The wonderful compactification for quantum groups

In this paper, we introduce a quantum version of the wonderful compactification of a group as a certain noncommutative projective scheme. Our approach stems from the fact that the wonderful compactification encodes the asymptotics of matrix coefficients, and from its realization as a GIT quotient of the Vinberg semigroup. In order to define the wonderful compactification for a quantum group, we adopt a generalized formalism of $\mathsf{Proj}$ categories in the spirit of Artin and Zhang. Key to our construction is a quantum version of the Vinberg semigroup, which we define as a $q$-deformation of a certain Rees algebra, compatible with a standard Poisson structure. Furthermore, we discuss quantum analogues of the stratification of the wonderful compactification by orbits for a certain group action, and provide explicit computations in the case of $\mathrm{SL}_2$.


Introduction
New geometric perspectives on familiar representation-theoretic objects often illuminate deep unifying structures among diverse phenomena, and lead to the resolution of long-standing algebraic problems. This paper synthesizes two distinct sources of insight in representation theory: (1) the study of a group 'at infinity,' that is, of compactifications of a group, and (2) the role of quantum groups as symmetries in noncommutative geometry. The interaction of these perspectives leads to results on the asymptotics of quantum groups, and to a deeper understanding of various central objects in representation theory. To place our work in context, we recall that a given semisimple algebraic group has a distinguished compactification, known as the 'wonderful' compactification. It is a projective variety, introduced by de Concini and Procesi, that links the geometry of the group with the geometry of its partial flag varieties and Levi subgroups [11]. In addition, it captures the equivariant degenerations of the group and encodes the asymptotics of matrix coefficients. Related to the wonderful compactification of a group is the associated Vinberg semigroup, introduced by Vinberg [32]. The latter is an affine variety that (generically) forms a multicone over the wonderful compactification. As such, the Vinberg semigroup can be regarded as a linear version of the wonderful compactification where various structures simplify, and where the rational degenerations of the group become more apparent.
The wonderful compactification has recently emerged as a powerful tool in geometric representation theory. For example, in the setting of p-adic groups, Bezrukavnikov and Kazhdan have illustrated how the wonderful compactification leads to a geometric understanding of the second adjointness theorem [6] (see also [28]). Drinfeld and Gaitsgory use the Vinberg semigroup in order to establish adjunctions that relate categories of D-modules on certain moduli stacks of bundles; their results contribute to the geometric Langlands program [13]. The Vinberg semigroup is also crucial in the proof of a dimension formula for a group version of affine Springer fibers, due to Bouthier [7]. The wonderful compactification has been used in the theory of character sheaves by several authors [5,16,18,24,30], while Lu, Yakimov, and others have studied the Poisson geometry of the wonderful compactification and related varieties [20,21].
Another, similarly fruitful, source of new perspectives in representation theory is the study of noncommutative geometry emanating from quantum groups. Since the inception of quantum groups by Drinfeld and Jimbo in the 1980s, much work has been devoted to the construction of q-deformations of classical varieties in order to understand categories of representations. These constructions often take the form of q-deformations of algebras (that is, as global rather than formal quantizations) and pivot on the structure of the quantum group as a q-deformation of the universal enveloping algebra of a Lie algebra. Just like quantum groups themselves, the q-deformations that appear in representation theory have remarkable connections to various other areas of mathematics. For example, there are strong parallels between the behavior of an object's q-deformation when q is a root of unity and the geometry of the classical object in positive characteristic.
The spirit of quantum geometric representation theory is embodied in work of Backelin and Kremnitzer on quantum flag varieties and differential operators, as well as related results by Lunts and Rosenberg and by Tanisaki [2,23,31]. Their work demonstrates that the category of quasicoherent sheaves on the flag variety of a reductive group admits a q-deformation that can be regarded as the category of sheaves on the quantum flag variety, and is a noncommutative projective scheme in the sense of Artin and Zhang [1]. Moreover, quantum versions of differential operators on flag varieties encode representations of quantum groups, giving rise to a quantum version of the Beilinson-Bernstein localization theorem.
In this paper, we take inspiration from previous work on quantum flag varieties in order to establish the wonderful compactification for quantum groups. Our approach is ultimately rooted in Peter-Weyl theorem and the asymptotics of matrix coefficients, and employs the formalism of noncommutative projective schemes. Moreover, we consider quantum versions of the wonderful compactification's stratification by G × G orbits and its associated Vinberg semigroup. Our perspectives relate directly to the aforementioned quantum flag varieties.
In the next section, we give an overview of the key properties of the wonderful compactification and the Vinberg semigroup. In Section 1.2, we explain the context behind our construction of the wonderful compactification for quantum groups. We list the main results of this paper in Section 1.3, before providing further motivation for our work in the form of on-going and future projects in Section 1.4.

What is the wonderful compactification?
Let G be a connected semisimple group over C, and let G ad = G/Z(G) denote the adjoint group of G. The wonderful compactification is a certain projective variety G ad that contains G ad as an open subvariety. We recall the precise definition of G ad in Section 3. For the purposes of this overview, here we simply highlight the key properties of G ad .
(1) The variety G ad is stratified by the orbits of a G × G action. These orbits are indexed by subsets I of the set Δ of positive simple roots.
(2) The orbit corresponding to I = Δ is a copy of the adjoint group G ad with G × G action by left and right multiplication. This is the unique open orbit.
(3) The orbit corresponding to I = ∅ is the square of the flag variety, G/B × B − \G, with G × G action induced by outermost left and right multiplication. This is the unique closed orbit.
(4) The other orbits are related to partial flag varieties, Levi subgroups, rational degenerations of G, and wonderful compactifications of groups of smaller rank. The rich structure of the orbits distinguishes the wonderful compactification from other compactifications † of G ad .
Example 1.1. The wonderful compactification of PSL 2 is CP 3 . However, for n 3, the wonderful compactification of PSL n is not CP n 2 −1 , and is somewhat more difficult to describe explicitly.
The wonderful compactification captures the asymptotics of matrix coefficients in the following way. The Peter-Weyl theorem asserts that the coordinate algebra O(G) is spanned by matrix coefficients of representations of G. The fact that the isomorphism classes of irreducible finite-dimensional representations of G are labeled by dominant weights leads to the definition of a (multi-)filtration on O(G) by the weight lattice Λ of G. We refer to this filtration as the Peter-Weyl filtration on O(G). (1) The (multi-)Rees algebra of O(G) for the Peter-Weyl filtration.
(2) The total coordinate ring, or Cox ring, of the wonderful compactification G ad . In particular, the Picard group of G ad is identified with Λ. ( We recall the definition of the Vinberg semigroup in Section 3. For now, one can think of V G as the affine variety associated to either of the isomorphic rings in (1) or (2) of Theorem 1.2. As the name indicates, the variety V G carries a canonical semigroup structure, and the group of units is, up to a finite group, the direct product of G with a maximal torus T . The weight lattice Λ of G can be realized as the character lattice of T , and so the Λ-grading on O(V G ) corresponds to an action of T on V G . precipitates the construction, for any given semisimple group, of a distinguished smooth stack that compactifies the group. If the group has trivial center, this stack coincides with the wonderful compactification.

What is the wonderful compactification for quantum groups?
A key tenet of algebraic geometry, due to Grothendieck, asserts that a space can be completely understood through its category of sheaves. Furthermore, q-deformations of the category of sheaves can be viewed as categories of sheaves on a (nonexistent) quantum version of the original space. Thus, replacing a space with a category provides more flexibility in producing deformations; this is starting point of much of noncommutative geometry.
Applying this philosophy to the case of the wonderful compactification, we seek a category QCoh q (G ad ) that forms a q-deformation of the category QCoh(G ad ) of quasicoherent sheaves on the wonderful compactification G ad . Here q is a nonzero complex number that is not a root of unity.
To obtain the desired q-deformation, we take inspiration from a result of Serre, which describes quasicoherent sheaves on a projective variety in terms of graded modules for the homogeneous coordinate ring. Specifically, let A = n∈Z A n be commutative graded ring, and let X be the associated projective scheme with twisting sheaf O(1). A graded A-module is called torsion if every element is annihilated by A N for some N . The quotient of the category of graded modules for A by the full subcategory of torsion modules produces an abelian category denoted Proj(A). There is a functor of graded global sections: Theorem 1.5 [29]. If A is finitely generated by elements of degree 1 over a field, then the functor Γ * of graded global sections is an equivalence of categories.
The definition of Proj(A) works just as well when A is a noncommutative graded ring. Moreover, rather than just considering rings graded by the integers, one can consider rings graded by the weight lattice Λ of a semisimple group G. For such a ring R, one can make sense of a torsion graded R-module, and form the quotient Proj(R) of the category of graded R-modules by the subcategory of graded torsion modules.
We now outline the construction of the quantum wonderful compactification.
(1) Use Theorem 1.2 to establish an equivalence between the category QCoh(G ad ) of quasicoherent sheaves on the wonderful compactification and the category Proj(O(V G )), where O(V G ) is the Λ-graded algebra of functions on the Vinberg semigroup. ( (3) Define the category of quasicoherent sheaves on the quantum wonderful compactification as the category QCoh q (G ad ) := Proj(O q (V G )).

Main results
Fix q to be a nonzero complex number that is not a root of unity. A version of the Peter-Weyl theorem holds for the quantum coordinate algebra O q (G) and we obtain a filtration on O q (G) by Λ, which we refer to as the Peter-Weyl filtration. Observe that the algebra O q (V G ) carries a grading by Λ. As it is a q-deformation of the total coordinate ring of the wonderful compactification, we make the following definition, which recovers the category of quasicoherent sheaves on G ad when q = 1. Definition 1.7 (Definition 4.2). The category of quasicoherent sheaves on the quantum wonderful compactification is defined as Recall that the variety G ad is stratified by the orbits of a G × G action, and these orbits are indexed by subsets of the set of positive simple roots. Given such a subset I, we write Orb I for the corresponding orbit, and Λ I for the sublattice of Λ spanned by the roots in I. There is a filtration on O q (G) by the quotient lattice Λ/Λ I , which is a coarser filtration than the Peter-Weyl filtration. Let gr I (O q (G)) denote the associated graded algebra.
A subset I of positive simple roots determines a parabolic subgroup P = P I of G and an opposite parabolic P − . These have a common Levi group L, whose Lie algebra is denoted l. Let u and u − be the Lie algebras of the unipotent radicals of P and P − . Theorem 1.9 (Theorem 4.13). There is an isomorphism of Λ/Λ I -graded algebras Consequently, the quantum orbits can also be described in terms of certain graded subalgebras of O q (G × G), which are given as invariants for a quantum group action. When I consists of all positive simple roots, the subalgebra in question is the 'diagonal' copy of O q (G). At the other extreme, when I = ∅, the multihomogeneous coordinate ring of the classical orbit G/B × B − \G is the algebra of functions on the asymptotic cone (G/N × N − \G)/T , which quantizes precisely to O q (G × G) Uq(n×t×n − ) . By Theorem 1.9, this algebra is isomorphic to the full associated graded algebra gr(O q (G)) for the Peter-Weyl filtration.

Motivation and future directions
This paper forms the first step in a program to understand the role of the wonderful compactification in quantum geometric representation theory. We remark on several goals in this program that provide motivation for the current work.
In joint on-going work with Ben-Zvi and Nadler, we aim to place the Beilinson-Bernstein localization theorem within the framework of the wonderful compactification and asymptotics of matrix coefficients. Precursors to our work appear in work of Ben-Zvi and Nadler, and of Emerton, Nadler, and Vilonen [4,14]. We expect some of our techniques, together with the newly introduced quantum wonderful compactification, to place the quantum Beilinson-Bernstein theorem within the same framework of asymptotics of matrix coefficients. This approach requires the development of the notion of quantum differential operators on the Vinberg semigroup, and an analogue of the Verdier specialization functor for quantum D-modules.
Another source of motivation is the development of the theory of quantum character sheaves as a contribution to the study of harmonic analysis on quantum groups. The theory of character sheaves for quantum groups will involve: (1) The construction of the appropriate q-deformation of the Hecke category of Borel equivariant D-modules on the flag variety. Related constructions appear in work of Backelin and Kremnitzer on quantum flag varieties [2].
(2) The elevation of various links between character sheaves and the wonderful compactification (see, for example, [5]) to the quantum level.
(3) An alignment with the quantum geometric Langlands program. In particular, we expect direct relations with topological field theories arising from quantum groups and their categories of representations [3].
Another future direction is the consideration of the case when the quantum parameter q = is a root of unity. In this case, the quantum coordinate algebra O (G) is finite dimensional over its center, and contains the unquantized coordinate algebra O(G) as a central sub-Hopf algebra. It is reasonable to expect the category QCoh (G ad ) to be the category of modules for a certain sheaf of algebras A on G ad , and that A has an Azumaya locus related to double Bruhat cells.

Outline
We now describe the contents of this paper. Section 2 contains preliminary material, including background on matrix coefficients for Hopf algebras (Section 2.1) and notation for algebraic groups and quantum groups (Section 2.2). Section 2.3 introduces the formalism of Proj categories for noncommutative, multi-graded algebras, including the notion of torsion graded modules, compatible Poisson structures, and the relation to quasicoherent sheaves. Section 3 gives an expository account of the construction of the Vinberg semigroup (Section 3.1) and wonderful compactification (Section 3.2). Although there are no new results in Section 3, our approach is somewhat more algebraic than that of other authors, and provides insight on the quantum case. We describe the stratification of the wonderful compactification by G × G orbits in Section 3.3, explain Poisson structures on the Vinberg semigroup and wonderful compactification in Section 3.4, and examine the case of SL 2 in Section 3.5.
Section 4 forms the heart of this paper. We begin the section with the construction of the quantum Vinberg semigroup and the quantum wonderful compactification (Section 4.1). We introduce filtrations on the quantum coordinate algebra O q (G) in Section 4.2 and explain the connection to quantum flag varieties. Finally, we use these filtrations to describe the quantum orbits in Section 4.3, which is perhaps the most technical section of this paper.
Section 5 examines the general constructions of Section 4 in the case of G = SL 2 . We include basic background (Section 5.1) followed by a description of the Peter-Weyl filtration for O q (SL 2 ) (Section 5.2). We state results on the quantum Vinberg semigroup and wonderful compactification for SL 2 in Section 5.3.

Preliminaries
This section collects background, notation, and other preliminary material that will be used in subsequent sections. The reader may skip this section on first reading and refer to it as necessary. Throughout the remainder of this paper, unless specified otherwise, the ground field is C.
Lemma 2.4. We collect the following basic properties of matrix coefficients.
In particular, the coproduct on

) As an H × H-module, H • is isomorphic to the directed union of the matrix coefficients for finite-dimensional irreducible H-modules M :
Let F be a family of finite-dimensional H-modules, and letF denote the closure of F under finite direct sums and tensor products.

Algebraic groups and quantum groups
Let G be a connected semisimple algebraic group over C with Lie algebra g. Definition 2.6. Define a partial order on Λ W by setting μ λ whenever λ − μ is a nonnegative multiple of positive simple roots. Similarly, we write λ < μ if λ μ and λ = μ. This partial order is referred to as the dominance ordering on the weight lattice Λ W .
The weight lattice Λ G of G is the character lattice X * (T ) of the maximal torus. We have inclusions of lattices: Λ R ⊆ Λ G ⊆ Λ W . The set of isomorphism classes of finite-dimensional irreducible representations of G are in bijection with points in the cone Λ + G = Λ + W ∩ Λ G of dominant weights for G. We denote by V λ the irreducible representation corresponding to λ ∈ Λ + G . Points in the interior of Λ + G are called regular dominant weights for G. In a context where the group G is fixed, we write Λ instead of Λ G .
Lemma 2.7. If V ν appears as an irreducible subrepresentation of the tensor product V λ ⊗ V μ , then ν λ + μ for the dominance ordering.
Given a subset I ⊆ Δ, denote by P I the parabolic subgroup of G whose Lie algebra p I is generated by b and the root vectors corresponding to the roots −α i for i ∈ I. Let U I be the unipotent radical of P I (with Lie algebra u I ), and let L I denote the subgroup of P I whose Lie algebra l I is generated by t and the root vectors corresponding to the roots ±α i for i ∈ I. Then L I is a maximal reductive subgroup of P I , and is called a Levi subgroup of G. We have P I = U I L I . Similarly, we define the opposite parabolic P − I and its unipotent radical U − I . Observe that P I ∩ P − I = L I . Write p − I and u − I for the corresponding Lie algebras. We will be interested in the so-called standard Lie bialgebra structure on g, described in [10, Example 1.3.8]. It is a coboundary Lie bialgebra structure determined by a certain skewsymmetric element r ∈ 2 g that satisfies the classical Yang-Baxter equation. Write O(G) for the coordinate algebra of G. For f ∈ O(G), we write X L f and X R f for the action of X ∈ g on f by left-and right-invariant vector fields, respectively.
Remark 2.9. The same formula gives a Poisson-Lie structure on G for any r ∈ 2 g that defines a coboundary Lie bialgebra structure on g. We note that when g is simple, all Lie bialgebra structures are coboundary [10, Example 2.1.7].
The standard Lie bialgebra structure on g admits a quantization, which leads to the quantized enveloping algebra U q (g). For the definition of U q (g), we follow [9, Chapter I.6; 19, Chapter 6]. Fix q ∈ C × . We will assume throughout that q is not a root of unity. Let C = (a ij ) be the Cartan matrix of g.
Proposition 2.11. The algebra U q (g) has the following Hopf algebra structure: Definition 2.12. We define the following subalgebras of U q (g): We will be interested exclusively in type 1 representations of U q (g). Let C q (g) denote the category of finite-dimensional U q (g)-modules of type 1.
Proposition 2.13. The category C q (g) is a semisimple rigid tensor subcategory of U q (g)-mod whose irreducible objects are in bijection with the set Λ + W of dominant weights of g.
For λ ∈ Λ + W , we write V λ for the corresponding irreducible representation of U q (g). Let C q (G) denote the subcategory of C q (g) generated under finite direct sums and tensor products by the irreducible representations whose highest weights lie in Λ + G . Using Lemma 2.5, we make the following definition.
Definition 2.14. The quantized coordinate algebra O q (G) of G is defined as sub-Hopf algebra of U q (g) • generated by the matrix coefficients of all modules in C q (G).
The algebra O q (G) is a flat deformation of the algebra O(G), and quantizes the Poisson-Lie structure on O(G) arising from the standard Lie bialgebra structure on g.
Lemma 2.15. If V ν appears as an irreducible subrepresentation of the tensor product V λ ⊗ V μ , then ν λ + μ in the dominance ordering on Λ W .

Λ-graded algebras
This section collects definitions and results concerning Proj categories for noncommutative rings graded by a lattice Λ. We present reformulations and refinements of constructions that appear in work of Artin and Zhang, and of Ginzburg, and are ultimately inspired by results of Serre [1,17,29].
Let Λ be a lattice, that is, a finitely generated torsion-free abelian group. Let Λ + be a subsemigroup of Λ (that is, a cone). We assume that Λ + has the following properties: (1) For any λ 1 and λ 2 in Λ, the intersection ( Definition 2.16. A Λ-graded algebra is a Λ-graded vector space R = λ∈Λ R λ over C equipped with an associative C-linear multiplication map R ⊗ R → R that restricts to a map R λ ⊗ R μ → R λ+μ for any λ, μ ∈ Λ. When Λ = Z and Λ + = Z 0 , we recover the usual notion of a graded algebra. The setting we will be most interested in is when Λ = X * (T ) is the weight lattice of G with respect to a maximal torus T , and Λ + is the cone of dominant weights.
Definition 2.17. Let R be a Λ-graded algebra.
(1) A graded left R-module is a Λ-graded vector space M = λ∈Λ M λ equipped with an action of R such the action map restricts to a map R λ ⊗ M μ → M λ+μ . The category of graded left R-modules is denoted Grmod(R) and its objects will henceforth be referred to simply as R-modules. ( The category of finitely generated graded left R-modules is denoted grmod(R), and it is a full subcategory of Grmod(R).
Remark 2.18. We will mostly be interested in the case where R is noetherian and locally finite, that is, R λ is finite dimensional over C for every λ ∈ Λ. When R is noetherian, the category grmod(R) is abelian.

Lemma 2.20. If M is finitely generated, then M is torsion if and only if there exists
General results on the localization of abelian categories (see [26,Sections 4.3,4.4]) allow us to make the following definition, and justify the subsequent proposition.
Proposition 2.24. The category Proj(R) is abelian. If R is noetherian, then the category proj(R) is abelian and noetherian.
We conclude this section with a discussion of the connection between Proj categories and quasicoherent sheaves on projective varieties. For the remainder of this section, suppose that R is a commutative Λ-graded algebra, and suppose that Λ = X * (T ) is the character lattice of a torus T . Thus, Spec(R) is a scheme with an action of the torus T .
Definition 2.25. The GIT quotient of Spec(R) by T at the character λ ∈ Λ, denoted Spec(R) / / λ T , is the projective scheme associated with the Z 0 -graded algebra n 0 R nλ : Notation 2.26. Let A be a Z 0 -graded ring. We write Proj(A) for the projective scheme associated to A, and Proj(A) for the category of graded A-modules modulo torsion modules. In other words, roman font indicates a space, while sans serif font indicates a category.

The wonderful compactification
In this section, we give an exposition of the construction of the Vinberg semigroup and the wonderful compactification. We present the main constructions in Sections 3.1-3.4, and devote Section 3.5 to a detailed discussion of the example of SL 2 .

The Vinberg semigroup
Fix a connected semisimple algebraic group G over C with Lie algebra g. We denote by U g the universal enveloping algebra of g, abbreviate the weight lattice Λ G by Λ, and adapt notation from Section 2.2.
defines an isomorphism of U g-bimodules.
While the Peter-Weyl theorem endows O(G) with a grading by Λ, the algebra structure on O(G) does not respect this grading. Instead, invoking the dominance order on Λ from Definition 2.6, we obtain a Λ-filtered algebra: The fact that we obtain a filtered algebra is a consequence of Lemma 2.7 and basic properties of matrix coefficients. We refer to this filtration on O(G) as the Peter-Weyl filtration, and note that it is not a filtration just by the cone Λ + of dominant weights. Indeed, there are nondominant weights λ, such that the set {μ ∈ Λ + | μ λ} is nonempty, and hence the subspace O(G) λ is nonzero. This is true, for example, for any positive simple root.
Let C[Λ] denote the group algebra of Λ as an abelian group; it is generated by formal variables z λ with relations z λ z μ = z λ+μ , for λ, μ ∈ Λ.
Henceforth, we use Rees(G) and O(V G ) interchangeably. Lemma 3.4 implies that V G is a semigroup with an action of G × G. One can show that the group of units of V G is the quotient of G × T by the antidiagonal multiplication action of the center Z(G) of G. is the abelianization map of [32]. We record the following basic observations.
• Since Λ = X * (T ) is the character lattice of the maximal torus T of G, the algebra of functions O(T ) on T is precisely the group algebra C[Λ].
• • The map π : Example 3.7. The Vinberg semigroup for SL 2 is the semigroup of two by two matrices, and the map π is the determinant map, as explained in Section 3.5.

The wonderful compactification
In this section, we describe two realizations of the wonderful compactification. The first is based on work of De Concini and Springer [12], as well as an exposition due to Evens and Jones [15], and proceeds as follows. Fix an irreducible representation V = V λ of G of regular highest weight λ ∈ Λ + . Consider the commutative diagram: where the left horizontal arrow is the action map, the right vertical arrow is the quotient by the free action of C × , and the remaining solid arrows represent obvious maps. The existence of the map ψ relies on the fact that the center Z(G) acts on V by scalars. A consideration of the weightspace decomposition of V is used to prove the following lemma:   Example 3.12. The wonderful compactification of PSL 2 is P 3 (see Section 3.5 for more details). However, for n 3, the wonderful compactification of PSL n is not P n 2 −1 .
To explain a second realization of the wonderful compactification of G ad , we first observe that, since Λ = X * (T ) is the character lattice of the maximal torus T of G, the Λ-grading on the Rees algebra O(V G ) endows V G with a T -action which commutes with the G × G-action. We have the following result: Proposition 2.27 implies the following corollary of the preceding theorem: Corollary 3.14. There is an equivalence of categories: Remark 3.15. The total coordinate ring, or Cox ring, of G ad is precisely O(V G ) (see [8]).

Orbits in the wonderful compactification
In this section, we describe the G × G orbits on the wonderful compactification. Fix a subset I ⊆ Δ and consider the corresponding parabolic subgroup P I , its opposite P − I , and its Levi L I . There are projection maps pr L : P I → L I and pr L − : P − I → L I , and each of these compose to a map valued in L ad I = L I /Z(L I ).
Proposition 3.16 [15]. We have: (1) the G × G orbits on G ad are in bijection with subsets of Δ. Write Orb I for the orbit corresponding to I ⊆ Δ. For subsets I and J of Δ, the containment Orb I ⊆ Orb J holds if and only if I ⊆ J; (2) there is a point in Orb I whose stabilizer is the subgroup (3) Let L ad I denote the wonderful compactification of the adjoint group L ad I of L I , and let Orb I denote the closure of Orb I in G ad . There are fibrations: Example 3.17. In the extreme cases, Orb Δ = G ad is the unique open orbit, and Orb ∅ = G/B × B − \G is the unique closed orbit. When G = SL 2 , there are only two G × G orbits on G ad = P 3 , and they are the extreme ones. In the SL 3 case, there are four orbits. The two nonextreme orbits each form PSL 2 -bundles over P 2 × P 2 , and the closure of each forms a P 3 -bundle over P 2 × P 2 .
Let e I be the point in A whose ith coordinate is zero if i / ∈ I and 1 otherwise. Recall the T -equivariant map π : V G → A defined in Section 3.1. The preimage π −1 (T · e I ) of the T -orbit of e I is an affine subvariety of V G with an action of T .
Proposition 3.18. Fix a regular dominant weight λ ∈ Λ + . For each subset I ⊆ Δ, the orbit Orb I is the GIT quotient of π −1 (T · e I ) by T at λ: Proof. The space A is the union of the T -orbits of e I , as I ranges over the subsets of Δ, and these orbits are pairwise disjoint. Therefore, the Vinberg semigroup V G is the union of the subvarieties π −1 (T · e I ). Since the action of T on V G commutes with that of G × G, each of these subvarieties carries an action of G × G. Using Theorem 3.13, we see that the wonderful compactification G ad is the union of the disjoint G × G-equivariant subvarieties π −1 (T · e I ) / / λ T , as I ranges over subsets of Δ. The result follows from the fact that the G × G orbits on G ad are in bijection with subsets of Δ (Proposition 3.16).
The algebra of functions O(π −1 (T · e I )) on π −1 (T · e I ) is a Λ-graded algebra. Proposition 2.27 implies the following corollary of the preceding proposition: Corollary 3.19. The category of quasicoherent sheaves on Orb I is equivalent to the Proj category for the Λ-graded algebra O(π −1 (T · e I )), that is:

Poisson structures
In this section, we describe how the standard Poisson-Lie structure on the group G leads to a Poisson structure on the Vinberg semigroup V G , on the wonderful compactification G ad , and on each orbit G × G orbit Orb I in G ad . We adopt the notation of Section 2.2.
Thus, the actions X R and X L preserve the space of matrix coefficients for a given representation. The result now follows from Proposition 2.8.  Remark 3.23. The methods of the proof can be used to show that, at least when G is simply connected, any Lie bialgebra structure on g leads to a Poisson structure on the Vinberg semigroup, on the wonderful compactification, and on any orbit. Poisson structures on the wonderful compactification have been studied extensively by Lu and Yakimov [20,21]. In [21], they consider a standard Poisson structure on G ad , which coincides with the one constructed in Proposition 3.22. Indeed, both are algebraic and restrict to the Sklyanin bracket on G ad .

The case of SL 2
In this section, we describe in detail the constructions of the previous sections for the case of SL 2 .
Identify the maximal torus of SL 2 with C × , the weight lattice Λ SL2 with the integers Z, the cone of dominant weights Λ + SL2 with the cone of nonnegative integers, and the root lattice Λ R with the sublattice 2Z generated by the positive simple root α 1 = 2. A dominant weight (that is, nonnegative integer) n is regular if and only if n is nonzero. The partial order on Z from Definition 2.6 becomes m n if and only if n − m is a nonnegative multiple of 2.
For any nonnegative integer n, write V n = Sym n (C 2 ) for the irreducible representation of SL 2 of highest weight n. As a special case of Lemma 2.7, we have that, if V k appears as an irreducible representation of the tensor product V n ⊗ V m , then k n + m.
The algebra of functions O(SL 2 ) is given by O(SL 2 ) = C[a, b, c, d]/(ad − bc = 1). This is a Hopf algebra with We will use the same notation for elements of C[a, b, c, d] and their images in O(SL 2 ).

Theorem 3.24 (Peter-Weyl Theorem). The map of matrix coefficients gives an
The proof of the following lemma is straightforward.
For example: 3 ) = Span{degree 3 and degree 1 monomials} Recall from Section 2.3 our convention that the notation 'Proj' in roman font indicates a projective scheme (that is, a space), while 'Proj' in sans serif font indicates a category.
Proposition 3.26. The associated graded algebra gr(O(SL 2 )) is the homogeneous coordinate ring of P 1 × P 1 . Consequently, we have an isomorphism of varieties: bc). The latter is isomorphic as a graded algebra to n Sym n (C 2 ) ⊗ Sym n (C 2 ) via the map a → x ⊗ w, b → x ⊗ z, c → y ⊗ w, and d → y ⊗ z, where x, y are the coordinate functions in the first copy of Sym • (C 2 ) and w, z are the coordinates in the second copy. Now, Sym n (C 2 ) ⊗ Sym n (C 2 ) is the space of global sections of the line bundle O(n, n) = O(1, 1) ⊗n on P 1 × P 1 . The line bundle O(1, 1) is ample since it is the pullback of O P 3 (1) under the Segre embedding P 1 × P 1 → P 3 . Therefore, there is an isomorphism O(1, 1) ⊗n )).
Proposition 3.27. We have the following.
(1) The Vinberg semigroup V SL2 is isomorphic to the semigroup Mat 2 of two by two matrices, and the action of C × on V SL2 coincides with the scaling action on Mat 2 .
(3) The wonderful compactification of PSL 2 is P 3 = Proj(O(V SL2 )). (4) The stratification of PSL 2 = P 3 into SL 2 × SL 2 orbits is given by: Proof. The algebra of functions on the Vinberg semigroup V SL2 is given by the Rees algebra O(V SL2 ) = n 0 O(SL 2 ) n z n . Lemma 3.25 and the relation z 2 = (az)(dz) − (bz)(cz) together imply that the subspace O(SL 2 ) n z n coincides with the span of monomials with k 1 + k 2 + k 3 + k 4 n (under the usual partial order on Z). Hence, O(V SL2 ) is a commutative algebra on the four generators az, bz, cz, dz, and no relations. It is straightforward to verify that the coproduct of O(V SL2 ) coincides with that of O(Mat 2 ). Since V SL2 = Spec(O(V SL2 )), this proves the first statement.
The second statement follows from the relation z 2 = (az)(dz) − (bz)(cz). Fix the regular dominant weight n = 1, regarded as a character of the maximal torus T = C × of SL 2 . The third statement follows from the computation: To prove the last statement, first note that each SL 2 × SL 2 orbit on Mat 2 \ {0} has the form det −1 (d) for some d ∈ C, where det : Mat 2 \ {0} → C is the determinant map. If d = 0, then one uses the C × -action on Mat 2 \ {0} to identify det −1 (d) with det −1 (1) = SL 2 . It follows that the disjoint union d∈C × det −1 (d) descends to a single SL 2 × SL 2 -orbit in PSL 2 , and it is isomorphic to PSL 2 . On the other hand, if d = 0, then we have: The second identification is given by sending pair of nonzero vectors (x 1 , x 2 ) and (y 1 , y 2 ) to the matrix whose (i, j) entry is x i y j for i, j ∈ {1, 2}. The C × action on Mat 2 \ {0} preserves this space, and the quotient is P 1 × P 1 .
We give another perspective on the SL 2 × SL 2 orbits on the wonderful compactification PSL 2 = P 3 = Proj(O(V SL2 )). First note that O(V SL2 ) contains z 2 , but does not contain z. We have corresponds to the (free) action on GL 2 by its center. Thus, A standard argument shows that O(V SL2 )/(z 2 ) = gr(O(SL 2 )). By Proposition 3.26, we have that Proj(O(V SL2 )/(z 2 )) = P 1 × P 1 . We see that the decomposition (3.2) becomes precisely the decomposition of Proposition 3.27.
Remark 3.28. The orbit P 1 × P 1 includes in PSL 2 = P 3 as the Segre embedding. We

The wonderful compactification for quantum groups
This section is the heart of the paper, in which we define the quantum coordinate algebra O q (V G ) of Vinberg semigroup and the category of sheaves on the quantum wonderful compactification. We adapt notation from previous sections, especially Section 2.

Main definitions
Let G be a connected semisimple algebraic group over C with Lie algebra g.
where the sum ranges over irreducible objects in C q (g) with highest weights in Λ + G = Λ + . The subspaces for λ ∈ Λ, endow O q (G) with the structure of a Λ-filtered algebra.
Proof. The first statement is a consequence of the definition of the quantum coordinate algebra O q (G) (Definition 2.14). General properties of matrix coefficients, together with Lemma 2.15 imply that the subspaces O q (G) λ form a filtration.
Definition 4.2. We make the following definitions: (1) the quantized coordinate algebra O q (V G ) of the Vinberg semigroup for G is defined as the Rees algebra for O q (G), that is, as the following Λ-graded subalgebra of O q (G) ⊗ C[Λ]: (2) the category of quasicoherent sheaves on the quantum wonderful compactification of G ad is given by Proof. The statements about O q (V G ) follow from Proposition 4.1 and the fact that O q (G) is a flat deformation of O(G), quantizing the standard Poisson-Lie bracket. The final assertion follows from Corollary 3.14 (but ultimately from [25,29]).
Recall from Definition 3.6 that we denote by A the spectrum of the polynomial subalgebra C[z αi ] of C[z λ ] generated by the elements z αi for i ∈ Δ. The fact that α i 0 for any positive simple root α i implies the following: The choice of positive simple roots endows A with a coordinate system. Over a point where all coordinates are nonzero, the fiber of the sheaf defined by O q (V G ) is isomorphic to O q (G). The fiber at a general point is a certain 'partial' associated graded algebra for O q (G); we describe these algebras in the next two sections.

Filtrations on the quantum coordinate algebra
In this section, we consider certain filtrations on the quantized coordinate algebra O q (G) of G, and describe the associated graded algebras. These associated graded algebras define the quantum orbits on the wonderful compactification.
Given a subset I ⊆ Δ of positive simple roots let Λ I = Z{α i | i ∈ I} ⊆ Λ R denote the sublattice of the root lattice spanned by the roots in I. Write [λ] or [λ] I for the image of λ ∈ Λ in Λ/Λ I . Define a partial order on Λ/Λ I by The coproduct Δ restricts to a map Δ : Proof. The fact that the subspaces O q (G) [λ]I form a filtration follows from general properties of matrix coefficients, together with Lemma 2.7. For λ ∈ Λ, the [λ] I -th graded piece of the associated graded algebra is given by The final claim is a consequence of Lemma 2.4(3).
At the other extreme, if I = Δ, then gr Δ (O q (G)) is isomorphic as an algebra to O q (G), and its grading coincides with the grading of O q (G) by the finite group Λ/Λ R (which can be naturally identified with the dual of the center Z(G) of G): Observe that O q (G) and gr I (O q (G)) are isomorphic as U q (g)-bimodules. The algebra gr I (O q (G)) is a 'partial' associated graded algebra, and its multiplication map can be described more explicitly as the composition of the ordinary multiplication map with the projection onto the partial sum of the images of those V ν ⊗ V * ν such that λ + μ − ν lies in Λ I .
Recall that e I denotes the point of A whose ith coordinate is zero if i / ∈ I and 1 otherwise (Section 3.3), and that there is a map π : V G → A whose preimage over the T -orbit of e I recovers the orbit Orb I as a GIT quotient (Proposition 3.18).  Proof. The first statement follows from standard Rees algebra formalism, noting that the algebra gr I (O q (G)) ⊗ C[Λ I ] carries a natural grading by (Λ/Λ I ) × Λ I = Λ. The second and third statements follow from the corresponding statements about O q (V G ) and the Vinberg semigroup, see Proposition 4.3.
Definition 4.11. For I ⊆ Δ, the category of quasicoherent sheaves on the quantum Orb I is given by We conclude this section by explaining the relation between the full associated graded algebra gr ∅ (O q (G)) = gr(O q (G)) and the quantum flag variety of Backelin and Kremnitzer [2]. The quantum coordinate algebra O q (G) for G is a comodule for the quantum coordinate algebra Theorem 4.12. There is an equivalence of categories

The quantum orbits
In this section, we provide a different description of the partial associated graded algebras gr I (O q (G)), and hence of the category of sheaves on each quantum orbit. Fix a subset I ⊆ Δ. Throughout, we abbreviate l I by l and u I by u. The map is an injective morphism of algebras, and we henceforth identify U q (u × l × u − ) with its image in U q (g × g). We consider the following two commuting actions of , given in terms of matrix coefficients.
• The 'internal' action: ( We consider the restriction of the internal action to the subalgebra U q (u × l × u − ). The resulting space of invariants O q (G × G) Uq(u×l×u − ) carries an external action of U q (g × g).
Theorem 4.13. For any I ⊆ Δ, there is a U q (g × g)-equivariant isomorphism of algebras The proof of this theorem requires some set-up.
Definition 4.14. Let V be an irreducible module for U q (g) with highest weight vector v 0 . We denote by V I the U q (l)-submodule of V generated by v 0 , that is, Lemma 4.15. We collect the following facts.
(1) The space V I is the sum of weight subspaces of V with weights that differ from the highest weight by Λ I .
(2) The subspace V I of V coincides with the U q (u I )-invariants: Proof. By definition, the space V I is obtained from v 0 by successive application of the operators F i for i ∈ I. Thus, the weight vectors that appear in V I are precisely those whose weights differ from the highest weight by elements of Λ I , and the first claim is established. It is straightforward to use the PBW theorem for U q (g) to show that V I ⊆ V Uq(uI ) . The opposite inclusion follows from the description of V I given in the first statement, thus proving the second claim. For the third claim, recall that, if v 0 is a highest weight vector in V, then the dual vector v * 0 is a lowest weight vector in V * . Now, . Another way to prove the third claim is to use the second claim and match weight subspaces. For the last claim, one considers the action of the K i in U q (l) and the action of the quantized enveloping algebra U q (l/z(l)) of the semisimple Lie algebra l/z(l), where z(l) denotes the center of l.
Proof. Write x = j v j ⊗ w j . Without loss of generality, we can assume that v j and w j are weight vectors of weights wt(v j ) and wt(w j ). Then wt(v j ) + wt(w j ) = ν for all j. We can write λ − wt(v j ) = i∈Δ n i α i and μ − wt(w j ) = i∈Δ m i α i for some integers n i , m i 0. Now, The far right-hand side lies in Λ I if and only if n i = m i = 0 for i / ∈ I, which in turn is true if and only if v j ∈ (V λ ) I and w j ∈ (V μ ) I for all j.
for the decomposition of the tensor product V λ ⊗ V μ into irreducibles. (1) as a U q (l)-module, the tensor product (V λ ) I ⊗ (V μ ) I is isomorphic to the direct sum of Proof. For each part of the lemma, we identify each of the two modules in question as a subspace of V λ ⊗ V μ and prove inclusions in both directions.
Suppose that x ∈ V λ ⊗ V μ is the highest weight vector of a U q (g)-submodule isomorphic to V ν , where λ + μ − ν ∈ Λ I . By Lemma 4.16, x lies in (V λ ) I ⊗ (V μ ) I . So the U q (l)-submodule generated by x is contained in (V λ ) I ⊗ (V μ ) I . We conclude that (V λ ) I ⊗ (V μ ) I contains, as a U q (l)-submodule, the direct sum of (V ν ) ⊕N ν λμ I for weights ν such that λ + μ − ν ∈ Λ I . It is an immediate consequence that the U q (g)-submodule W of V λ ⊗ V μ generated by (V λ ) I ⊗ (V μ ) I contains the direct sum of V ⊕N ν λμ ν for weights ν that λ + μ − ν ∈ Λ I . We have established one inclusion for each of the two claims.
For the opposite inclusions, observe that (V λ ) I ⊗ (V μ ) I is U q (p I )-stable, so any element in W lying outside of (V λ ) I ⊗ (V μ ) I is obtained by applying the action of F i for i ∈ Δ \ I to elements in (V λ ) I ⊗ (V μ ) I . Hence, the resulting elements have lower weights than those in (V λ ) I ⊗ (V μ ) I . It follows that all highest weight vectors of W are contained in (V λ ) I ⊗ (V μ ) I , and this establishes the second inclusion of the second claim. The second inclusion of the first claim now follows from Lemma 4.16, which implies that any weight vector of (V λ ) I ⊗ (V μ ) I has weight ν satisfying λ + μ − ν ∈ Λ I .
Proof of Theorem 4.13. The invariants in O q (G) for the right (respectively, left) action of U q (u I ) (respectively, U q (u − )) can be expressed as: The residual right action of U q (l) on O q (G) Uq(uI ) is on the second factor, and the residual left action of U q (l) on Uq(u − I ) O q (G) is on the first factor. Therefore, we have the following isomorphisms of U q (g)-bimodules: Thus, a different basis for this space is given by e j ⊗ e (μ) where j = 1, . . . , dim((V λ ) I ), and = 1, . . . , dim((V μ ) I ). Consequently, the above expression is equal to: where the multiplication in the last two expressions occurs in Example 4.18. In the extreme cases we have The former is the quantized coordinate algebra of the asymptotic semigroup AsG of G; the asymptotic semigroup is defined in [27,33].
Remark 4.19. We make the following remarks.
(1) Bezrukavnikov and Kazhdan observe the classical version of the result of Theorem 4.13 in [6, Remark 2.9].
(2) See [15, Section 2.5] for results analogous to Lemma 4.15 in the classical case.

The case of SL 2
In this section, we describe the constructions of Section 4 in the case when G = SL 2 .
5.1. The algebras O q (SL 2 ) and U q (sl 2 ) Fix q ∈ C × . We assume that q is not a root of unity. The following discussion is adapted in part from [9].
Definition 5.1. The quantum 2 × 2 matrix algebra is the bialgebra O q (Mat 2 ) generated over C by elements a, b, c, d with relations ab = qba ac = qca bc = cb bd = qdb cd = qdc ad − da = (q − q −1 )bc, and with coalgebra structure given by We use the same notation for elements of O q (Mat 2 ) and their images in O q (SL 2 ) and O q (GL 2 ). Lemma 5.3. The bialgebra structure on O q (Mat 2 ) descends to a bialgebra structure on O q (SL 2 ) and O q (GL 2 ). Each of the latter bialgebras is a Hopf algebra with antipode is given by: Definition 5.4. The quantized enveloping algebra U q (sl 2 ) of sl 2 is generated by elements E, F, K ±1 subject to the relations We omit the definitions of the coalgebra structure and antipode on U q (sl 2 ) from this summary (but see Proposition 2.11). For any nonnegative integer n, there is a unique finite-dimensional irreducible U q (sl 2 )-module of type 1, which we denote V n .

The Peter-Weyl filtration on O q (SL 2 )
In the previous section, we gave a definition of O q (SL 2 ) without reference to matrix coefficients; one can show that definition coincides with the more general Definition 2.14. More precisely: Theorem 5.5 [9,Theorem I.7.16]. The sub-Hopf algebra of U q (sl 2 ) • generated by the matrix coefficients of the representations V n is isomorphic to the Hopf algebra O q (SL 2 ) of Definition 5.2. Consequently, there is an isomorphism of U q (sl 2 )-bimodules:

Thus, we have
and no inclusions between the two strings. The following lemma is a straightforward verification; the proof of each part is parallel to its classical analogue.
Lemma 5.7. We have the following.
(1) The spaces of Definition 5.6 define a filtration on O q (SL 2 ): (2) The coproduct Δ restricts to a map (3) For n ∈ Z, consider the subspace of the free algebra C a, b, c, d spanned by monomials words of length k where k n (under the usual partial order on Z) and k ≡ n mod 2. The image of this subspace under the quotient map C a, b, c, d → O q (SL 2 ) is precisely O q (SL 2 ) n .
Definition 5.8. We define the following algebras.
• Let Sym k q (C 2 ) denote the kth graded piece of the algebra C x, y / xy − qyx , and set P 1 q × P 1 q = k Sym k q (C 2 ) ⊗ Sym k q (C 2 ).
• The associated graded algebra of O q (SL 2 ) is defined as the Z-graded algebra gr(O q (SL 2 )) = n O q (SL 2 ) n /O q (SL 2 ) <n .
Setting q = 1 in the definition of P 1 q × P 1 q , we obtain the homogeneous coordinate ring of P 1 × P 1 . Note that P 1 is the flag variety for SL 2 .
Proposition 5.9. The associated graded gr(O q (SL 2 )) is isomorphic to P 1 q × P 1 q .
Proof. Observe that there are isomorphisms where the second isomorphism is given by Here (x, y) and (u, w) denote the coordinates on the first and second copies of C 2 , respectively. Proof. The proof is a routine computation. Note that O q (V SL2 ) is generated as an algebra by O q (SL 2 ) 0 = C · 1 and O q (SL 2 ) 1 z = {az, bz, cz, dz}. We set Δ(az) = az ⊗ az + bz ⊗ cz, Δ(bz) = az ⊗ bz + bz ⊗ dz, etc.  Proof. The proof is analogous to the proof in the classical case. By Lemma 3, an element of O q (SL 2 ) n can be represented by a word in a, b, c, d of length k where k n (in the usual partial order on Z) and k ≡ n mod 2. Similarly, an element of O q (SL 2 ) n z n can be represented by such a word together with a factor of z n . Using the commutation relations between the