ) = ∅ for all 0 < q < (cid:3) Remark 7.6.
Given the ∗ -character χ q as above with associated central character x q = x χ q , it would beinteresting to know explicitly the value of the associated λ x q ∈ Λ ≫ ν,q ( x q ) . It is not clear to us at the momenthow to determine this value without going into the computation of eigenvalues of the corresponding operator ρ χ q ν ( a ρ,q ) . Symmetric spaces of the form U × U/ diag( U ) . Let as before u be a compact semisimple Lie algebrawith Dynkin diagram Γ built on the index set I . Consider the direct sum u ⊕ u with its Dynkin diagram Γ × Γbuilt on the set I × I , and consider the associated ungauged twisting datum ν = ( τ, +) where τ ( r, s ) = ( s, r ).We can then interpret O q ( Z ν ) = O q ( G R \\ G R × G R ) . The associated ∗ -algebra O q ( Z reg ν ) is generated by elements x ( r, , x (0 ,r ) and a ( ω,χ ) where the x ( r, and x (0 ,s ) satisfy the usual quantum Serre relations, where any of the x ( r, and x (0 ,s ) commute and where a ( ω + ω ′ ,χ + χ ′ ) = a ( ω,χ ) a ( ω ′ ,χ ′ ) , a ∗ ( ω,χ ) = a ( χ,ω ) ,a ( ω,χ ) x ( r, = q − ( ω + χ,α r ) x r a ω , a ( ω,χ ) x (0 ,r ) = q ( ω + χ,α r ) x (0 ,r ) a ( ω,χ ) and x ( r, x ∗ ( s, − q − ( α r ,α s ) x ∗ ( s, x ( r, = − δ r,s q r − q − r , x (0 ,r ) x ∗ (0 ,s ) − q − ( α r ,α s ) x ∗ (0 ,s ) x (0 ,r ) = − δ r,s q r − q − r ,x ( r, x ∗ (0 ,s ) − x ∗ (0 ,s ) x ( r, = δ r,s q r a (0 , − α r ) q r − q − r . On the other hand, consider for u the twisting datum µ = (id ,
0) on I with associated ∗ -algebra O q ( Z reg µ ).Then we see that we have an embedding of ∗ -algebras O q ( Z reg µ ) → O q ( Z reg ν ) , a ω a ( ω,ω ) , x r x ( r, . Proposition 7.7.
We have λ = q ρ ∈ Λ ≫ ν .Proof. Let M λ be the Verma module of O q ( Z reg ν ) at λ with highest weight vector ξ λ , and let V λ be itsirreducible quotient. An easy computation shows that all the ( x ∗ ( r, + x ∗ (0 ,r ) ) ξ λ are also highest weightvectors in M λ , and hence x ∗ (0 ,r ) ξ λ = − x ∗ ( r, ξ λ in V λ . It follows that the vector ξ λ ∈ V λ is already cyclic for the ∗ -algebra generated by the x ( r, andthe a ( ω,ω ) . Since the latter is an isomorphic copy of the ∗ -algebra O q ( Z reg µ ), it follows by uniqueness thatthe O q ( Z reg ν )-invariant inner product on V λ must coincide with the one coming from its representation of O q ( Z reg µ ). The latter is however positive-definite by [DeC13, Theorem 2.7]. (cid:3) et x = x λ ∈ G R ,q \ G R ,q × G R ,q /U q be the associated central character of O q ( Z ν ). We will show that O q ( O νx ) ∼ = O q ( U ) are isomorphic as ∗ -algebras, and identify the corresponding coaction of O q ( U ) ⊗ O q ( U ). As the computations are somewhattedious, and as the results are probably known to specialists already in one form or another, we will notspecify all details.We start with recalling the following results from [ST09], to which we refer for the ideas behind the compu-tations. Up to a regauging (and change of conventions), the element Y in the lemma below corresponds tothe half-twist element of [ST09]. We follow the same conventions as [DCM18]. Lemma 7.8.
Let T w ∈ Q ̟ End( V ̟ ) be the braid group operator corresponding to the longest element w in the Weyl group, and let C ∈ Q ̟ End( V ̟ ) be the operator given by Cξ = q (wt( ξ ) , wt( ξ ))+(wt( ξ ) ,ρ ) Let S = e πiρ ∨ , where ρ ∨ ( α r ) = 1 for all r ∈ I , and define Y = S CT w . Then the following hold: (1) R = ( Y ⊗ Y )∆( Y ) − . (2) Y = S v − , where v ∈ Q ̟ End( V ̟ ) is the standard central ribbon element satisfying v ̟ = q − ( ̟,̟ +2 ρ ) , R R = ∆( v )( v − ⊗ v − ) , (3) Y ∗ = R ( Y ) = Y S , where R is the unitary antipode. (4) Ad( Y ) is an anti-comultiplicative ∗ -algebra automorphism with Ad( Y ) K ω = K − τ ( ω ) , Ad( Y ) E r = − q r F τ ( r ) , Ad( Y ) F r = − q − r E τ ( r ) , where τ is the involution on Γ determined by the − Ad( w ) . Proposition 7.9.
The map χ = χ K : O q ( Z ν ) → C , Z ( ξ ⊗ η, ξ ′ ⊗ η ′ )
7→ h ξ ⊗ η, K ( ξ ′ ⊗ η ′ ) i where K = ( Y − ⊗ op ( Y ) − ( Y S v − ⊗ v − ) , defines a ∗ -character on O q ( Z ν ) .Proof. Using the above properties of Y , this follows by an easy, albeit tedious computation following thedescription of the ∗ -compatible K -matrix in [DCM18], after noting that the quasi- K -matrix of [BK15b]coincides with (id ⊗ Ad( K ρ Y )) e R , with e R ∈ U q ( n ) ˆ ⊗ U q ( n − ) the quasi- R -matrix. (cid:3) From [DCM18, Theorem 4.40], it follows that the associated ∗ -homomorphism ρ χν : O q ( Z ν ) → O q ( U ) ⊗O q ( U )has as its range the right coideal dual to the left coideal written as C = R ( e B ) in [DCM18], which in thecase at hand is easily seen to be given as C = (Ad( Y ) ⊗ id)∆ op ( U q ( u )) ⊆ U q ( u ) ⊗ U q ( u ) . The dual coideal is then given by O q (diag( U ) \ ( U × U )) = { U τ ( ̟ ) ⊗ ̟ ( X i Y K ρ e i ⊗ e i , ξ ⊗ η ) | ̟ ∈ P + } , here we identify V τ ( ̟ ) ∼ = V ̟ with the usual conjugate Hilbert space structure and U q ( u )-representationgiven by xξ = R ( x ) ∗ ξ . Restricting the above ∗ -subalgebra of O q ( U ) ⊗ O q ( U ) to functionals on C ⊗ U q ( u ),we obtain an isomorphism of ∗ -algebras O q ( U ) ∼ = O q (diag( U ) \ ( U × U )) U ̟ ( ξ, η ) U τ ( ̟ ) ⊗ ̟ ( X i Y K ρ e i ⊗ e i , ( Y ∗ ) − K − ρ ξ ⊗ η ) . Under this isomorphism, the right coaction of O q ( U ) ⊗ O q ( U ) is transported to the coaction e ∆ diag ( f ) = f (2) ⊗ S ( f (1) ) ◦ Ad( Y ) ⊗ f (3) . Let now Z be the span of the elements 1 ⊗ Z ̟ ( ξ, η ) in O q ( Z ν ) corresponding to the component 1 ⊗ O q ( U ),and consider similarly the span Z of the elements Z ̟ ( ξ, η ) ⊗
1. From for example (7.1) one sees that Z Z = O q ( Z ν ) = Z Z . Moreover, it is easily seen that ρ χν (1 ⊗ Z ̟ ( ξ, η )) = U ̟ ( Y ξ, η ) , ρ χν ( Z ̟ ( ξ, η ) ⊗
1) = U ̟ ( Y η, ξ ) ∗ , where the second formula easily follows from the first using that ρ χν is a ∗ -homomorphism.We are now ready to prove the following theorem. Theorem 7.10.
The O q ( U ) ⊗ O q ( U ) equivariant ∗ -homomorphism ρ χν factors through π q ρ , inducing a ∗ -isomorphism O q ( O νx qρ ) ∼ = O q ( U ) . Proof.
We see from the above that ρ χν ( a (0 ,̟ ) ) = q − ( ̟,̟ ) b ̟ , b ̟ = U ̟ ( Y ξ ̟ , ξ ̟ ) . Localizing at b ̟ , b ∗ ̟ , we obtain a ∗ -algebra O loc q ( U ) that is the image of O q ( Z reg ν ). In fact, O loc q ( U ) isisomorphic by ρ χν to the ∗ -algebra e O q ( Z µ ) generated by the a ( ̟,̟ ′ ) , the x ( r, and the x ∗ ( r, , see e.g. [DC18].It now follows immediately from the argument in the construction of π q ρ that the latter ∗ -representationmust factor over O loc q ( U ). (cid:3) Remark 7.11.
Once this identification is made, we note that the specific instance of Theorem 6.5 in thiscase was obtained in [RY01] . Quantization of the real Grassmannian spaces SU ( N ) /S ( U ( m ) × U ( n )) . To end, we return tothe example we considered in the introduction. Put u = su ( N ), and consider the Hermitian symmetric pair s ( u ( m ) ⊕ u ( n )) ⊆ su ( N ). It can be described by the Vogan diagram (id , ǫ ) where ǫ r = ( − δ r,m . m N − In this case, the set Λ ≫ ν was already determined in [DeC13]. Proposition 7.12. [DeC13, Theorem 2.7]
Let ν = (id , ǫ ) with ǫ r = ( − δ r,m . Then λ ∈ Λ ≫ ν if and only if λ = q − γ with γ = P N − r =1 k r ̟ r for k r ∈ N when r = m and k m ∈ R . Remark 7.13.
There is a missing assumption in the statement of [DeC13, Theorem 2.7] , in that it onlyapplies to hermitian symmetric spaces. Indeed, the theorem relies crucially on [JT02, Proposition 5.13] ,which needs the property that the distinguished simple root appears only with single multiplicity in all otherroots, a property satisfied precisely in the hermitian symmetric case. ote now that in this case W ν = W = S N , while W + ν = S m × S n . The decomposition W = W − × W + isthen the standard one where w = w − w + if w − is the element of minimal length in the W + -coset class of w .Thus W − consists of those w ∈ S N with w ( i ) ≤ w ( j ) for all 1 ≤ i ≤ j ≤ m and all m + 1 ≤ i ≤ j ≤ N . Weobtain then the following theorem. Theorem 7.14.
Let ν = (id , ǫ ) with ǫ r = ( − δ r,m , and let λ ∈ Λ ≫ ν be given as in Proposition 7.12. Let x = x λ be the associated character. Then L ∞ q ( O νx ) ∼ = ⊕ w ∈ W − B ( H w · ǫ λ ) . This gives a quantum analogue, in the measure-theoretic setting, of the decomposition in (0.4). Of course,this measure-theoretic setting does not see the lower-dimensional cells, for which finer C ∗ -algebraic techniquesneed to be used. A full understanding of C q ( O νx ) is however out of reach at the moment. Remark 7.15.
The ∗ -algebra O q ( Z reg ν ) is in this case just a copy, up to taking square roots of the Cartanpart, of U q ( sl ( N, C )) with a peculiar ∗ -structure. This means that most of the algebraic machinery to study U q ( sl ( N, C )) can also be used to study O q ( Z reg ν ) . In particular, one can use the computations of [UTS90] tocompute explicit Gelfand-Zeitlin bases for Verma modules of O q ( Z reg ν ) for any ungauged twisting data. It isthen not hard to deduce from this the precise form of Λ ≫ ν . Appendix . Computations in rank 2
In this appendix we classify the highest weight ∗ -representations of O q ( Z reg ν ) for the case of a rank 2 Liealgebra u with ν = ( τ, ǫ ), ǫ ungauged and τ non-trivial. We write the associated simple roots α , α so that τ (1) = 2. We further write ǫ = ǫ = ǫ ∈ R .A highest weight ∗ -representation is completely determined by the scalar λ = λ P ( ̟ ) ∈ C × . Since we knowalready that the class of λ ’s appearing this way is stable under multiplication with the circle group H g , cf.Lemma 5.12, we can restrict to consdering λ >
0. On the level of O q ( Z reg ν ) this means we may impose theextra relation a ̟ = a ̟ = a ∗ ̟ . We denote the resulting quotient by O q ( e Z reg ν ) and write a = a ̟ .A.1. Twisted conjugacy classe of type A × A with nontrivial automorphism. Let u = su (2) ⊕ su (2).The associated ∗ -algebra O q ( e Z reg ν ) is generated by elements a ± , x , x with a self-adjoint and ax = q − x a, x x ∗ = q − x ∗ x + ( q − − q ) − , (A.1) ax = q − x a, x x ∗ = q − x ∗ x + ( q − − q ) − , (A.2) x x ∗ = x ∗ x + ǫqq − q − a − , (A.3)and x x = x x . (A.4) Lemma A.1.
The following relations hold for k, n ∈ N : x ( x ∗ ) n = q − n ( x ∗ ) n x + q − n +1 q − n − q n ( q − − q ) ( x ∗ ) n − ,x ( x ∗ ) k = ( x ∗ ) k x − ǫq − k +2 q − k − q k ( q − − q ) ( x ∗ ) k − a − ,x ( x ∗ ) n = ( x ∗ ) n x − ǫq − n +2 q − n − q n ( q − − q ) ( x ∗ ) n − a − ,x ( x ∗ ) k = q − k ( x ∗ ) k x + q − k +1 q − k − q k ( q − − q ) ( x ∗ ) k − . Proof.
By straightforward induction. (cid:3) s in the proof of Lemma 5.8, let M λ be the Verma module at weight λ with highest weight vector ξ λ . Put ξ kn = ( x ∗ ) k ( x ∗ ) n ξ λ . Also the following lemma follows by straightforward induction.
Lemma A.2.
The generators act as follows on the ξ kn : aξ kn = λq k + n ξ kn , x ∗ ξ kn = ξ k,n +1 , x ∗ ξ kn = ξ k +1 ,n ,x ξ kn = q − n +1 q − n − q n ( q − − q ) ξ k,n − − ǫλ − q − k − n +2 q − k − q k ( q − − q ) ξ k − ,n ,x ξ kn = − ǫλ − q − k − n +2 q − n − q n ( q − − q ) ξ k,n − + q − k +1 q − k − q k ( q − − q ) ξ k − ,n . Lemma A.3.
For each t ∈ N , the space of highest weight vectors for x at weight λq t is one-dimensional,and given by C η ( t )0 where η ( t )0 = P k + n = t c n ξ k,n with c = 1 and c n = ǫλ − q − t +2 q − t + n − − q t − n +1 q − n − q n c n − , ≤ n ≤ t. Proof.
The recursive formula follows at once from Lemma A.2. (cid:3)
Put now η ( t ) n = ( x ∗ ) n η ( t )0 and consider the unique invariant hermitian form on M λ with h ξ λ , ξ λ i = 1. Lemma A.4.
The η ( t ) n are mutually orthogonal and form a basis of M λ .Proof. Orthogonality follows straightforwardly from invariance of the scalar product and weight considera-tions. Also the linear independence of the η ( t ) n follows directly from weight considerations. They must henceform a basis since the dimension of the weight spaces of their span matches that of M λ itself. (cid:3) Lemma A.5.
The following formulas hold: x η ( t )0 = q − t +1 q − t − q t ( q − − q ) (1 − | ǫ | λ − q − t +4 ) η ( t − ,x ∗ η ( t )0 = η ( t +1)0 − ǫλ − q − t +1 η ( t )1 . Proof.
We note that x η ( t )0 must again be a highest weight vector for x , hence a scalar multiple of η ( t − .The precise coefficient follows from comparing the coefficient of ξ t − , in both expressions.By weight considerations x ∗ η ( t )0 must be a linear combination of η ( t +1)0 and an element ζ t in the span of theelements η ( t ′ ) n for t ′ ≤ t . Since x ζ t = x x ∗ η ( t )0 = − ǫqq − − q a − η ( t )0 = − ǫλ − q − t +1 q − − q η ( t )0 , it follows again from weight considerations that we must have ζ t = − ǫλ − q − t +1 η ( t )1 . The coefficient for η ( t +1)0 is found by comparing coefficients of ξ t +1 , . (cid:3) Theorem A.6.
There exists a unitarizable highest weight module at weight λ if and only if ǫ = 0 or, in case ǫ = 0 , λ is of the form λ = | ǫ | / q − n for n ∈ N . roof. As in the proof of Lemma 5.8, a unitarizable highest weight module at weight λ exists if and only ifthe invariant hermitian form on M λ is positive semi-definite. This will be the case if and only if c t,n = h η ( t ) n , η ( t ) n i ≥ t, n . Now since the x , x ∗ satisfy the Heisenberg relations, it is easily seen that we only need to checkthat c t = c t, ≥ t . However, from Lemma A.5 we see that c t = h η ( t )0 , x ∗ η ( t − i = h x η ( t )0 , η ( t − i = q − t +1 q − t − q t ( q − − q ) (1 − | ǫ | λ − q − t +4 ) c t − . Thus c t ≥ t if and only if λ is as indicated in the theorem. (cid:3) A.2.
Twisted conjugacy classe of type A with nontrivial automorphism. Consider u = su (3) withtwisting datum ν = ( τ, ǫ ) where τ = id. We use the notation as explained at the beginning of the Appendix,and consider the ∗ -algebra O q ( e Z reg ν ). It is generated by elements a ± , x , x with a self-adjoint and ax = q − x a, x x ∗ = q − x ∗ x + ( q − − q ) − , (A.5) ax = q − x a, x x ∗ = q − x ∗ x + ( q − − q ) − , (A.6) x x ∗ = qx ∗ x + ǫq / q − q − a − , (A.7)and such that, with x = x x − qx x , x x = q − x x , (A.8) x x = qx x . (A.9) Lemma A.7.
The following further relations hold in O q ( e Z reg ν ) : ax = q − x a, ax ∗ = q x ∗ a, x x ∗ = q − x ∗ x , (A.10) x x ∗ = q − x ∗ x + qx ∗ − q / ǫx ∗ a − , (A.11) and x x ∗ = q − x ∗ x + ( q − − q ) x ∗ x + q ( q − − q ) − + | ǫ | q q − − q a − . (A.12) Proof.
By direct computation. (cid:3)
Definition A.8.
We define B to be the ∗ -algebra generated by elements a ± , x , x with a selfadjoint,satisfying the relations (A.5) , (A.8) , (A.10) , (A.12) . Clearly, we can define a notion of highest weight module and Verma module for B . Lemma A.9.
For any λ > , the Verma module for B at λ is unitarizable.Proof. Let ξ m,n = ( x ∗ ) m ( x ∗ ) n ξ λ . Then straightforward computations show that x ∗ ξ m,n = q m ξ m,n +1 , x ∗ ξ m,n = ξ m +1 ,n , x ξ m,n = q − m − n +1 q − n − q n ( q − − q ) ξ m,n − ,x ξ m,n = q − m − n +2 q − m − q m ( q − − q ) (1 + | ǫ | λ − q − m +3 ) ξ m − ,n . It follows easily from this that the ξ m,n are mutually orthogonal with respect to the invariant sesqui-linearform on the Verma module, and that the ξ m,n have positive norm squared. (cid:3) Note that the image of B in its Verma module representation must then also be the image in the Vermamodule at corresponding weight of O q ( e Z reg ν ) through its natural embedding. emma A.10. Let λ > , and let M λ be the Verma module for O q ( e Z reg ν ) at λ . Put ξ kmn = ( x ∗ ) k ( x ∗ ) m ( x ∗ ) n ξ λ . Then x ξ kmn = q k − m − n +1 q − n − q n ( q − − q ) ξ k,m,n − − ǫλ − q − m − n + q − k − q k ( q − − q ) ξ k − ,m,n ,x ∗ ξ kmn = q − k + m ξ k,m,n +1 − q − q − k − q k q − − q ξ k − ,m +1 ,n ,x ξ kmn = q − k +1 q − k − q k ( q − − q ) ξ k − ,m,n + q − k +1 q − m − q m q − − q ξ k,m − ,n +1 − ǫλ − q − k − m + q − n − q n ( q − − q ) ξ k,m,n − − ǫλ − q − k − m − n + q − m − q m q − − q ξ k +1 ,m − ,n ,x ∗ ξ kmn = ξ k +1 ,m,n ,x ξ kmn = q − k − m − n +2 ( q − m − q m )(1 + | ǫ | λ − q − k − m +3 )( q − − q ) ξ k,m − ,n + q − m − n ( q − k − q k )( q − n − q n )(1 + | ǫ | λ − q − k − m +3 )( q − − q ) ξ k − ,m,n − − ǫλ − q − k − m − n + ( q − k − q k )( q − m − q m )( q − − q ) ξ k − ,m − ,n +1 − ǫλ − q − k − m − n + ( q − k − q k )( q − k +1 − q k − )( q − − q ) ξ k − ,m,n x ∗ ξ kmn = q k ξ k,m +1 ,n . Proof.
This follows from some tedious but straightforward computations using the defining relations. (cid:3)
Theorem A.11. If ǫ = 0 , then there exists a unitarizable highest weight module for all weights λ > . If ǫ = 0 , this is the case if and only if λ = | ǫ | q − n for some n ∈ N .Proof. Let M λ be the Verma module at weight λ . We claim that there exists, up to a scalar multiple, aunique non-zero vector η ( t )00 at weight q t λ which is highest weight for B . Indeed, if η t = X k +2 m + n = t c k,m ξ kmn is such a vector, where again ξ kmn = ( x ∗ ) k ( x ∗ ) m ( x ∗ ) n ξ λ , it follows easily from the commutation relationsin the previous lemma that the c k,m must, by the vanishing condition for x , satisfy the relations c k,m = ǫλ − q − k − m + q − k − − q k +1 q − t + k +2 m − q t − k − m c k +1 ,m , ≤ k + 2 m < t. (A.13)Writing out the vanishing under x and using (A.13), we also obtain c k,m +1 = −| ǫ | λ − q − t +3 ( q − k − − q k +1 )( q − t + k +2 m +1 − q t − k − m − )( q − − q )( q − m − − q m +1 )(1 + | ǫ | λ − q − t +2 m +5 ) c k +1 ,m , ≤ k + 2 m < t − . From this, it is easily seen that any value of c t, determines a unique solution. In the following, we fix asolution η ( t )00 with c t, = 1.Let now V ( t ) be the B -module spanned by η ( t )00 , and write η ( t ) mn = ( x ∗ ) m ( x ∗ ) n η ( t )00 . Then the invarianthermitian form on M λ must restrict to a scalar multiple, possibly negative or zero, of the unique normalizedpositive B -invariant inner product on V ( t ) . Since the V ( t ) have different highest weights under B , it followsthat the η ( t ) mn must be mutually orthogonal for different indices. It is then sufficient to check that h η ( t )00 , η ( t )00 i ≥ , ∀ t ∈ N . ow the vector x η ( t )00 must again be a highest weight vector for B , if it is non-zero. Calculating the coefficientat ξ t − , , , we find by another tedious computation that x η ( t )00 = q − t +1 ( q − t − q t )(1 + | ǫ | λ − q − t +4 )( q − − q ) (1 + | ǫ | λ − q − t +5 ) (1 − | ǫ | λ − q − t +4 ) η ( t − . On the other hand, from weight considerations it follows that x ∗ η ( t )00 = η ( t +1)00 + ξ t , ξ t ∈ ⊕ tk =0 V ( k ) . Hence h η ( t )00 , η ( t )00 i = h η ( t )00 , x ∗ η ( t − i = h x η ( t )00 , η ( t − i = q − t +1 ( q − t − q t )(1 + | ǫ | λ − q − t +4 )( q − − q ) (1 + | ǫ | λ − q − t +5 ) (1 − | ǫ | λ − q − t +4 ) h η ( t − , η ( t − i , and it follows that h− , −i is positive semi-definite if and only if λ is of the form prescribed in the statementof the theorem. (cid:3) References [BK15b] M. Balagovi´c and S. Kolb, Universal K-matrix for quantum symmetric pairs,
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Vrije Universiteit Brussel, Vakgroep Wiskunde en Data Science
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