Twisted Hochschild homology of quantum flag manifolds and K\"ahler forms

We study the twisted Hochschild homology of quantum flag manifolds, the twist being the modular automorphism of the Haar state. We prove that every quantum flag manifold admits a non-trivial class in degree two, with an explicit representative defined in terms of a certain projection. The corresponding classical two-form, via the Hochschild-Kostant-Rosenberg theorem, is identified with a K\"ahler form on the flag manifold.

An important motivation for studying the twisted Hochschild homology of C q [U/K S ] in degree two is given by Kähler forms. Indeed, classically the flag manifolds U/K S are Kähler and hence admit these two-forms with special properties. The study of analogues of Kähler forms in the quantum setting is important from the point of view of non-commutative complex and Kähler geometry, see for instance [BeSm13,ÓBu16,ÓBu17].
We now come to the main results of this paper. Our first result shows the non-triviality of twisted Hochschild homology in degree two for any quantum flag manifold.
Theorem. Given any quantum flag manifold C q [U/K S ], there exists a non-trivial class [C(P)] ∈ HH θ 2 (C q [U/K S ]) represented by an explicit element C(P) ∈ C q [U/K S ] ⊗3 . The explicit form involves a certain projection P with entries in C q [U/K S ]. Next, the representative C(P) admits an appropriately defined classical limit for q → 1. Under the Hochschild-Kostant-Rosenberg theorem, this classical element corresponds to a two-form on U/K S . Our second result shows that this is a Kähler form.
Theorem. Let ω P ∈ Ω 2 (U/K S ) be the form corresponding to [C(P)] under the classical limit.
Then ω P is a Kähler form on U/K S .
We will actually construct the Kähler form ω P entirely in the classical setting, in a way which is suitable for comparison with the quantum case.
Finally let us mention the connection with another common approach to differential forms on quantum spaces, due to Woronowicz [Wor89]. In this approach, given an algebra with an action of a compact quantum group, we introduce the structure of a differential calculus on the given algebra, together with various requirements about the action of the compact quantum group. In general we have many inequivalent choices of differential calculi, but the situation is much better for the quantum irreducible flag manifolds C q [U/K S ]. In this case it turns out that there is a unique analogue of the de Rham complex, which enjoys essentially all the classical properties, as shown by Heckenberger and Kolb in [HeKo06]. Within this setting, there is also a notion of Kähler forms introduced in [ÓBu17]. Quantum irreducible flag manifolds were shown to admit Kähler forms in this sense in [Mat19]. Finally we will show that these can also be identified with the forms ω P in the classical limit.
The structure of the paper is as follows. In Section 1 we recall various preliminaries on Lie algebras and quantum groups and fix some notations. In Section 2 we summarize some results from [Mat20] on twisted Hochschild homology of quantum flag manifolds. In Section 3 we prove that all quantum flag manifolds admit non-trivial classes in degree two. In Section 4 we introduce some Kähler forms on classical flag manifolds, from a point of view suitable for comparison with the quantum case. Finally, in Section 5 we compare the classical limit of the quantum classes with the Kähler forms obtained before.

Notations and preliminaries
In this section we fix most of our notations and recall various preliminary notions. These include Lie algebras and parabolic subalgebras on the classical side, and quantized enveloping algebras and coordinate rings on the quantum side. In particular we try to adopt notations that illustrate the link between the two sides as much as possible.
1.1. Lie algebras and Lie groups. Let g be a complex simple Lie algebra with fixed Cartan subalgebra h. We denote by ∆(g) the roots of g with respect to h, and by ∆ ± (g) a choice of positive/negative roots. We write {α i : i ∈ I} for the simple roots and {ω i : i ∈ I} for the fundamental weights. We denote by (·, ·) the bilinear form on h * obtained from the Killing form, rescaled in such a way that (α, α) = 2 for the short roots.
We denote by u the compact real form of g. It is given by Ri(e α + f α ).
Here and in the following i ∈ C will denote the imaginary unit. Corresponding to the Lie algebras g and u, we have the connected, simply-connected Lie groups G and U. We will also denote by T the maximal torus of U.
1.2. Parabolic subalgebras and flag manifolds. Let S be a subset of the simple roots, that S ⊂ I. Corresponding to this choice, we define the roots In terms of these we define the Levi factor corresponding to S by It is a reductive Lie subalgebra containing the Cartan subalgebra. We also define ∆(n ± S ) := ∆ ± (g)\∆ ± (l S ). In other words, the set ∆(n + S ) contains the positive roots which are not in ∆(l S ), and similarly for ∆(n − S ). In terms of these roots we define n ± S := α∈∆(n ± S ) g α .
These are also Lie subalgebras of g, which we call the positive and negative nilradical corresponding to S. They are l S -modules with respect to the adjoint action. In terms of the previously defined subalgebras we have the decomposition g = n + S ⊕ l S ⊕ n − S . We call p S := l S ⊕ n + S the standard parabolic subalgebra corresponding to S. Here standard parabolic means that p S is a subalgebra containing the standard Borel subalgebra In the case S = ∅ we have p S = b. We also note that g/p S ∼ = n − S as l S -modules. In terms of the compact real form u, we will consider k S := l S ∩ u, m S := (n + S ⊕ n − S ) ∩ u. We have that k S is a Lie subalgebra of u, but this is not the case for m S .
Corresponding to these Lie subalgebras, we have the (generalized) flag manifolds. These are homogeneous spaces of the form G/P S , where P S is the parabolic subgroup with Lie algebra p S . This definition shows that they are complex manifolds. It is also possible to give a realization within U, the compact real form of G, by the diffeomorphism G/P S ∼ = U/K S , K S := P S ∩ U.
The realization U/K S makes it clear that they are compact manifolds. Finally there is also a projective realization of the flag manifolds, which we will recall later.
Some notable subclasses are the following. For S = ∅ we have the homogeneous spaces G/B ∼ = U/T , called the full flag manifolds. At the other extreme we have the irreducible flag manifolds, corresponding to the case S = I\{t} with α t having multiplicity 1 in the highest root of g. These include the Grassmannians, for instance.
1.3. Quantized enveloping algebras. We will use the conventions of [KlSc97], which will be our main reference for this part. Given 0 < q < 1, the quantized enveloping algebra U q (g) is a Hopf algebra deformation of the enveloping algebra U(g) defined as follows. It has , with r = rank(g), and relations as in [KlSc97, Section 6.1.2]. In particular, the comultiplication, antipode and counit are given by Given λ = r i=1 n i α i we will write K λ = K n 1 1 · · · K nr r . Let ρ be the half-sum of the positive roots of g. Then we have S 2 (X) = K 2ρ XK −1 2ρ for any X ∈ U q (g). We will also consider a * -structure on U q (g), which in the classical case corresponds to the compact real form u. We can take for instance i . The precise formulae are not so important here, as any equivalent * -structure will work equally well for our purposes. We will write U q (u) := (U q (g), * ) when we consider U q (g) endowed with the * -structure corresponding to the compact real form.
We will also consider a quantum analogue of the Levi factor l S , following [StDi99]. The quantized Levi factor U q (l S ) is defined by Here · denotes the subalgebra generated by the given elements in U q (g). It is easily verified that U q (l S ) is a Hopf subalgebra. Moreover it is a Hopf * -subalgebra with * corresponding to the compact real form. Taking the * -structure into account we write U q (k S ) := (U q (l S ), * ). In the special case S = ∅ we write instead U q (t) := K i : i ∈ I with its * -structure.
1.4. Quantized coordinate rings. The quantized coordinate ring C q [G] is defined as a subspace of the linear dual U q (g) * . We take the span of all the matrix coefficients of the finite-dimensional irreducible representations V (λ) (see below). It becomes a Hopf algebra by duality in the following manner: given X, Y ∈ U q (g) and a, b ∈ C q [G] we define Moreover it becomes a Hopf * -algebra by setting a * (X) := a(S(X) * ).
We have a left action ⊲ and a right action ⊳ of U q (g) on C q [G] given by Using the action of U q (g) on C q [G] we can define quantum analogues of the (generalized) flag manifolds. The quantum flag manifold C q [U/K S ] is defined by Notice that C q [U/K S ] = C q [U] Uq(k S ) , the invariants with respect to the action of U q (k S ). In the special case S = ∅ we will write C q [U/T ] := C q [U] Uq(t) , as this case corresponds to the the quantum analogue of the full flag manifold U/T . 1.5. Matrix coefficients. The representation theory of U q (g) is essentially the same as that of U(g), hence of g. In particular we have the analogue of the highest weight modules V (λ) for any dominant weight λ, which we will denote by the same symbol. In any case, given a finite-dimensional representation V , we define its matrix coefficients by These elements span C q [G], according to the description given above.
In particular we will be interested in unitary representations. We say that an inner product Here we use the * -structure of U q (u). It is well-known that an U q (u)-invariant inner product exists on every representation V (λ), and it is unique up to a constant. Let {v i } i be an orthonormal weight basis of V (λ) with respect to (·, ·), and write λ i for the weight of v i . Also denote by {f i } i the dual basis of V (λ) * . With this notation we set We will often omit the superscript V (λ) from the notation, as we will mostly work with one fixed representation in the following. Note that we have This state is not a trace, but instead we have the property where θ is the modular automorphism corresponding to the Haar state.
The modular automorphism has a simple expression in terms of the action of U q (u) on In particular, for the unitary matrix coefficients u i j we have θ(u i j ) = q (2ρ,λ i +λ j ) u i j , where λ i denotes the weight of the basis element v i .

Some results on twisted Hochschild homology
In this section we recall some basics of (twisted) Hochschild homology, as well as some results obtained in [Mat20]. In particular we discuss how to construct certain 2-cycles on quantum flag manifolds in terms of appropriate projections.
2.1. Hochschild homology. Hochschild homology is a homology theory for associative algebras, which we consider here to be over C. The main reference for this section is [Lod13]. Let A be an associative algebra and M be an a n + (−1) n a n m ⊗ a 1 ⊗ · · · ⊗ a n−1 .
It satisfies b 2 = 0, hence we have corresponding homology groups denoted by H • (A, M). We will also use the notation HH • (A) := H • (A, A). Hochschild homology can also be defined in terms of derived functors as There is a corresponding dual cohomology theory, whose groups are denoted by H n (A, M).
A natural choice of bimodule is given by M = A, in which case we talk about the Hochschild homology of A. Here we will focus on the twisted bimodules M = σ A, which are defined as follows: as a vector space we have M = A, but the bimodule structure is given by a · b · c = σ(a)bc, where σ ∈ Aut(A). In this case we will use the notation HH σ • (A) := H • (A, σ A) and refer to it as the twisted Hochschild homology of A. Notice that we could as well introduce a twist for the right multiplication, but as bimodules this gives nothing new.
In the case when A is the algebra of functions on some smooth space X, the Hochschild homology of A is related to the differential forms on the space X. This is the Hochschild-Kostant-Rosenberg theorem [HKR62], see also [Lod13,Theorem 3.4.4]. Recall that for a commutative unital algebra A, we have the A-module of differential forms Ω • (A) := • A Ω 1 (A) constructed from the module of Kähler differentials Ω 1 (A).
Here the algebra structure on HH • (A) is given by the shuffle product, which relies on commutativity of A. There is also a continuous version of this theorem, essentially due to Connes [Con85], which allows to consider smooth forms as opposed to algebraic ones.
Finally we note that, at the level of chains, the map A ⊗n+1 → Ω n (A) is given by a 0 ⊗ a 1 ⊗ · · · a n → a 0 da 1 ∧ · · · ∧ da n .
2.2. Some results. We will now focus on the quantum flag manifolds C q [U/K S ]. We will recall some results on their twisted Hochschild homology and cohomology from [Mat20].
be an orthonormal weight basis with respect to an U q (u)-invariant inner product, and write λ i for the weight of v i . Denote by u i j = (u V (λ) ) i j the unitary matrix coefficients. Given a, b ∈ {1, · · · , N}, we define the The * -structure is extended to matrices by the conjugate transpose, that is . We also define the quantum trace by The matrices M b a behave like matrix units, as shown in [Mat20, Proposition 3.3]. Proposition 2.2. The matrices {M b a } a,b are linearly independent and satisfy 3. These matrices are denoted by N b a in [Mat20]. In the cited paper we also considered the matrices M b a given by (u m i ) * u n j , but we will not use them here. In particular, the elements P a := M a a are self-adjoint projections of "quantum rank one". We will use them to construct certain twisted Hochschild 2-cycles on C q [U].
Remark 2.4. It is worth noting that we have P a ∈ Mat(C q [U/T ]), essentially by construction. On the other hand, we need additional conditions to get P a ∈ Mat(C q [U/K S ]).

Let us consider the elements
Here the quantum trace is extended in the obvious way, namely Recall that θ denotes the modular automorphism of C q [U], which acts by θ(u i j ) = q (2ρ,λ i +λ j ) u i j . The following result can be found in [Mat20, Proposition 5.1].
Proposition 2.5. The element C(P a ) ∈ C q [U] ⊗3 is a 2-cycle in the (normalized) twisted Hochschild complex, hence it defines a class for all entries, then we also have a class [C(P a )] ∈ HH θ 2 (C q [U/K S ]). Remark 2.6. Here C(P a ) is a modification of the usual Chern character Ch n : K 0 (A) → H λ 2n (A) given by Ch n (P ) = Tr(P ⊗2n+1 ), where H λ • denotes the cyclic homology of A. This can be easily modified to the twisted case by using the quantum trace, as opposed to the usual trace. On the other hand, the factor 2P a − 1 and the property Tr q (P a ) = q (2ρ,λa) guarantee that we map lands in Hochschild homology, as opposed to cyclic homology.
Next we would like to check whether the class [C(P a )] is non-trivial. To do this we can introduce an appropriate cohomology class and to show that the corresponding pairing is non-zero. Given a ∈ I, consider the linear functional η a : C q [U] ⊗3 → C given by It is easy to check that, due to the properties of the counit, these linear functionals define (twisted) cohomology classes, as shown in [Mat20, Proposition 5.5].
Proposition 2.7. The restriction of η a to C q [U/K S ] gives a cohomology class [η a ] ∈ HH 2 θ (C q [U/K S ]). Remark 2.8. Observe that this is true for the restriction of η a to any quantum flag manifold C q [U/K S ]. On the other hand, these functionals do not give classes in HH 2 θ (C q [U]). Finally we look at the pairing between [η a ] and [C(P b )]. The result can be expressed entirely in terms of representation-theoretic data, as shown in [Mat20, Proposition 6.11].
Our aim in the next section will be to produce some non-trivial classes in HH θ 2 (C q [U/K S ]). In order to do this, we will proceed in two steps: (1) define a projection P 1 ∈ Mat(C q [U/K S ]), giving a class [C(P 1 )] ∈ HH θ 2 (C q [U/K S ]), (2) prove that it is non-trivial by showing that [η a ], [C(P 1 )] = 0 for some a ∈ I.

Non-trivial classes on quantum flag manifolds
In this section we will construct some non-trivial classes in HH θ 2 (C q [U/K S ]). The first step will be to construct appropriate projections P 1 ∈ Mat(C q [U/K S ]). This will make use of a certain irreducible representation, which in the classical case is used to give a projective realization of U/K S . With these projections at hand and the results of the previous section, it will be fairly straightforward to show that we get non-trivial classes.
3.1. The projections. It is well-known that the flag manifold U/K S can be realized as a U-orbit in a projective space, see for instance [CaSl09, Section 3.2.8]. The projective space here is P(V (ρ S )), where V (ρ S ) is the irreducible representation with highest weight In the quantum case we proceed along these lines by considering the corresponding irreducible representation V (ρ S ) of U q (u). Let {v i } i be an orthonormal weight basis of V (ρ S ) with respect to a U q (u)-invariant inner product. We write u i j = (u V (ρ S ) ) i j for the unitary matrix coefficients. For notational convenience, we will assume from now on that v 1 is a highest weight vector of V (ρ S ), hence of corresponding weight λ 1 = ρ S .
With the notation as above, we define the elements (3.1) Notice that (P 1 ) i j = p i j , using the notation of the previous section. Our goal will be to show that p i j ∈ C q [U/K S ]. First we will need the following lemma. Lemma 3.1. We have F i v 1 = 0 for all i ∈ S.
Proof. This works as in the classical case, but we provide a proof for completeness. Suppose that F i v 1 = 0, which implies that F i v 1 has weight ρ S − α i . Recall that the Weyl group acts transitively on the weights of an irreducible representation. Denoting by s α (λ) = λ − 2(λ,α) (α,α) α the reflection of λ with respect to α, we find that s α i (ρ S − α i ) = s α i (ρ S ) + α i . Moreover, since (ρ S , α i ) = 0 for i ∈ S by definition of ρ S , we obtain s α i (ρ S − α i ) = ρ S + α i . But this is impossible, since ρ S is the highest weight, hence we must have F i v 1 = 0.
We are now ready to construct the invariant projections.
Proof. Since p i j = (M 1 1 ) i j , all claims follow from Proposition 2.2 except for p i j ∈ C q [U/K S ]. For this it suffices to show that p i j is invariant under the generators of U q (k S ).
, it is clear that E k ⊲ u i 1 = 0 for k ∈ S (this is true for any k, since v 1 is a highest weight vector). Next we have F k ⊲ u i 1 = 0 for k ∈ S, by Lemma 3.1. On the other hand we have On the other hand, using the results above, we have for any k ∈ I that 3.2. Non-triviality. We have just constructed a projection P 1 with entries

By Proposition 2.5 and Proposition 3.2, we have a corresponding class
[C(P 1 )] ∈ HH θ 2 (C q [U/K S ]). Finally we want to show that this class is non-trivial.
From this we conclude that [C(P 1 )] is non-trivial.
Thus we have shown that, for every quantum flag manifold C q [U/K S ], we have a non-trivial class [C(P 1 )] ∈ HH θ 2 (C q [U/K S ]). These classes have an appropriate classical limit for q → 1. Then, according to the Hochschild-Kostant-Rosenberg theorem, they will correspond to some differential two-forms on U/K S . We will investigate this aspect in the following.
Recall that all flag manifolds U/K S are Kähler manifolds. In particular, they admit twoforms with some special properties, the Kähler forms. In the next section we will construct some Kähler forms on U/K S with the goal of comparing them with the classes [C(P 1 )].

Kähler forms on (classical) flag manifolds
In this section we will provide a construction of some Kähler forms on the flag manifolds U/K S . While there are many possible approaches, our aim is to proceed in a way that is wellsuited for comparison with the quantum classes from the previous section. The construction will use projections analogous to those used in the quantum case.

4.1.
Notations. Before getting into the construction, let us quickly explain the notations we will employ. These will parallel those we have already used the quantum case.
Recall that the Lie groups G and U, as well as the Lie algebras g and u, all act in a compatible way in a given finite-dimensional representation V . The representation of G will be holomorphic, while the representation of U will be unitary with respect to an appropriate inner product. We will mainly consider the algebra of matrix coefficients C We will occasionally consider the matrix coefficients c V f,v as functions on G according to the same formula. In the same way we can make sense of c V f,v (X) for X in g or u. Now let {v i } i be an orthonormal weight basis with respect to the given inner product on V . Let us also denote by {f i } i the dual basis of V * . Then we have c V f i ,v j (g) = (v i , gv j ). Corresponding to this choice we will employ the notation u i j := c V f i ,v j , that is u i j (g) = (v i , gv j ). We will omit the index V , as the representation V will be fixed in the following. Recall that, given a function f : U → C, its conjugate f is defined by f (g) := f (g). Since the matrix with entries u i j (g) is unitary for every g ∈ U, we obtain the identity . Finally let us consider the action of the Lie algebra g on c V f,v . Considering g as derivations at 1 ∈ G, it turns out that X(c V f,v ) = f (Xv), where g acts on V by its representation. 4.2. The projections. In this subsection we will construct a projection P for each flag manifold U/K S , in full analogy with the construction given in Section 3.1.
Corresponding to the subset S ⊂ I, we consider the dominant weight We have an irreducible representation V (ρ S ) of highest weight ρ S . Let {v i } i be an orthonormal weight basis as above. We assume that v 1 is a highest weight vector (of weight ρ S ).
As we have already mentioned, the action of P S preserves the line Cv 1 , see for instance [CaSl09, Section 3.2.8]. This defines a character ξ ρ S : P S → C × by This restricts to a character of K S = P S ∩ U, which we denote by the same symbol.
Our notation ξ ρ S for the character corresponding to V (ρ S ) is consistent with this one. Now using the unitary matrix coefficients u i j of V (ρ S ) we define (4.1) We will now derive some properties satisfied by the functions p i j . Lemma 4.2. We have p i j ∈ C ∞ (U/K S , C). Moreover we have the identities Proof. We need to show that p i j (gk) = p i j (g) for all g ∈ U and k ∈ K S . We compute Here we have used that the representation is unitary. Therefore p i j (gk) = u i 1 (gk)u j 1 (gk) = |ξ ρ S (k)| 2 u i 1 (g)u j 1 (g) = p i j (g). The other properties are easy to check.
We denote by P the matrix with entries p i j . By the previous lemma, this is an orthogonal projection of rank 1 with entries in C ∞ (U/K S , C).

Line bundles.
In this subsection we will interpret the functions u i 1 as sections of a line bundle over G/P S ∼ = U/K S . The material here will be used only tangentially in the following, but it gives an interesting geometric perspective, as well as connecting the construction we will use with other ways to obtain Kähler forms on flag manifolds.
Recall that the Lie group G can be considered as a principal P S -bundle over the flag manifold G/P S (similarly for the compact description U/K S ). Given a representation of G on a vector space V , we can form the associated vector bundle G × P S V . The points of this bundle are the equivalence classes of G × V with respect to the relation This is a holomorphic vector bundle if the given representation is holomorphic. The sections of this bundle can be identified with the functions f : G → V such that f (gp) = p −1 f (g), g ∈ G, p ∈ P S .

Now let us consider the holomorphic line bundle
where P S acts on C by p · z = ξ ρ S (p) −1 z. Observe that this is a holomorphic representation, since the character ξ S comes from the holomorphic representation of G on V (ρ S ).
Lemma 4.3. The functions u i 1 : G → C are holomorphic sections of L −ρ S . Proof. It is clear that they are holomorphic functions, since u i 1 (g) = (v i , gv 1 ) for g ∈ G and the representation of G on V (ρ S ) is holomorphic. To show that they are sections of L −ρ S we observe that pv 1 = ξ ρ S (p)v 1 implies u i 1 (gp) = ξ ρ S (p)u i 1 (g) for p ∈ P S . Remark 4.4. The minus sign in the definition of L −ρ S (that is, using the character ξ −1 ρ S instead of ξ ρ S ) is due to the fact that we take the positive Borel subgroup, as opposed to the negative one, which is the more common choice when stating the Borel-Weil theorem. For a formulation using this convention see for instance [Sep07,Theorem 7.58].
It is possible to proceed along these lines to obtain a Kähler form on U/K S , as we will now sketch. Equipping the line bundle L −ρ S with a connection, its curvature gives a (1, 1)-form η (a representative of the first Chern class of U/K S ). Then η is Kähler if the line bundle L −ρ S is positive, which in turn is equivalent to L −ρ S being ample by Kodaira's embedding theorem. But it is known that L −ρ S is ample by results of Borel-Weil, see for instance [Sno89, Theorem 6.5] (keeping in mind the opposite convention for the Borel subgroup).
However we will not proceed this way, since this description is not particularly well-suited for comparison with the quantum setting. Instead, we will define the candidate Kähler form in terms of the projection P introduced before, using the Chern character. 4.4. Differential forms. Let P be the projection with entries p i j ∈ C ∞ (U/K S , C) from (4.1). Corresponding to this projection, we define a two-form on U/K S bỹ ω := Tr(P · dP ∧ dP ) = i,j,k p i j dp j k ∧ dp k i .
Here and in the following we will adopt some obvious matrix-type notation. Our aim will be to show thatω is, up to a constant, a Kähler form on U/K S . Before getting into that, let us motivate this choice from a suitably non-commutative point of view. By Lemma 4.2 the matrix P with entries p i j ∈ C ∞ (U/K S , C) is a projection of rank one. Hence we have a projective C ∞ (U/K S , C)-module of rank one which, according to the Serre-Swan theorem, corresponds to a complex line bundle over U/K S . Moreover this bundle admits a Hermitian structure, due to the fact that P is orthogonal. More importantly, this line bundle admits a natural connection defined in terms of the projection P , namely the Levi-Civita one. Its curvature can be computed using the Chern character and coincides with the two-formω defined above, up to a factor. For more on this point of view, see for instance [Kar87, Chapter 1] and [Lod13, Chapter 8].
We will now show some basic properties ofω. We remark that the fact that it is closed is a general result of Chern-Weil theory, but we give a short proof for completeness.
Lemma 4.5. The two-formω on U/K S satisfies the properties: (1) it is closed, Proof.
(1) The exterior derivative ofω is the 3-form given by dω = i,j,k dp i j ∧ dp j k ∧ dp k i = Tr(dP ∧ dP ∧ dP ).
(2) The left translation L g : U → U given by L g h = gh induces a map L g : U/K S → U/K S , denoted by the same symbol. A form ω on U/K S is left U-invariant if L * g ω = ω for every g ∈ U. In other words, for any g ∈ U we must have L * g ω gh = ω h for every h ∈ U/K S . For the matrix coefficients u i j , considered as functions on U, we have Here π denotes the representation of U on V (ρ S ). From these identities we immediately obtain that the pullback of the functions p i j is given by Using this fact, it is easy to check that we have the identity Since pullbacks are compatible with the wedge product and commute with the exterior derivative, we conclude that L * gω gh =ω h and henceω is left U-invariant. 4.5. Complex decomposition. In this section we explore the consequences of the u i 1 being holomorphic sections of a line bundle over U/K S . We begin with a simple lemma.
Proof. These can be easily derived from Lemma 4.2 together with ∂(u i 1 ) = 0 and ∂(u j 1 ) = 0, where the last two identities follow from u i 1 being holomorphic. For instance we have Remark 4.7. There is an exact analogue of these identities for the quantum irreducible flag manifolds in terms of the Heckenberger-Kolb calculus, see [Mat19, Lemma 5.2].
We will now rescaleω by setting ω := −iω = −i Tr(P · dP ∧ dP ). (4.2) We recall that i ∈ C denotes the imaginary unit. The main reason for this rescaling is to make ω into a real form (and also positive definite, as we will see later on).
Since ω = −iω, we conclude that ω is a (1, 1)-form with the claimed expression. Next we show that ω is real, that is ω = ω. Using the identity p i j = p j i we compute So far we have shown that ω is a closed, real (1, 1)-form on U/K S , which is also U-invariant. To conclude that it is a Kähler form we still need to show that it is positive definite. 4.6. Positive-definiteness. As ω is U-invariant, it suffices to show that it is positive definite at the origin o = K S , namely the identity coset in U/K S . Recall that, since U/K S ∼ = G/P S , we can identify the holomorphic tangent space at o with g/p S ∼ = n − S and the anti-holomorphic tangent space with n + S . In other words, the former is the span of the root vectors {f α } α∈∆(n + S ) , while the latter is the span of the root vectors {e α } α∈∆(n + S ) . We will denote by f ⋆ α and e ⋆ α the corresponding dual elements in their respective cotangent spaces.
From the above discussion and the fact that ω is a (1, 1)-form, it follows that it must take the following form at the origin To show that ω is positive definite it suffices to show that the matrix with entries c αβ is positive definite. This is what we will check in the following. First we will require the following simple lemma.
Lemma 4.9. Let α ∈ ∆(n + S ). Then v α = (ρ S , α) −1/2 f α v 1 is a vector of norm 1. Proof. Recall that v 1 is a highest weight vector of weight ρ S . Hence we have Now we observe that, since ρ S = i∈I\S ω i and α ∈ ∆(n + S ), we have (ρ S , α) > 0. Hence we can consider the vector v α = (ρ S , α) −1/2 f α v 1 . To show that it has norm 1 we compute Up to now the choice of the orthonormal weight basis {v i } i for V (ρ S ) was arbitrary. We will now require that the basis contains all the vectors v α for α ∈ ∆(n + S ) as in the lemma above. Notice that this requirement makes sense, as they all have norm 1.
Theorem 4.10. We have that ω is a Kähler form on U/K S . Moreover Proof. We have already shown that ω is a closed, real (1, 1)-form. Hence it suffices to show that ω is positive definite (that is, the corresponding symmetric bilinear form is positive definite). As ω is U-invariant, it suffices to show this at the origin of U/K S . As we have already discussed, the (1, 1)-form ω at the origin has the following form To show that ω is positive definite it suffices to show that the matrix with entries c αβ is positive definite. Observe that c αβ = −i ω o (f α , e β ).
We will now determine c αβ using the expression ω = −i i,j,k p i j dp j k ∧ dp k i . We will consider o = K S as being 1 ∈ U, as the result will not depend on the chosen representative. Then In the last step we have used that p i j (1) = δ i j , which follows from p i j (1) = (v i , v j ). Moreover, since p i j = u i 1 u j 1 and f α is a derivation at the identity, we have Here we have used the fact that f α (u j 1 ) = 0 by weight reasons. Similar computations show that e β (p j i ) = δ j 1 e β (u i 1 ). Therefore we obtain the expression First we will consider f α (u i 1 ) = (v i , f α v 1 ). Since f α v 1 is proportional to v α , it is clear that this is zero unless v i = v α (as the chosen basis is orthonormal). In this case we have Similarly consider e β (u i 1 ) = −(v i , f β v 1 ). Again this is zero unless v i = v α , in which case e β (u i 1 ) = −(ρ S , α) 1/2 . Hence we conclude that c αβ = 0 for α = β, while c αα = (ρ S , α), α ∈ ∆(n + S ).

Comparison in the classical limit
In this last section we will consider an appropriate classical limit of the classes [C(P 1 )] ∈ HH θ 2 (C q [U/K S ]) constructed in Section 2. We will show that the corresponding classical two-forms can be identified with the Kähler forms on U/K S constructed in Section 4. 5.1. Remarks on the classical limit. Informally the quantized enveloping algebra U q (u) reduces to the enveloping algebra U(u) for q → 1, namely the classical limit. This specialization can be made precise with a bit of care, but for our purposes we only need to know that it makes sense at the level of representations, which is rather easy to show.
Next, we have defined the quantized coordinate ring C q [U] as the algebra of matrix coefficients of finite-dimensional U q (u)-representations. Hence for q → 1 it reduces to the algebra of matrix coefficients of finite-dimensional U(u)-representations.
Finally we want to identify the algebra above with C[U] ⊂ C ∞ (U, C), the algebra of matrix coefficients of finite-dimensional U-representations. This is simply done by , where X ∈ U(u) and g ∈ U. Observe that this identification makes sense, since U and u act in a compatible way in each finite-dimensional representation V . 5.2. The comparison. Consider now the representative C(P 1 ) ∈ C q [U/K S ] ⊗3 of the class [C(P 1 )] ∈ HH θ 2 (C q [U/K S ]). It corresponds to an element C(P 1 ) ∈ C[U/K S ] ⊗3 under the classical limit, as explained above. Denote by ω P 1 ∈ Ω 2 (U/K S ) the corresponding two-form obtained from the Hochschild-Kostant-Rosenberg map.
Proof. The representative C(P 1 ) ∈ C q [U/K S ] ⊗3 is given explicitly by The elements p i j ∈ C q [U] from (3.1) correspond to the elements p i j ∈ C[U] from (4.1) under the classical limit. Hence we obtain Now ω P 1 is the image of C(P 1 ) under the Hochschild-Kostant-Rosenberg map, which is given by a 0 ⊗ a 1 ⊗ · · · a n → a 0 da 1 ∧ · · · ∧ da n . We obtain the two-form ω P 1 = i,j,k (2p i j − δ i j )dp j k ∧ dp k i ∈ Ω 2 (U/K S ).
Observe that the second term is zero by graded-commutativity, since i,j dp i j ∧ dp j i = − i,j dp j i ∧ dp i j = 0.
Hence ω P 1 can be rewritten as ω P 1 = 2 i,j,k p i j dp j k ∧ dp k i ∈ Ω 2 (U/K S ).
This coincides with ω as defined in (4.2), up to a constant. But we know that the latter is a Kähler form on U/K S according to Theorem 4.10, which gives the claim.
Remark 5.2. For a quantum irreducible flag manifold C q [U/K S ], we have constructed in [Mat19] a Kähler form in the sense of [ÓBu17] using the differential calculus of [HeKo06]. It is of the form i,j,k q (2ρ,λ i ) p i j dp j k ∧ p k i , up to a constant. Hence, by the same argument as above, it can be identified with a Kähler form on the irreducible flag manifold U/K S .