Pseudo-parabolic category over quaternionic projective plane

Quaternionic projective plane $\mathbb{H} P^2$ is the next simplest conjugacy class of the symplectic group $SP(6)$ with pseudo-Levi stabilizer subgroup after the sphere $\mathbb{S}^4\simeq \mathbb{H} P^1$. Its quantization gives rise to a module category $\mathcal{O}_t\bigl(\mathbb{H} P^2\bigr)$ over finite-dimensional representations of $U_q\bigl(\mathfrak{s}\mathfrak{p}(6)\bigr)$, a full subcategory in the category $\mathcal{O}$. We prove that $\mathcal{O}_t\bigl(\mathbb{H} P^2\bigr)$ is semi-simple and equaivalent to the category of quantized equivariant vector bundles on $\mathbb{H} P^2$.


Introduction
With every point t of a maximal torus T of a simple complex algebraic group G one can associate a full subcategory O t in the BGG category O of the corresponding quantum group, U q (g).This subcategory is additive and stable under the tensor product with the category Fin q (g) of finitedimensional (quasi-classical) U q (g)-modules.Its objects are submodules in tensor products of V ∈ Fin q (g) with a distinguished base module M of highest weight λ depending on t.In generic situation, the locally finite part of End(M) is an equivariant quantization A of the coordinate ring of C t = Ad G (t), the conjugacy class of t.If O t is semi-simple, then its objects can be regarded as "representations" of quantum equivariant vector bundles on Ad G (t).According to the famous Serre-Swan theorem [S, Sw], global sections of vector bundles on an affine variety form finitely generated projective modules over its coordinate ring and vice versa.Finitely generated projective right A-modules equivariant with respect to U q (g) can be viewed as quantum equivariant vector bundles.They constitute a Fin q (g)-module category, Pr q (A, g).
Equivalence of Fin q (g)-module categories O t and Pr q (A, g) is established via functors acting on objects as Pr q (A, g) ∋ Γ → Γ ⊗ A M ∈ O t and O t ∋ N → Hom • C (M, N) ∈ Pr q (A, g), where the circle designates the locally finite part with respect to the U q (g)-action.The module M is absent in the classical picture as there is no faithful irreducible representation of a classical commutative coordinate ring.
Quantization of vector bundles is a natural extension of the deformation quantization programme for Poisson manifolds [BFFLS].Vector bundles on non-commutative spaces are of interest in the K-theory [Sheu], non-commutative geometry [C], and non-commutative quantum field theory [DN].There is one more area of their applications in connection with quantum symmetric pairs and universal K-matrices, [Let1,Kolb].If the class C t is a symmetric space, then there is a one-dimensional representation of A (a classical point on quantized C t ).It satisfies the reflection equation [KS] defining a coideal subalgebra U q (k ′ ) ⊂ U q (g).Then A can be realized as the subalgebra of U q (k ′ )-invariants in the Hopf algebra of functions on the quantum group that is dual to U q (g).In the classical limit, U q (k ′ ) turns into the centralizer U(k ′ ) of a point t ′ ∈ C t , which is conjugate to the centralizer U(k) of the point t.
The representation theory of U q (k ′ ) is a challenge since t ′ ∈ T (which is fixed for a quantum group) and the triangular decomposition of U q (g) is not compatible with that of U q (k ′ ), [Let1,Let2].The category O t , if semi-simple, plays the role of a bridge between Pr q (A, g) and the category of finite-dimensional U q (k ′ )-modules via a chain of equivalences.This is discussed in details in [M4] for quantum spheres.
Remark that an associated vector bundle in the classical geometry is obtained via induction functor from a finite dimensional representation of the stabilizer subgroup, which is a relatively simple thing.In the non-commutative world the picture is quite opposite.It is surprisingly easier to construct an apparently more complex vector bundle, and arrive at the fiber via specialization at the (quantum) initial point, if any.This transition is demonstrated for projective spaces in [M6].
In the present paper we study the category O t for G = SP (6) and t ∈ T one of 6 points with the stabilizer ≃ SP (4) × SP (4) (they belong to two isomorphic conjugacy classes).In this case, C t is the quaternionic projective plane HP 2 which enters one of the two infinite series, HP n , of rank 1 non-Hermitian symmetric conjugacy classes.The other series comprises even spheres and has been studied in [M4].However, the approach of [M4] (as well of the last section in [M5]) is special for S 2n and cannot be extended any further.The method we demonstrate here on the example of HP 2 works for any semi-simple conjugacy class comprising elements of finite order (e.g.symmetric conjugacy classes).This method reduces the question of semi-simplicity of O t to simplicity of M.
We prove that the module M is irreducible in the case of HP 2 and explicitly construct an orthonormal basis with respect to the contravariant form on it.Our approach is based on viewing M as a module over U q (l) ⊂ U q (g), where l ≃ gl(2) ⊕ sp(2) is the maximal reductive Lie subalgebra in k such that U(l) is quantized as a Hopf subalgebra in U q (g).This is the content of Section 2.
In Section 3, we prove semi-simplicity of the category O t .It is an illustration of the complete reducibility criterion for tensor products of highest weight modules based on a contravariant form and Zhelobenko extremal cocycle [M3,M5,Zh].We show that for every finite-dimensional quasi-classical U q (g)-module V the tensor product V ⊗ M is completely reducible and its simple submodules are in a natural bijection with simple k-submodules in the classical g-module V .This way we establish equivalence of O t and Fin(k) as Abelian categories.
In Section 4 we present a classical point on quantum HP 2 , i.e. a one-dimensional representation of A. It is a numerical solution of the reflection equation that satisfies other relations of quantized C[HP 2 ].Therein we describe the coideal subalgebra U q (k ′ ).
In the last Section 5 we establish equivalence of the category O t with the category Pr q (A, g).
1.1 Quantum group U q sp(6) and basic conventions In this paper, g = sp(6), k = sp(4) ⊕ sp(2) and l = gl(2) ⊕ sp(2).There are inclusions g ⊃ k ⊃ l of Lie algebras, which we describe by inclusions of their root bases as follows.Both k and l are reductive subalgebras of maximal rank, i.e. they contain the Cartan subalgebra h of g.Fix the inner product on h such that the long root has length 2. All positive roots of g are expressed in an orthonormal basis of weights 2, and α 3 = 2ε 3 form the basis of simple roots Π g = Π.The basis of simple roots of k is Π k = {α 1 , 2α 2 + α 3 , α 3 }.Note that the root 2α 2 + α 3 is not in Π g , so k is not a Levi subalgebra in g.On the contrary, l is the maximal subalgebra in k that is Levi in g.Its basis of simple roots is Π l = {α 1 , α 3 }.
For two elements x, y of an associative algebra and a scalar a we write [x, y] a = xy − ayx.
We say that x and y quasi-commute if [x, y] a = 0 for some a ∈ C, and call the algebra quasicommutative if this holds for all pairs of its generators.
The quantum group U q (g) is a C-algebra with unit parameterized by a complex number q, which is assumed not a root of unity, [ChP].It is generated by simple root vectors e i , f i (Chevalley generators), and invertible Cartan generators q h i , i = 1, 2, 3.The elements q ±h i generate a commutative subalgebra U q (h) in U q (g) isomorphic to the polynomial algebra on a torus.They obey the following commutation relations with e i , f i : for all i, j = 1, 2, 3. Non-adjacent positive Chevalley generators commute while adjacent generators satisfy quantum Serre relations [e i , [e i , e j ] q ] q = 0, i, j = 1, 2, i = j, [e 2 , [e 2 , [e 2 , e 3 ] q 2 ]] q2 = 0, [e 3 , [e 3 , e 2 ] q2 ] q 2 = 0, where q = q −1 .Similar relations hold for the negative Chevalley generators on replacement f i → e i , which extends to an involutive algebra automorphism of U q (g) with σ(q h i ) = q −h i .
A comultiplication defined on the generators by makes U q (g) a Hopf algebra.The assignment q h i → 1, e i → 0, f i → 0 extends to the counit homomorphism U q (g) → C, then antipode γ acts on the generators by q h i → q −h i , e i → −e i q −h i , It is an anti-algebra and anti-coalgebra automorphism of U q (g).
The composition ω = σ • γ is an involutive automorphism of U q (g) that preserves comultiplication and flips multiplication.
The Serre relations are homogeneous with respect to the U q (h)-grading via its adjoint action on U q (g).They are determined by the corresponding weight, so we refer to a particular relation by its weight in what follows.
We remind that a total ordering on the set of positive roots is called normal if any α ∈ R + presentable as a sum α = µ + ν with µ, ν ∈ R + lies between µ and ν.A reductive Lie subalgebra l ⊂ g of maximal rank is called Levi if it has a basis Π l of simple roots which is a part of Π.
Then there is an ordering such that every element of R + g/l is preceding all elements of R l .In this paper, l designates the subalgebra gl(2) ⊕ sp(2) as agreed upon earlier.
With a normal ordering one can associate a system { fα } α∈R + ⊂ U q (g − ) of elements such that ordered monomials in fα form a PBW-like basis in U q (g − ).In particular, the algebra U q (g − ) is freely generated over U q (l − ) by ordered monomials in fα with α ∈ R + g/l .In the classical limit, the elements fα form a basis of root vectors in g − .For a detailed construction of such a basis, the reader is referred to [ChP].
By Λ g we denote the root lattice of g, i.e. a free Abelian group generated by fundamental weights relative to the fixed polarization of R. The semi-group of integral dominant weights is denoted by Λ + g .All U q (g)-modules are assumed diagonalizable over U q (h).A non-zero vector v of a U q (h)-module V is said to be of weight µ ∈ h * if q hα v = q (α,µ) v for all α ∈ Π + .Vectors of weight µ span a subspace in V denoted by V [µ].The set of weights of V is denoted by Λ(V ).
Infinitesimal character of a U q (h)-module is defined as a formal sum µ∈Λ(V for all µ and ch(V ) < ch(W ) if this inequality is strict for some µ.
By all q we mean all not a root of unity; almost all q stands for all except for a finite set of values.
2 Base module for HP 2 In this section we study a U q (g)-module M that generates the category of our interest.We prove its irreducibility and construct an orthonormal basis with respect to a contravariant form on it.
Let κ denote the half-sum of the positive roots of k.Regard roots (more generally, integaral weights) as characters of the maximal torus T of the group G (the torus has been fixed and its Lie algebra is h participating in the construction of U q (g)).Define base weight λ ∈ h * as one featuring the property q 2(λ,α) = α(t)q 2(κ−ρ,α) , for all α ∈ Π g , where α(t) is the value of root α on the initial point t ∈ T .It is the eigenvalue of the operator Ad t on the corresponding root space in g and, in particular, α(t) = 1 once α ∈ Π k .
Remark that λ is evaluated on squared Cartan generators in the above equality.Therefore base weight is not uniquely determined by the point t but up to a choice of sign in ± α(t) for each α ∈ Π g .One can pick up any for λ, but we additionally assume q (λ,α) = 1 for all α ∈ Π l = Π g ∩ Π k .This is consistent with the conditions on λ because (κ, α) = (ρ, α) = 1 for such α.The rational for this will be explained later.
We fix the initial point t by It is easy to check that f δ commutes with f 3 and e 3 , cf. [M7].Let Mλ denote the Verma module with highest weight λ and define M as the quotient of Mλ by its submodule generated by singular vectors The module M supports quantization of the conjugacy class HP 2 in the sense that its quantized coordinate ring C q [HP 2 ] can be represented as a U q (g)-invariant subalgebra in End(M).
Its explicit formulation in terms of generators and relations is given in Section 4.
As l is a Levi subalgebra in g, its universal enveloping algebra is quantized to a Hopf subalgebra U q (l) ⊂ U q (g).The module M is a quotient of the parabolic Verma module of the same weight, by the submodule generated by (the image of) f δ 1 λ .It follows that M is locally finite over U q (l), [M5].We will study M regarding it as a U q (l)-module; then our additional requirements (λ, α) = 0 for α ∈ Π l will keep us within the category of quasi-classical U q (l)-modules (deformations of classical U(l)-modules.Note that M is not quasi-classical for entire U q (g).
Remark 2.1.Note that M contains a base module for the quantum 4-sphere, [M7].It is generated by the highest vector, over the natural quantum subgroup U q sp(4) ≃ U q so(5) in U q sp(6) .We have used this fact when constructing the singular vectors f δ 1 λ and f 3 1 λ in the Verma module Mλ .We will further refer to results on S 4 in our study of higher pseudo-parabolic modules over HP 2 in Section 3.
2.1 U q (l)-module structure of M It turns out that highest vectors of finite dimensional U q (l)-submodules in M belong to a subalgebra ≃ U q sl(3) ⊂ U q (g), which we describe next.
Remark that the subalgebra U q (m) results from a Lusztig transformation of the subalgebra with the simple root basis α 1 , α 2 , see Appendix.
Proof.Both e 1 and e 3 commute with f 2 , so we check their interaction with f θ .An easy calculation λ is annihilated by e 1 and e 3 , by virtue of (A.21).
Corollary 2.4.The vector f θ belongs to the normalizer of the left ideal J.
Proof.Indeed, f δ f θ ∈ J by (A.21).Furthermore, f θ 1 λ generates a finite-dimensional U q (l)submodule in M. Since (λ − θ, α i ) = 0 for i = 1, 3, this submodule is trivial, hence f 1 f θ and where a = q (α i +...+α j ,α j+1 ) .Then Serre relations imply since f δ commutes with f 2 and f 3 .It will be also of use to write these formulas as Proof.Pushing f 3 and then f 1 to the right in f 1 f 3 f k 2 we find it equal to where we have used (2.1).Expressing f 12 f 23 and f 2 f 1 f 23 on the right through f 2 f ξ and f θ modulo J we prove the lemma.
Proof.It is sufficient to check that the U q (l)-submodule L is invariant under U q (g − ) as it contains 1 λ .That is so if and only it is f 2 -invariant.
The elements f ij with i < j quasi-commute with f 1 and Therefore Notice that f 12 quasi-commutes with every power of f 2 while f 23 quasi-commutes with it modulo J because f δ ∈ J commutes with f 2 and f 3 .Therefore we can further push them to the right until they hit f θ -s and then apply (2.2).This way we prove f 12 B ⊂ L and f 23 B ⊂ L, with the help of Corollary 2.4.
Furthermore, push f ξ to the right in the third term until it hits Then for all k, l ∈ Z + we get This never turns zero because q 2λ 2 = q 2(λ,ε 2 −ε 3 ) = q 2(λ,ε 2 ) = −q −2 .For k > 0, the operator e 1 e 3 annihilates the first term in (2.4) and returns f l+1 2 f k−1 θ 1 λ , up to a non-zero scalar multiplier, on the second.Proceeding this way we obtain (e ) unless l + k = 0. Hence these vectors are highest for different U q (l)-submodules in M and none of them is singular for U q (g).
In summary, M is isomorphic to the natural It is semi-simple and multiplicity free.In the classical limit, the subalgebra of (C 2 )] is a polynomial algebra in two variables generated by the principal minors of the coordinate matrix, see e.g.[GW].In the quantum case, the space of U q (l + )invariants in M is isomorphic to a polynomial algebra in quasi-commuting variables f 2 , f θ .
Corollary 2.8.The infinitesimal character of the base module M equals α∈R

Orthonormal basis in M
A symmetric bilinear form (., .) on a U q (g)-module V is called contravariant if (xv, w) = (v, ω(x)w) for all x ∈ U q (g) and all v, w ∈ V .Recall that every highest weight module over a reductive quantum group has a unique contravariant form with respect to the involution ω normalized to 1 on the highest vector.In this section we construct an orthonormal basis in M, with the help of the subalgebras U q (l) and U q (m).It can be constructed as the Gelfand-Zeitlin basis in every U q (l)-submodule L l,k ⊂ M, up to a common factor equal to the norm of the highest vector of L l,k .Thus the problem essentially reduces to calculation of those norms.That is done within a U q (m)-submodule in M generated by 1 λ because the space of U q (l + )-invariants is in that submodule.
Proposition 2.9.Set λ θ = (λ, θ).Then the assignment (l, k) Proof.The boundary conditions easily follow from the basic relations of U q (m).Uniqueness can be checked by an obvious induction on l + k.To prove the recurrence relation permute f k θ and f l 2 , then in the resulting matrix element q −lk 1 λ , e k θ e l 2 f k θ f l 2 1 λ push one copy of e 2 to the right: This calculation is actually done in U q (m).In particular, we used (2.3) and [f 2 , f θ ] q = 0.
Proposition 2.10.The matrix element c (2.5) Proof.Let fθ ∈ U q (g − ) be the vector obtained from f θ by the substitution q −1 → q.Using the formula (A.18), replace f θ with q −2 fθ in the left argument.Then c l,k equals One can express the right hand side through cl,k = 1 λ , e k θ e l 2 f l 2 f k θ 1 λ and check that cl,k defined by (2.5) satisfies the conditions of Proposition (2.9).
Corollary 2.11.The system y l,k i,j , is an orthonormal basis with respect to the contravariant form on M.
3 Category O t (HP 2 ) While the base module M supports a representation of C q [HP 2 ], it generates a family of modules which may be regarded as "representations" of more general quantum vector bundles.This interpretation is only possible if all such modules are completely reducible: then they give rise to projective modules over C q [HP 2 ].They appear as submodules in tensor products V ⊗ M (representing a trivial vector bundle), for every V from the category Fin q (g) of finite-dimensional quasi-classical U q (g)-modules.Therefore the key issue is complete reducibility of tensor products We solve this problem in the present section using a technique developed in [M3, M5].

Complete reducibility of tensor products
Suppose that V and Z are irreducible modules of highest weight.Each of them has a unique, upon a normalization, nondegenerate contravariant symmetric bilinear form, with respect to the involution ω : U q (g) → U q (g).Define a contravariant form on V ⊗ Z as the product of the forms on the factors.Then the module V ⊗ Z is completely reducible if and only if the form on V ⊗ Z is non-degenerate when restricted to the span of singular vectors (V ⊗ Z) + .Equivalently, if and only if every submodule of highest weight in V ⊗ Z is irreducible, [M3].
For practical calculations, it is convenient to deal with the pullback of the form under an isomorphism of (V ⊗ Z) + with a certain vector subspace in V (alternatively, in Z) which is defined as follows.Let I − Z ⊂ U q (g − ) be the left ideal annihilating a vector 1 ζ ∈ Z of highest weight ζ, and I + Z = σ(I − Z ) a left ideal in U q (g + ).Denote by V + Z ⊂ V the kernel of I + Z , i.e. the subspace of vectors killed by There is a linear isomorphism between V + Z and (V ⊗ Z) + assigning a singular vector u = v ⊗ 1 ζ + . . . to any weight vector v ∈ V + Z .Here we suppressed the terms whose tensor Z-factors have lower weights than ζ.Note that the isomorphism V + Z → (V ⊗ Z) + is "almost" U q (h)-equivariant: it shifts weights by ζ.
The pullback of the contravariant form under the map V + Z → (V ⊗ Z) + can be expressed through the contravariant form −, − on V as θ(v), w , for a certain linear map θ on V + Z with values in its dual space.We call it extremal twist defined by Z.In this paper, the contravariant form on V is always non-degenerate when restricted to V + Z , so we can write θ ∈ End(V + Z ).This operator is related with the extremal projector p g , which is an element of a certain extension Ûq (g) of U q (g), [KT].It is constructed as follows.
It acts by the assignment q → q α = q (α,α) 2 , e → ẽα , f α → fα , q h → q hα , where e, f and q h are the standard generators of U q sl(2) and the twiddled elements are root vectors constructed via Lusztig automorphisms, [ChP].For ψ ∈ h * , set p g (ψ), to be an ordered product where p α (z) is the image of under ι α .For generic ψ, the operator p g (ψ) is well defined and invertible on every finitedimensional U q (g)-module.The specialization p g = p g (0) is an idempotent satisfying e α p g = 0 = p g f α for all α ∈ Π.This idempotent is called extremal projector.
The element p g (ψ) gives rise to a rational trigonometric operator function of weight in every weight U q (g)-module that is locally nilpotent over U q (g − ).
Theorem 3.1 ( [M5]).Suppose that the map p g (0) : In the case of our concern, p g = p g (0) is well defined, cf.Proposition 3.2 below.However, the operator p g (ζ) may have poles as a function of ζ.The above theorem implies that such poles are removable.In the special case of the fundamental module V = C 6 all weights in V + Z are multiplicity free.Then det(θ) ∝ α∈R + µ∈Λ(V ) θ α µ up to a non-zero factor, with Here l µ,α is the maximal integer k such that ẽk α V + [µ] = {0} for ẽα = ι α (e).We compute θ in the next section.

Extremal twist and extremal projector
In this section we calculate the determinant of the extremal twist defined by the base module M using its relation to extremal projector and show that it does not vanish at all q.
Denote simple positive roots of the Lie subalgebra k ⊂ g by That is straightforward for F i factors through a parabolic Verma module relative to U q (l): it is the quotient of Mζ by the submodule generated by {f is+1 s 1 ζ } s=1,3 .Therefore M i is locally finite over U q (l), [M4].We use the same notation 1 ζ for the highest vector in M i .Denote by F is+1 s ∈ U q (g − ) the Shapovalov elements, i.e. the images of singular vectors F is+1 s 1 ζ under the natural isomorphisms U q (g − ) ≃ Mζ , and set Note with care that, contrary to F i 2 +1 2 , the elements F i 2 +1 2 are not powers of F 2 .
From now to the end of the section we fix V = C 6 , the smallest fundamental module of U q (g).
Up to non-zero scalar factors, the action of U q (g + ) on V is described by a graph where the vectors v ±i of weights ±ε i , i = 1, 2, 3, form an orthonormal basis with respect to the contravariant form.The diagram for the U q (g − )-action is obtained by reversing the arrows in (3.9).We find from the diagram that ker(E s ) equals , such a factor is proportional to q x + q −x for some x ∈ Q and does not vanish because q is not a root of unity.Therefore all factors p α (t) for such α are regular at z = (ρ, α ∨ ).Moreover, the extremal projector of the subalgebra U q (g α 2 ) is well defined on V ⊗ 1 ζ taking it to ker e 2 .
Thus the first condition of Theorem 3.1 is satisfied.The second condition will be secured by the following calculation.
In the next section we shall see that the kernels Ṽ + i parameterise irreducible decompositions in a pseudo-parabolic category associated with HP 2 .
3.3 Pseudo-parabolic category O t (HP 2 ) and its structure In this section we define the pseudo-parabolic category over HP 2 , prove its semi-simplicity and describe simple objects, based on the results of the previous section.
Denote by O t (HP 2 ) a full subcategory in the category O whose objects are submodules in W ⊗ M, where W ∈ Fin q (g) is a quasi-classical finite-dimensional module over U q (g).It is a module category over Fin q (g) because for every submodule N ⊂ W ⊗ M and U ∈ Fin q (g), the We denote by Fin(k) the tensor category of finite-dimensional k-modules.It is a module category over Fin(g) via the restriction functor.
Let M i denote the irreducible quotient of M i (we will later prove that they coincide at almost all q).We call it pseudo-parabolic Verma module of the corresponding highest weight.
We define V + i as the kernel of the left ideal The subspace V + i is isomorphic to the span of singular vectors in V ⊗ M i , in compliance with discussion of Section 3.1.In principle, Ṽ + i might be bigger than V + i but we shall see that they coincide for almost all q (for all if dim V = 6. From now until Corollary 3.9 we assume that V = C 6 .Let X i ∈ Fin(k), with i ∈ Z 3 + , denote the finite-dimensional k-module of highest weight ξ = 3 s=1 i s µ s .For each i ∈ Z 3 + , introduce a set of triples Ĩ( i) ⊂ Z 3 + : where those with negative coordinates are excluded.Elements of Ĩ( i) parameterize irreducible k-submodules in V ⊗ X i : their components are coordinates of highest weights in the basis of fundamental weights {µ s } 3 s=1 .Let Fin(g ↓ k) denote the subcategory of k-modules that are submodules in modules from Fin(g).
Proof.Since Fin(g) is generated by V as a tensor category, it is sufficient to prove that for each i ∈ Z 3 + the k-module X i is in some tensor power of V .We do it by induction on Suppose that the statement is proved for all X i with | i| = m 0. Fix an index i with | i| = m + 1 and let ℓ be the minimal s ∈ {1, 2, 3} such that i s > 0. We will separately consider two cases depending on the value of ℓ.
In the case of ℓ = 2 we consider a pair of vectors j, k ∈ Z 3 + via j s = i s −δ 2s and k s = i s +δ 1s −δ 2s for s = 1, 2, 3. Since | j| = m, the module X j is in Fin(g ↓ k) by the induction assumption.Now observe from (3.11) that k ∈ Ĩ( j) and i ∈ Ĩ( k).Therefore X k and X i are in Fin(g ↓ k).This completes the proof.
Proof.Notice that the case of i > 1 is easy because the kernels coincide with the whole V .The case i = 1 is an elementary calculation based on the diagram (3.9).
Corollary 3.6.The vector space Proof.First of all observe that ∩ 3 s=1 ker(e is+1 βs ) ≃ ∩ 3 s=1 ker(E is+1 βs ) because all weights in V are multiplicity free.Then the statement is due to the isomorphism Ṽ + i ≃ ∩ 3 s=1 ker(e is+1 βs ) because the right-hand side is in bijection with the span of singular vectors in the k-module V ⊗ X i .
Proposition 3.7.The category O t (HP 2 ) is semi-simple, and its simple objects are U q (g)- at all q in a punctured neighbourhood of 1 (that might depend on i and µ).Then ch(M j ) = ch(X j )ch(M) at all q as M j is simultaneously a quotient of a Verma module and is a submodule in V ⊗(m+1) ⊗ M, which are both flat at all q including q = 1.Induction on m proves 1) for all M j .
To prove 2), we use the equality Ĩ( i) = I( i) we have already established.That is, for each weight η of a singular vector in the k-module V ⊗ X i the pseudo-parabolic module of highest weights λ + η does appear in V ⊗M i (uniquely since all weights in V are multiplicity free).Again induction on m such that V ⊗m ⊗ M ⊃ M i along with Proposition 3.4 secures 2).
Corollary 3.9.For every V ∈ Fin q (g) and for all i, j ∈ Z 3 + , there is an isomorphism Proof.The equality ch(V ⊗ M i ) = j∈I ch(M j ), where the summation is over an irreducible decomposition of V ⊗M i , implies ch(V ⊗X i ) = j∈I ch(X j ), thanks to Proposition 3.8.Therefore the k-module ⊕ j∈I X j is isomorphic to V ⊗ X i and the assertion follows.
Now we summarise the main result of the paper.
Theorem 3.10.1. O t (HP 2 ) is semi-simple for all q.
2. For all q, O t (HP 2 ) is equivalent to the category Fin(k).
3. Simple objects in O t (HP 2 ) are exactly pseudo-parabolic Verma modules, for almost all q.
Proof.The category O t (HP 2 ) is clearly additive.To prove the first statement, observe that a module V from Fin q (g) can be realized as a submodule in a tensor power of C 6 .Then apply Propositions 3.4 and 3.7.
We know from Propositions 3.7 and 3.8 that simple objects of O t (HP 2 ) are exactly M i , i ∈ Z 3 + .Let us prove that for M i ≃ M i for all but a finite number of values of q.Indeed, a module of highest weight is irreducible if and only if its contravariant form is nondegenerate or, alternatively, it has no singular vectors.Weights of singular vectors may be only in the orbit of the highest weight under the shifted action of the Weyl group.Let W ⊂ M i and W ⊂ M i denote the sums of weight spaces whose weights are in that orbit.It is sufficient to check non-degeneracy of the form only on W . Since W is finite dimensional, there is an alternative: either the form is degenerate for all q or or it is not at some and therefore almost all q.From (3.13) we see that W ≃ W in an open neighbourhood of 1. Therefore the form is non-degenerate on W and hence on M i for almost all q as required.
Note that the set of exceptional q where M i ≃ M i may depend on a module.We nevertheless conjecture that it is empty for all i, as is the case for the base module.
4 The algebra C q [HP 2 ] and Reflection Equation In this section we give a more detailed description of the quantized polynomial ring A = C q [HP 2 ] and its one-dimensional representation.This is a special case of a general construction, and the reader is referred to [M2, M6] for details.
Let π be the representation homomorphisms of U q (g) to End(V ), V ≃ C 6 .Pick up a basis It commutes with the coproduct of every element in U q (g).Denote by P the flip of the tensor factors in C 6 ⊗ C 6 and fix a U q (g)-invariant braid matrix S ∈ End(C 6 ) ⊗ End (C 6 ).
Note that R = P S needs not to be image of the particular R entering Q: e.g. one can take R = (π ⊗ π)(R −1 21 ).One can choose π and R as in [FRT].It is known that C q [G] can be realized as the locally finite part of the adjoint U q (g)-module.
It is a subalgebra in U q (g) generated by entries of the matrix (π ⊗ id)(Q).The image of C q [G] in End(M) is a flat deformation of a quotient of C[G] by the defining ideal of HP 2 .That is a maximal proper invariant ideal in C[G], whence the image is a quantization of C[HP 2 ], see [M2] for details.
The algebra A is generated by the entries {Q ij } 6 i,j=1 , which satisfy Equations of the first line are understood in End (C 6 ) ⊗ End (C 6 ) ⊗ End(M) and the subscripts label the End(C 6 )-factors.They are equations of C q [G], a deformation of C[G] that is equivariant under the conjugation action of G on itself.The last two equations fix the quantized conjugacy class HP 2 .This is the full set of relations defining A.
There is a one-dimensional representation χ : In the classical limit, the matrix A goes over to a point t ′ ∈ HP 2 where the Poisson bracket vanishes.
The matrix A defines an embedding of A in the restricted Hopf dual to U q (g) that we denote by T .A description of the algebra T can be extracted from [FRT].Let T = (T ij ) 6 i,j=1 denote its matrix of generators.This matrix is invertible with (T −1 ) ij = γ(T ij ), where γ is the antipode of T .One has two commuting left and right translation actions of U q (g) on T expressed through the Hopf paring and the comultiplication in T by h ⊲ a = a (1) (h, a (2) ), a ⊳ h = (a (1) , h)a (2) , a ∈ T , h ∈ U q (g).
They are compatible with multiplication on T making it a U q (g)-bimodule algebra.
The assignment Q ij → (T −1 AT ) ij defines an equivariant homomorphism A → T , where T is viewed as a U q (g)-module under the left translation action.It is an embedding by similar deformation arguments as with the case of A ⊂ End(M).The character χ factors through the composition A → T → C, where the right arrow is the counit ǫ.
The entries of the matrix K = (id ⊗ π)(R 12 )A 2 (id ⊗ π)(R 21 ) ∈ U q (g) ⊗ End (C 6 ) generate a left coideal subalgebra U q (k ′ ) ⊂ U q (g).It is a deformation of U(k ′ ) with k ′ ≃ k being the Lie algebra of the centralizer of t ′ .
One can check that a ⊳ b = ǫ(b)a for all b ∈ U q (k ′ ) and a ∈ A. We argue that A exhausts all of the subalgebra of U q (k ′ )-invariants, for generic q.Indeed, the latter is ∩ 6 i,j=1 ker K ′ ij where K ′ ij = K ij − ǫ(K ij ) ∈ k ′ mod (q − 1).Restricted to every isotypic component of the Peter-Weyl decomposition of T , the kernel cannot increase in deformation.

Quantization of equivariant vector bundles on HP 2
In this section, we will interpret O t (HP 2 ) as a category of "representations" for quantum vector bundles on HP 2 .
In the classical algebraic geometry, global sections of vector bundles on a variety are finitely generated projective modules over its coordinate ring.If a group G acts on the bundle coherently with the base, the vector bundle is called equivariant.Algebraically it means that G acts on global functions by automorphisms, G acts on global sections, and the multiplication between functions and sections is equivariant.
In the case of homogeneous space G/K, a vector bundle Γ(G/K, X) is characterized by a finite dimensional K-module X over the initial point.It can be realized as the space of K- For a reductive pair G ⊃ K, the Peter-Weyl decomposition C[G] = [V ] V ⊗ V * gives the isotypic component of an irreducible module V in Γ(G/K, X); it is V ⊗ Hom K (X, V ).This is the classical input that we are going to mimic in our approach to quantization.
We have already argued that the base module M supports a faithful representation of A as a subalgebra in the locally finite part End • (M) of linear operators on M. Similarly we claim that the locally finite part Hom • (M, M i ) of the U q (g)-module of linear maps from M to M i is a quantization of the vector bundle Γ(HP 2 , X i ) with fiber X i .Note that Hom • (M, M i ) is a natural equivariant right End • (M)-module via the composition of linear maps.
Proof.Since M and M i are irreducible along with their dual modules of lowest weight, equivariant maps from V to Hom(M, M i ) are in bijection with equivariant maps from Hom(M * i , M * ) to V * , for every V ∈ Fin q (g).We have a version of Corollary 3.9 for dual modules and we can write Hom Uq(g) Hom(M * i , M * ), V * ≃ Hom Uq(g) The rightmost term is isomorphic to Hom k (X i , V ) as V is completely reducible over k.Thus the isotypic component of V in Hom • (M, M i ) is a deformation of the isotypic component of its classical counterpart in Γ(HP 2 , X i ).
In particular, setting M i = M we conclude that End • (M) has the same module structure as A. This implies that, for q = 1, the algebra A exhausts all of End • (M).We will give a recipe for construction of Hom • (M, M i ) in what follows.For each V ∈ Fin q (g) an invariant projector dominant (with respect to k) weight ξ = 3 s=1 i s µ s with i = (i s ) 3 s=1 ∈ Z 3 + and set ζ = ξ + λ.The Verma module Mζ of highest weight ζ and highest vector 1 ζ has singular vectors F is+1 s 1 ζ , where Fs = f s , s = 1, 3, and invariants in C[G]⊗X under right translations.The group G acts on Γ(G/K, X) ≃ (C[G]⊗X) K by left translations.