A wonderful embedding of the loop group

Advances in Mathematics 313 (2017) 689–717

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Advances in Mathematics www.elsevier.com/locate/aim

A wonderful embedding of the loop group Pablo Solis Caltech, Pasadena CA, United States

a r t i c l e

i n f o

Article history: Received 16 June 2015 Received in revised form 9 October 2016 Accepted 12 October 2016 Communicated by Tony Pantev Keywords: Loop groups Affine Lie algebras Moduli of G bundles on curves Embeddings of reductive groups Representation theory Spherical varieties

a b s t r a c t I describe the wonderful compactification of loop groups. These compactifications are obtained by adding normalcrossing boundary divisors to the group LG of loops in a reductive group G (or more accurately, to the semi-direct product C×  LG) in a manner equivariant for the left and right C×  LG-actions. The analogue for a torus group T is the theory of toric varieties; for an adjoint group G, this is the wonderful compactification of De Concini and Procesi. The loop group analogue is suggested by work of Faltings in relation to the compactification of moduli of G-bundles over nodal curves. © 2016 Elsevier Inc. All rights reserved.

1. Introduction The wonderful compactification of a complex semisimple adjoint group G was constructed and named by De Concini and Procesi in [7]. The compactification is a smooth projective variety containing G as a dense open sub variety and the boundary is a normal crossing divisor whose structure is determined by the root datum of G. The wonderful compactification has seen a wide range of applications particularly in spherical geometry. Here we construct an analogue of the wonderful compactification for the loop group LG of G. The loop group is the group of maps from a punctured formal disc to the E-mail address: [email protected]. http://dx.doi.org/10.1016/j.aim.2016.10.016 0001-8708/© 2016 Elsevier Inc. All rights reserved.

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group G; it’s C-points are G(C((z))) where C((z)) is the field of formal Laurent series. The embedding we construct is denoted X af f ; strictly speaking X af f is not compact but it does have many nice properties making it worthy of being called the loop group analogue of the wonderful compactification. The main application of our construction concerns the moduli stack MG (C) of G bundles on a family of nodal curves C. This moduli stack is not compact and we use X af f to compactify this stack in [18].1 In fact, one of the earliest attempted applications of De Concini and Procesi’s wonderful compactification of G was precisely to compactify MG (C), or rather its coarse moduli space of semistable bundles. For the special case of G = GLn (C) see the work of Kausz [12] and Seshadri and Nagaraj [16]. There are compactifications for other semisimple groups but this has not lead to a satisfactory construction of a compact moduli space of bundles over families of nodal curves. We claim a compactification G of G is insufficient to compactify MG (C) in general. Moreover, we claim the embedding X af f does provide enough additional data to compactify MG (C). We discuss briefly a modular interpretation of X af f in section 5.4. A thorough development of the modular interpretation behind X af f is addressed in [18]. The motivation to compactify MG (C) comes from work of Frenkel, Teleman and Tolland [23] where a compactification of the moduli of C× -bundles is used to define Gromov–Witten invariants for [pt/C× ]. The index theorem of Teleman and Woodward [20] suggest similar invariants could be defined with an adequate completion of MG (C). The parallels between X af f and the wonderful compactification Gad suggests that other constructions related to Gad may also have loop analogues. In [24], Vinberg gave an alternative construction of Gad using a monoid SG , often called the Vinberg monoid. This construction was generalized by Thaddeus and Martens [15] to provide stacky compactifications for any split reductive group. In light of Theorem 5.1, it is natural ask if there is a loop group analogue of the Vinberg monoid. The recent work of [17] makes this even more compelling. The wonderful compactification has also been a key object of study in geometric representation theory. For example the compactification can be used to define the HarishChandra transform defined on the category of D-modules on G. Similar constructions may exist for D modules on LG. Some hints of this appear in [5]. Another possible area for exploration is elliptic Springer theory [3]. It is an analogue of Lustig’s character sheaves for loop groups aimed at providing a geometric construction of character sheaves for p-adic groups. One wants to study sheaves on conjugacy classes in LG. Using Baranovsky and Ginzburg’s theorem in [1] this can be framed as studying sheaves on the moduli space MG,E of semistable bundle on an elliptic curve, preferable as the latter is finite dimensional. 1 We only show one can obtain a complete moduli stack; to get something separated a stability condition must be imposed.

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As conjugacy classes in LG are the same as Δ(LG) orbits in LG, we ask if Δ(C× LG) orbits in X af f admit a modular interpretation which is a degeneration of the Baranovsky and Ginzburg interpretation. An affirmative answer could be useful in formulating a theory of character sheaves for loop groups. The referee has pointed out that De Concini and Procesi’s wonderful compactification has been used recently in work of Bezrukavnikov–Kazhdan [4] to give a geometric description of Bernstein’s second adjointness for p-adic groups. It is thus reasonable to think that a similar construction using X af f could be used to lift this construction to the geometric setting. 1.1. LG and main results For any ring R we form the ring R((z)) of formal Laurent series with coefficients in R. ∞ Elements of R((z)) are formal sums i=i0 ri z i with ri ∈ R and where the start of the sum i0 can be any integer. If G is an algebraic group over C then the loop group is a functor from C-algebras to groups which assigns to a C-algebra R the group G(R((z))) of R((z)) valued points of G. G Example 1. Consider   = SL2 (C) and R = C. Then the group LSL2 (C) consists of 2 × 2 a b matrices γ = where the entries are in C((z)) and ad − bc = 1. These can be c d equivalently represented as formal sums γ(z) =

∞  i=i0



 ai ci

bi di

zi

ai , bi , ci , di ∈ C.

The determinant condition translates into a sequence of polynomial conditions on the coefficients ai , bi , ci , di . Any matrix group G ⊂ GLn (C) admits a similar description. LG is not a scheme but rather an ind-scheme; an increasing union of schemes LG = ∪k≥0 (LG)k with each (LG)k infinite dimensional. For example, given a faithful representation G ⊂ GLn (C) one can take (LG)k to be formal sums as in the example with i0 = −k. LG is studied through its representations. The remarkable fact is that LG admits a class of projective representations which behave in many ways like the finite dimensional representations of a semisimple group. These are projective representations of LG, or  of LG by C× . Any such equivalently honest representations of a central extension LG representation V is infinite dimensional but by introducing an additional C× to form  V decomposes into a direct sum of finite dimensional weight Gaf f (C) := C×  LG, spaces for a maximal torus in Gaf f (C).  Therefore it is Gaf f (C) := C×  LG(C) whose representation theory mimics that of a finite dimensional semisimple group. The group Gaf f (C) is called a Kac–Moody group

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and it is associated to an affine Dynkin diagram in the same way a semisimple group is associated to a Dynkin diagram. To see the parallel between the wonderful embedding of LG and the wonderful compactification of G we briefly recall a construction of the latter. Strictly speaking the latter is a compactification of Gad := G/Z(G) where Z(G) is the center of G. Fix a regular dominant weight λ and let V (λ) be the associated highest weight representation of G then set Gad = G × G[id] ⊂ PEnd(V (λ)).

(1)

Then Gad is called the wonderful compactification of Gad . In an analogous fashion we begin with a regular dominant weight λ (notation explained in section 4) of Gaf f and using the associated representation V (λ) we construct an ind-scheme PEndind (V (λ)) and our desired embedding is X af f = Gaf f × Gaf f [id] ⊂ PEndind (V (λ)). f af f Then X af f is an ind-scheme that contains Gaf /Z(Gaf f ) = C×  LG/Z(G) ad := G as a dense open sub ind-scheme and carries an equivariant action of C×  LG × C×  LG.

Theorem 5.1. Let G be a simple, connected and simply connected group over C with f maximal torus T and set r = dim T . The ind-scheme X af f contains Gaf ad as a dense open sub-ind scheme and further (a) X af f is formally smooth and independent of the choice of regular dominant weight λ = (0, λ, l). f (b) The boundary X af f − Gaf ad is a Cartier divisor with r + 1 components D0 , . . . , Dr . (c) Each Di is formally smooth and ∪i Di is locally isomorphic to a product S × Z where Z is the union of the hyper planes in Ar+1 . f  af f (d) The torus Tad := C× ×T /Z(G) ⊂ Gaf which ad has a partial compactification in X    r+1 ∼ A is an affine toric variety Tad,0 = and there is a bijection between Tad × Tad orbits in Ar+1 and L G × L G orbits in X af f . In particular, the L G × L G orbit closures are given by subsets I ∈ {0, . . . , r} such that to I we associate ∩i∈I Di . Parts (a), (b), (d) are direct analogues for X af f of properties of the Gad (with the only mismatch being that smoothness of Gad is replaced by formal smoothness of X af f ). Morally part (c) says that the boundary ∪ri=0 Di is a normal crossing divisor which is also a property of Gad .    The toric variety Tad,0 is an open subset of the full closure Tad . Then Tad is described by a fan with infinitely many cones coming from the Weyl alcove decomposition:

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 ∼  Proposition 5.1. Tad,0 is smooth and its = C[t−α0 , . . . , t−αr ] ∼ = Ar+1 . In particular Tad,0  fan is given by the cone on the negative Weyl alcove −Al0 ; the fan Tad is given by the   cone on the Weyl aclove decomposition of tR = Lie(Tad )R .

The statement about the Weyl alcove is a direct generalization that the closure Tad of Tad = T /Z(G) ⊂ Gad in Gad is a toric variety with fan the Weyl chamber decomposition of Lie(T )R . Moreover Tad also appears as the closure of a generic T orbit in G/B. It is  not quite the case that Tad also appears as the closure of a generic C× × T /Z(G) orbit in the flag variety of LG but an ind-type interpretation of this result does hold; see [19]. Let Orb(I) denote L G×L G orbit corresponding to I. Each I determines a parabolic or parahoric subgroup PI ⊂ LG with a Levi decomposition PI = LI UI . Each LI is a finite dimensional reductive group. Proposition 5.2. For I ⊂ {0, . . . , r} let Orb(I) be the L G ×L G orbit in X af f according to Theorem 5.1(d). Then there is a surjective map π : Orb(I) → Gaf f /PI × Gaf f /PI− and all the fibers are isomorphic to LI,ad . The above result draws further parallels between X af f and the wonderful compactification of Gad . 1.2. Contents In section 3, we define the loop group LG as well as two closely related groups C× LG and Gaf f as well as polynomial versions of these groups. We describe their Lie algebras, define the Weyl alcove decomposition and discuss parahoric subgroups associated to co-characters of a maximal torus. In particular, for any field K we state the Birkhoff decomposition of the set LG(K). The special cases K = C and K = C((t)) are used in subsequent results. Subsection 3.1, takes a closer look at the Lie algebra Lsl2 of LSL2 . No results are proved however it does detail by example how copies of sl2 ⊂ Lsl2 can be used to study highest weight representations of LSL2 . The same strategy works for general G and is used in the proof of Proposition 4.3. Hopefully the example also makes the notation more transparent. Section 4 discusses three types of vector spaces: countable dimensional highest weight representations V of Gaf f , the dual V ∗ , and End(V ). In general V is an ind-variety, V ∗ is an infinite dimensional scheme, and End(V ) is only a sheaf of sets however it contains an ind-scheme Endind (V ) which will suffice for the construction of X af f . We also discuss the two versions of flag varieties for LG corresponding to opposite parahorics subgroups P, P − : LG/P is a projective ind-variety and embeds in PV and LG/P − is an infinite dimensional schemes that embeds in PV ∗ but is not proper.

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In section 5, we consider the Gaf f × Gaf f orbit of the identity [id] in PEndind (V ), the closure of this orbit is X af f . We state and prove Theorem 5.1 we also compute P ic(X af f ) (Proposition 5.8) and show it is free of rank r +1 with r = rank(G). We discuss the orbit closures in X af f and prove Proposition 5.2. We also discuss the polynomial version X poly of X af f . The main difference is the boundary of X poly is proper whereas the boundary of X af f is not. In the final subsection we discuss the modular interpretation of one of the orbits in X af f . 2. Notation All groups will be defined over C. For example, we write SLr to indicate the C-scheme 2 {det = 1} ⊂ ArC . We write H ⊂ G for a closed sub algebraic group of G over C. We make the following definition for any field k but in most cases we take k = C. Let Aff k be the site of k-algebras equipped with the f ppf -topology. Following [2], we define a k-space as a sheaf of sets on Aff k and a k-group is defined as a sheaf of groups. A morphism between k-spaces F → G is a map of sheaves. The category of schemes over k forms a full subcategory of the category of k-spaces. Many of the k-spaces we consider will be ind-schemes: increasing union of schemes ∪i Xi with the additional data of closed immersions Xi → Xi+1 . Other k-spaces come from vector spaces. Let V be an abstract vector space over k. Consider V as a k-space via V (R) := V ⊗k R. We denote by V ∗ the k-space homR (V (R), R), similarly End(V ) becomes a k-space via End(V )(R) := EndR (V (R)). The group of units in End(V ) gives a k-group GL(V ). Let Gm ⊂ GL(V ) denote the scalars and take P GL(V ) to be the quotient k-group GL(V )/Gm . Precisely, P GL(V ) is the sheaf associated to the presheaf R → GL(V )(R)/Gm (R). If V is countable dimensional then any choice of ordered basis determines an equivalent ind-scheme structure V = ∪i Vi [13, 4.1.3(4)]. Then PV = ∪i PVi is a projective indvariety. Moreover if V is countable dimensional then V ∗ is a scheme as V ∗ = Spec S • (V ) where S • (V ) is the symmetric algebra on V . If V is given degree 1 then S • (V ) is graded and we set PV ∗ = Proj S • (V ). In section 4 we make a similar definition of PEndind (V ). 3. Preliminaries on loop groups Let G be a simple algebraic group over C with π1 (G) = π0 (G) = 1. The loop group LG is the C-group given by R → LG(R) := G(R((z))). We call elements of LG(R) loops. The functor LG has an ind-scheme structure. For G = SLr this is shown in [2, 1.2]; in this case loops are matrices and the ind-scheme structure LSLr = ∪k (LSLr )k is defined so that (LSLr )k (R) consists of loops γ such that both γ, γ −1 have entries in R((z)) with ∞ poles of order ≤ k; that is, each entry is of the form i=−k ai z i where each ai ∈ R.

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For general G one gets an ind scheme structure on LG via an embedding G → SLr . The ind-scheme structure does not depend on the choice of representation [14, 3.7]. We will soon define two closely related ind-C-groups L G and Gaf f . They are related f af f  af f by the identification Gaf by ad = L G/Z(G) where Gad represents the quotient of G its center and Z(G) is the center of G. Precisely L G(R) := Gm (R)  LG(R) where  indicates that uγ(z)u−1 := γ(uz) for u ∈ Gm (R), γ ∈ LG(R); we call this Gm action loop rotation. Finally, Gaf f is a central extension of L G by Gm : 1 → Gm → Gaf f → L G → 1. The group Gaf f is constructed in [13, sect. 13]. A description of Gaf f can be given in terms of the affine grassmannian GrG ; the general properties of GrG are discussed in section 4 but for the moment it suffices to know that GrG is a projective ind-variety which is a homogeneous space for LG. Moreover GrG has an ample line bundle O(1) which generates P ic(GrG ) = Z. Then there is a central extension  → LG → 1 1 → Gm → LG ∼  such that LG(R) = {(g, τ )|g ∈ LG(R), τ : g ∗ O(1) −→ O(1)}. The loop rotation action  and Gaf f = Gm  LG.  As we are working over C we’ll often write C× for Gm . lifts to LG  The group L G is considered an ind-scheme by taking L G = ∪i (C×  (LG)i ). Similarly Gaf f surjects onto L G with fiber C× . The inverse image (Gaf f )i of C× (LG)i defines an ind-scheme structure Gaf f = ∪i (Gaf f )i . The maximal torus of Gaf f is T af f := C× × T × C× c where the subscript c indicates af f af f C× is central in G . Characters of T are denoted as μ = (n, μ, l) ∈ Z ⊕ ΛT ⊕ Z; c we use μ when stating the values n, l are not essential. Sometimes we abbreviate T  := f  = T  /Z(G) ⊂ Gaf C× × T ⊂ L G and Tad ad . Let g = Lie(G). The Lie algebra Lg of LG is an ind-C-space defined by

Lg(R) = {γ ∈ LG(R[]/2 ))|γ = 1 mod }. We similarly have the Lie algebras L g(C) and gaf f (C) of L G, Gaf f which are ind-C-spaces. In particular, Lg(C) = g ⊗C C((z)), gaf f (C) = Cd ⊕ Lg(C) ⊕ Cc where c is central d and  [d, − ] acts by z dz . The full Lie algebra structure can be described explicitly. If t X s

∈ Cd ⊕ Lg(C) ⊕ Cc then ⎞ ⎞ ⎛ ⎞⎤ ⎛ d1 d2 0 ⎟ ⎢⎜ ⎟ ⎜ ⎟⎥ ⎜ ⎣⎝ Xz n ⎠ , ⎝ Y z m ⎠⎦ = ⎝ [X, Y ]z n+m + d2 nXz n + d1 mY z m ⎠ , c1 c2 δn,−m X, Y ⎡⎛

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where , is the Killing form. With Lie(T ) = t, then taf f := Cd ⊕ t ⊕ Cc is a Cartan subalgebra. Let g = N− ⊕ t ⊕ N be the triangular decomposition of g then we have the triangular decomposition for gaf f : gaf f = n− ⊕ taf f ⊕ n,  n− = N− g ⊗ z −1 C[z −1 ],  n=N g ⊗ zC[[z]].

(2)

The weights appearing here are the roots of Gaf f also called affine roots; they are all of the form z k X with X ∈ g. If α1 , . . . , αr are the simple roots of G then the simple affine roots of Gaf f are α1 = (0, α1 , 0), . . . , αr = (0, αr , 0), α0 = (1, −θ, 0) where θ is the longest root of G. The positive roots of L G are those of the form (k, α, 0) with k > 0 or with k = 0 and α positive. We set tQ = hom(C× , T ) ⊗Z Q. The roots define linear forms on Q ⊕ tQ and restricting to 1 ⊕ tQ we obtain affine linear forms on tQ . For α = 0 we can define affine hyperplanes in tQ via Hk,α = {ζ ∈ tQ |α(ζ) = −k} = {ζ| (1, ζ, 0), (k, α, 0) = 0}. The complement of all the Hk,α is the Weyl alcove decomposition of tQ . Let W = s1 , . . . , sr be the Weyl group of G. Let αj∨ be the simple co-roots of g. Because π1 (G) = 0 we have that the αj∨ span hom(C× , T ) and si (αj∨ ) := αj∨ − αi (αj∨ )αi∨ defines an action of W on hom(C× , T ). ∨ The affine co-roots are αi∨ = (0, αi∨ , 0) and α∨ 0 = (0, θ , 1). In the same way the group af f × × ∼ W = s0 , . . . , sr = W hom(C , T ) acts on hom(C , T af f ) and preserves the lattice 1 ⊕ hom(C× , T ) ⊕ 0. In particular W af f acts on 1 ⊕ tQ and permutes the Weyl alcoves; hom(C× , T ) ⊂ W af f acts by translation on 1 ⊕ hom(C× , T ). A fundamental domain is given by the positive Weyl alcove: Al0 := {ζ ∈ 1 ⊕ tQ |αi (ζ) ≥ 0, i = 0, . . . , r}. The vertices ωi∨ of Al0 are defined by αj (ωi∨ ) = δi,j . The vertices determine so called parahoric subgroups which appear in our description of the boundary of X af f ; we define parahorics now. Let p1 : T  = C× × T → C× be the projection. For any co-character η : C× → T  we say η is positive if p1 ◦ η > 0 and negative if p1 ◦ η < 0 and nonzero if it is either positive or negative. A nonzero η defines a C-group as follows:   Pη (R) = γ ∈ L G(R)| lim η(s)γη(s)−1 ∈ L G(R) . s→0

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The limit condition can also be phrased as: there exists n > 0 such that η(s)γη(s)−1 ∈ L G(R[s1/n ]). In particular, Pη is defined even for η ∈ hom(C× , T  ) ⊗Z Q and in particular for η ∈ Al0 . For G = SLr the condition η(s)γη(s)−1 ∈ L G(R[s1/n ]) is equivalent to the vanishing of certain matrix entries hence the Pη give sub ind-C-groups of L SLr . Choosing an embedding G → SLr we conclude the same for general G. For convenience we refer to the Pη as simply subgroups. A parahoric subgroup of L G is any subgroup P  ⊂ L G that is conjugate by an element of L G(C) to some Pη . We set Pη = Pη ∩ (1 × LG) and P ⊂ LG is a parahoric subgroup if it is conjugate to some Pη ; if η is positive then we write Pη− := P−η . The groups Pη come with a natural Levi decomposition Pη (R) = Lη (R)Uη (R) where Lη (R) = {γ ∈ Pη (R)| lim η(s)γη(s)−1 = γ}, s→0

Uη (R) = {γ ∈

Pη (R)|

lim η(s)γη(s)−1 = 1},

(3)

s→0

and in an analogous manner we obtain Levi decompositions of Pη− . The group Lη is a finite dimensional reductive group and if η is positive Uη is an infinite dimensional group scheme; if η is negative then Uη is and ind-variety. Now let us return to the vertices ωi∨ ∈ Al0 . For r = rk(G) and I ⊂ {0, . . . , r} we set  1 ∨ ωI∨ = #I i∈I ωi ∈ Al0 and define PI = PωI∨ ,

PI− = P−ωI∨ ,

(4)

for I = ∅ we set PI = LG. Example 2. Take G = SLr and I = {0}. Then ωI∨ (s)γ(z)ωI∨ (s)−1 = γ(sz). Thus PI = C×  G(R[[z]]) and PI (R) = G(R[[z]]). For another example take I = {0, . . . , r} and let B ⊂ G be the Borel subgroup of upper triangular matrices. Then B(R) := P{0,...,r} (R) = {γ(z) ∈ L+ G(R)|γ(0) ∈ B(R)}. The group B is also called an Iwahori subgroup. Example 2 generalizes. For any reductive G we have P{0} (R) = G(R[[z]]) and we − typically write P{0} = L+ G; also P{0} = G(R[z −1 ]) =: L− G(R). Fixing a Borel B ⊂ G gives a similar description of B = P{0,...,r} as B(R) = {γ(z) ∈ L+ G(R)|γ(0) ∈ B(R)} and − similarly for B − := P{0,...,r} . We let U be the unipotent factor in the Levi decomposition (3) of B and define U − similarly. From the point of view of flag varieties L+ G is the loop analogue of a maximal parabolic subgroup of G and B is a loop analogue of B ⊂ G. The above applies to Lpoly G(R) = G(R[z ± ]). One simply replaces R((z)) and R[[z]] with R[z ± ] and R[z] throughout.

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Fix a set I and opposite parahorics PI , PI− . It is known that there is a discrete set WI and a surjection W af f → WI such that LG(C) =

P

− I (C)wPI (C).

w∈WI

For I = {0} we have WI = hom(C× , T ) and the above decomposition is known as the Birkhoff factorization of LG(C). Another special case is I = {0, . . . , r} in which case WI = W af f . The definition of WI is given in [13, 1.3.17] and the proof of the decomposition is [13, 5.2.3(g)]. It turns out these factorizations hold more generally. By [13, sect. 5.2], there is a factorization result for any group that satisfies a series of axioms [13, Def. 5.2.1] (refined Tits system). In [21, sect. 15.2], Tits gives a different set of axioms which generalize refined Tits system and in [22, pg. 570(c)] Tits shows that split Kac–Moody groups over any field satisfy the axioms in [21, sect. 15.2]. It follows that for any field K we have LG(K) =

P

− I (K)wPI (K).

(5)

w∈WI

3.1. Lsl2 : an extended example We give here a partial description of some highest weight representations of Lsl2 . This section can be skipped but maybe helpful to get some intuition about the general case. For the sake of simplicity we work with polynomial versions Lsl2 = sl2 ⊗C C[z ± ], f slaf = Cd ⊕ Lsl2 ⊕ Cc. 2

Let X, Y, H be the standard basis for sl2 with [H, X] = 2X, [X, Y ] = H and [H, Y ] = −2Y . Let , denote the Killing form normalized so that H, H = 2.; in this case X, Y = 1. f For any v ∈ sl2 set v(n) = v ⊗ z n . Then c, d, X(n), H(m), Y (l) form a basis for slaf 2 . Write in particular, sl2 (n) for sl2 ⊗C z n , then we have the triangular decomposition     af f sl2 = ⊕n<0 sl2 (n) ⊕ CY Cd ⊕ CH ⊕ Cc ⊕n>0 sl2 (n) ⊕ CX = n− ⊕ taf f ⊕ n+ , and the restriction of the bracket to Lsl2 ⊕ Cc is [A(n), B(m)] = [A, B](n + m) + nδn,−m A, B . We have [X(n), Y (−n)] = H + nc and X(n), H + nc, Y (−n) generate a copy denoted sn f of sl2 in slaf 2 .

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Highest weight representations for LSL2 are labeled by characters (0, n, l) of taf f with n ≤ l. The defining properties of the highest weight representation V with highest weight (0, n, l) is that there is a vector v ∈ V such that n+ v = 0 (H + c)v = (n + l)v and V is generated by the images of v under the repeated action of n− . The vector v is a highest weight vector for sm of highest weight n + ml. In particular V contains arbitrarily large sl2 representations. Further, if v is any other weight vector in v then there is an integer j such that X(j)v = 0 and so v is a highest weight vector for sm for any m ≥ j. 4. Representation theory and flag varieties We discuss a class of projective representations of L G that have similar properties to highest weight representations of a semisimple group. These representations come from honest representations of Gaf f and are labeled by weights λ = (0, λ, m) of the maximal af f torus T  × C× . c ⊂G If ω1 , . . . , ωr are the fundamental weights of G then the fundamental weights of Gaf f are ω 0 = (0, 0, 1), ω 1 := (0, ω1 , 1) . . . , ω r := (0, ωr , 1). A dominant weight is any weight r of the form i=0 ni ω i with ni ≥ 0. A dominant weight is regular if all ni > 0. The integer l is called the level. Associated to any dominant weight λ = (0, λ, l) is a highest weight representation V (λ) of the group Gaf f (C). The representation V (λ) is infinite dimensional but decomposes  under T  ×C× c into a direct sum of finite dimensional weight spaces V (λ) = n,μ V(n,μ,l) . Any vector v(0,λ,l) in V(0,λ,l) is called a highest weight. One choice of ind structure on  V = V (λ) is to take V = ∪i Vk with Vk = j≤k V(j,μ,l) ; however this choice is not essential. r If λ = i=0 ni ω i define I(λ) = {i|ni > 0} ⊂ {0, . . . , r}. r Proposition 4.1. Set λ = (0, λ, l) = i=0 ni ωi and I = I(λ). Let P = PI and V = V (λ). Let Z(Gaf f ) denote the center of Gaf f . Then If μ = (n, μ, l) is any other wight of V then λ − μ is a sum of positive roots. If λ is regular then λ − αi is a weight of V for all i. The stabilizer of the weight space Vλ in PV is P . If (0, λ, l) is regular then P = B.   The morphism LG P → P(V ) given by γP → γVλ is injective and gives LG P the  structure of a projective ind variety; in particular LG P is closed in P(V ).  (f) The action of Gaf f on P(V ) factors through a faithful action of Gaf f Z(Gaf f ) = L G/Z(G).

(a) (b) (c) (d) (e)

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(g) The action map Gaf f × V → V is a morphism of ind-schemes and descends to a morphism of ind-schemes L G × PV → PV . Proof. Statements (a)–(f) are proven in [13]. For (a)–(b) see [13, 2.2.1]; for (c)–(e) see [13, 7.1.2]; for (f) see [13, 13.2.8]. Now we prove (g). In [2, appendix 7] it is shown that the representation V of Gaf f is algebraic in the sense that there is a morphism of C-groups LG → P GL(V ); this can be lifted to a morphism of C-spaces Gaf f → GL(V ) and Gaf f × V → V ; see also [9]. Any morphism from a scheme into an ind-scheme factors through a subscheme of the ind-scheme therefore for each k there is an n(k) such that we have a morphism of schemes (Gaf f )k × Vk → Vn(k) . This descends to (L G)k × PVk → PVn(k) . 2 4.1. Dual representation and LG/P − Let V be a countable dimensional vector space over k. Then as explained in section 2, we have infinite dimensional schemes V ∗ and PV ∗ . The group GL(V ) acts on each graded piece of the symmetric algebra S • (V ) giving an action of GL(V ) on V ∗ which descends to an action of P GL(V ) on PV ∗ . Let us now take k = C and V to be a highest weight representation. Let [v] ∈ PV (C) be the class of the highest weight vector. We have seen the LG orbit of [v] is an ind flag variety LG/P . Projection onto the highest weight space defines a point [v ∗ ] of PV ∗ (C). It is natural to expect the LG orbit of [v ∗ ] to be the flag variety LG/P − . To set this up, let V be a projective highest weight representation of LG. The map of C-groups LG → P GL(V ) gives an action LG × PV ∗ → PV ∗ . The point [v ∗ ] ∈ PV ∗ (C) gives rise to an R point also denoted [v ∗ ] for every C algebra R. Define a C-space Stab([v ∗ ]) ⊂ LG by setting Stab([v ∗ ])(R) to be the stabilizer of [v ∗ ] in LG(R). Stabilizing [v ∗ ] is a closed condition so Stab([v ∗ ]) is a closed ind-subgroup: Stab([v ∗ ]) = ∪k Stab([v ∗ ])k with Stab([v ∗ ])k ⊂ (LG)k . Now suppose V = V (λ) with λ a dominant weight and I = I(λ); recall I(λ) was defined before Proposition 4.1. Proposition 4.2. Let λ be a dominant weight and set I = I(λ) and P = PI , P − = PI− and let U , L, U − be the factors in the Levi decomposition of P, P − (e.g. (3)). Let V = V (λ) and [v ∗ ] the class of a lowest weight vector in PV ∗ (C). Then (1) (2) (3) (4)

Stab([v ∗ ]) = P − . The multiplication map P − × U ∼ = U − × P → LG is an open immersion. The LG orbit of [v ∗ ] carries a scheme structure. The orbit of [v ∗ ] gives an injective map LG/P − → PV ∗ which is injective on tangent spaces.

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Proof. (1): By [13, 6.1.16, 6.1.17] the group LG is generated by standard parahoric subgroups Pi for i ∈ {0, . . . , r}. Let Pi = Li Ui be the Levi decomposition. Then Ui ⊂ U and as Lie(U) acts nontrivially on [v ∗ ] we see that Stab([v ∗ ]) is generated by Stab([v ∗ ]) ∩ Pi ⊂ Li ⊂ Lpoly G hence Stab([v ∗ ]) ∩ Lpoly G = Stab([v ∗ ]). By 4.1(c) we get Stab([v ∗ ]) = P −. (2): Let R be a C algebra and g ∈ LG(R). From Proposition 4.1 we have Stab([v]) = P (R). The element v ∗ ∈ V ∗ (R) defines a function from PV (R) → {0, 1}. The set of g ∈ LG such that v ∗ [gv] = 0 is an open subscheme Ω. In particular g stabilizes the element [v ∗ ⊗ v] ∈ P(V ⊗ V ∗ ). Thus Ω is the image of the multiplication map Stab([v ∗ ])(R) × Stab([v])(R) → LG which is P − (R)P (R). As P − (R) ∩ P (R) = L(R) we get Ω ∼ = U− × L×U ∼ = U− × P ∼ = P− × U. (3): We use the fact that U P − is open in LG thus we can consider U[v ∗ ] as an open subscheme of LG[v ∗ ] isomorphic to U . We can translate this scheme structure to cover LG[v ∗ ]. (4): We observe LG acts transitively on LG/P − so it is enough to check the condition at the identity. Then Tid LG/P − ∼ = Lg/Lie(P − ) ∼ = Lie(U ) and the injectivity of this map follows because each root subgroup of U acts nontrivially on [v ∗ ]. 2 Neither LG/P − or PV ∗ are proper over C. However, we can show LG/P − → PV ∗ is a closed immersion. Proposition 4.3. Let λ be a dominant weight and set I = I(λ) and P − = PI− . Let V = V (λ) and [v ∗ ] the class of a lowest weight vector in PV ∗ (C). Then the orbit of [v ∗ ] determines a closed immersion i : LG/P − → PV ∗ . Proof. We have that LG/P − → PV ∗ is a monomorphism so we just need to show it is proper. We have that i is separated because it is a monomorphism and PV ∗ is separated. So we just need to show i is complete; i.e. it satisfies the existence part of the valuative criterion for properness. Set R = C[[t]] and K = C((t)). Suppose we have a diagram Spec K

Spec R

g

LG/P −

PV ∗

By passing to an nth root of t we can lift g to a point g ∈ LG(C((t1/n ))). We can establish the result if we can show g extends to a C[[t1/n ]] points of LG/P − . This is equivalent to showing gP − (C((t1/n ))) contains a C[[t1/n ]] points of LG. Keeping track of the root t1/n is immaterial so by abuse of notation we work only with R and K-points. The action of g is [v ∗ ] → [v ∗ ]g −1 and by the Birkhoff factorization (5) we can write g = uwp− ∈ U (K)W af f P − (K) with U the unipotent factor of P . Therefore we can

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assume g = uw. Moreover U ∼ = Spec C[x1 , x2 , . . . ] so a point u ∈ U (K) is an arbitrary sequence of coordinates in K. Finally we claim that [v ∗ ]w−1 u−1 extends to an R point of PV ∗ if and only if there are only finitely many coordinates of u that are not in R. If there are infinitely many coordinates with values a ∈ K − R then u is a product of terms exp aXn,α with n arbitrarily large. Arguing as in section 3.1, for n >> 0 the vector [v ∗ ]w−1 is a lowest weight vector of copies sl2 generated by Xn,α , X−n,−α , [Xn,α , X−n,α ] in PV ∗ (K). This implies that [v ∗ ]w−1 u−1 is mapped into PV ∗ (K) with coordinates with arbitrarily negative exponents in t and hence this K point cannot extend to an R points of PV ∗ . If there are finitely many coordinates of u that are not in R then we can factor u = u upoly with upoly ∈ U(K) ∩ Lpoly G(K) and u ∈ U(R). Then [v ∗ ]w−1 u−1 poly is a point − of the projective ind-variety Lpoly G(K)/P (K) and thus extends to an R point. That is, there is a u ∈ upoly wP − (K) with u ∈ Lpoly G(R). Then u u ∈ uwP − (K) and u u ∈ LG(R). 2 4.2. Endomorphism spaces Let V be a highest weight representation of Gaf f . Given an ind-scheme structure V = ∪i Vi the action map Gaf f × V → V becomes a morphism of ind-schemes. In particular ∀n we have a morphism of schemes (Gaf f )i × Vi → Vn(i) . Notice in particular for any i, j ≥ 0 we have a morphism of schemes (Gaf f )i × Vj ⊂ (Gaf f )(i+j) × Vi+j → Vn(i+j) Definition 4.4. The sequence of integers (n(i))i≥0 determine an ind-subscheme of End(V ). Namely Endind (V ) = ∪i Endind i (V ) where Endind i (V )(R) = {φ ∈ End(V )(R) : φ(Vj (R)) ⊂ Vn(i+j) (R)} Apriori Endind i (V ) is just a sub C-space of End(V ) however we have Lemma 4.5. The C space Endind i (V ) is represented by an affine scheme and the union ∪i Endind (V ) is an affine ind-scheme. i Proof. Define grVi = Vi /Vi−1 and when i > j define grVji = Vi /Vj . Then elements of End(V )(R) can be depicted as N × N matrices: ⎛

a0,0 ⎜ a1,0 ⎝ .. .

a0,1 a1,1 .. .

⎞ ··· ···⎟ ⎠ .. .

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where ai,j ∈ homR (grVj,R , grVi,R ). Then for example elements of Endind 0 (V ) are of the form ⎛ ⎞ a0,1 a0,2 ··· a0,0 ⎟ ⎜ . .. ⎜ .. . ···⎟ ⎟ ⎜ ⎟ ⎜ .. ⎜ ⎟ . ···⎟ ⎜ an(0),0 ⎟ ⎜ .. ⎜ ⎟ ⎜ 0 ⎟ an(1),1 . ⎟ ⎜ ⎟ ⎜ .. ⎟ ⎜ . 0 an(2),2 ⎝ ⎠ .. . 0 Which we can identify with the infinite product Endind 0 (V ) = hom(V, Vn(0) ) ×



n(i)

hom(V, grVn(i−1) ),

i>0

where

⎛

⎞ a0,0 a0,1 ··· ⎜ ⎟ .. hom(V, Vn(0) ) = ⎝ ... . ···⎠ an(0),0 an(0),0 · · ·  n(i) hom(V, grVn(i−1) ) = 0 · · · 0 an(i),i an(i),i+1

···



Each piece is affine because for any finite dimensional vector space W we have hom(V, W ) = Spec S • (V ⊗ W ∗ ) where S • (−) denotes symmetric algebra and moreover the infinite product of affine schemes is an affine scheme. We similarly have  n(i+j) Endind hom(V, grVn(i−1+j) ). j (V ) = hom(V, Vn(j) ) × i>0 ind It is also evident that each inclusion Endind j (V ) → Endj+1 (V ) is a closed immersion. 2 ind Definition 4.6. Let S(V, j) be the coordinate ring of Endind j (V ) so that Endj (V ) = • Spec S(V, j). The ring is the co product of S (V ⊗ W ) where W ranges over the finite n(i+j) dimensional grVn(i−1+j) . There is a natural grading with elements in V ⊗ W being in degree 1. Therefore we can define PEndind j (V ) = Proj S(V, j) and we obtain another ind-scheme PEndind (V ) = ∪j PEndind (V ). j

5. The embedding We let G be a simple, connected and simply connected algebraic group over G. Using representation theory of the associated Kac–Moody group Gaf f we construct an embedding of Gaf f /Z(Gaf f ) = L G/Z(G).

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Fix a dominant weight λ = (0, λ, l) of Gaf f and let V = V (λ) be the associated highest weight representation. In Definition 4.4 we constructed an ind-scheme Endind (V ). This ind-scheme is not preserved by left and right multiplication by GL(V ) × GL(V ), however via the composition Gaf f ×Gaf f → GL(V )×GL(V ) there is a well defined action Gaf f ×Gaf f ×Endind (V ) → Endind (V ). By Proposition 4.1(g) this is moreover a morphisms of ind-schemes. The map Gaf f → GL(V ) descends to a map L G → P GL(V ). In Definition 4.6 we also defined an ind-scheme PEndind (V ) and we have a map a : L G × L G × PEndind (V ) → PEndind (V )

(6)

which is in particular a morphism of ind-schemes. Let [id] ∈ PEndind (V )(C) denote the class of the identity. f Lemma 5.1. The L G × L G orbit of [id] is an ind-scheme isomorphic to Gaf ad .

  Proof. By Proposition 4.1(f), the stabilizer of [id] is Z(G) × Z(G) Δ(L G) therefore f Gaf ad and the orbit of [id] are identified at the level of points. It remains to show the f orbit has an ind-scheme structure isomorphic to Gaf ad . For any algebraic group G let m : G × G → G be the map (g1 , g2 ) → g1 g2−1 . Then the fibers of m are Δ(G) and if U ⊂ G is any subset then m−1 (U ) = (U, 1)Δ(G). m f The above applied to the composition L G × L G −→ L G → Gaf ad and the open f  cell U − Tad U ⊂ Gaf ad shows that   (U − T  U, 1) · Z(G) × Z(G)Δ(L G) is open in L G × L G. Consequently we can take (U − T  U, 1)[id] as an open ind subscheme in the orbit of [id] isomorphic to U − Tad U. Translating this open set we get af f   ∼ L G × L G[id] = Gad as ind-schemes. 2 5.1. Definition of X af f and main theorem We require now that V is the highest weight representation associated to a regular dominant weight. We show in Theorem 5.1 that another choice of a regular dominant weight yields an isomorphic embedding. f = L G/Z(G) in By Lemma 5.1 the L G × L G orbits of [id] embeds Gaf ad af f ind  PEnd (V ). We will write Gad instead of L G/Z(G) to emphasize the parallel with the wonderful compactification. The embedding is a morphism of ind-scheme therefore for every k ≥ 0 we have a map f ind of schemes (Gaf ad )k → PEndn(k) (V ) We define an ind-scheme X af f as follows

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f ind (X af f )k := (Gaf ad )k ⊂ PEndn(k) (V )  X af f := (X af f )k

705

(7)

k

The (ind)-scheme PV, PV ∗ , PEndind (V ) all carry a line bundle O(1). Let v be the highest weight vector of V . The classes [v] ∈ PV , [v ∗ ] ∈ PV ∗ and [v ⊗ v ∗ ] ∈ PEndind (V ) are sections of O(1) and their nonvanishing define open ind-subschemes P[v] V = {[v] = ∗ 0}, P[v∗ ] V ∗ {[v ∗ ] = 0}, PEndind [v⊗v ∗ ] (V ) = {[v ⊗ v ] = 0}. Define (X0af f )k = (X af f )k ∩ PEndind [v⊗v ∗ ],n(k) (V ) X0af f = ∪k (X0af f )k

(8)

The main theorem we prove about X af f is Theorem 5.1. The basic strategy is to reduce to questions about X0af f which is easier to study. The action of the central C× ⊂ Gaf f acts trivially on X af f so the Gaf f × Gaf f action on X af f factors through an action of L G × L G which is what we work with. We have morphisms ψv : PEndind [v⊗v ∗ ] (V ) → P[v] V [φ] → [φ(v)] ∗ ψv∗ : PEndind [v⊗v ∗ ] (V ) → P[v ∗ ] V

[φ] → [v ∗ ◦ φ] Lemma 5.2. The map ψv∗ × ψv : X0af f → P[v] V × P[v∗ ] V ∗ is U − × U equivariant and the image is U − × U. Proof. The group U − preserves P[v] V and U preserves P[v∗ ] V ∗ hence U − × U preserves af f PEndind . That ψv∗ × ψv is equivariant is readily verified. [v⊗v ∗ ] (V ) and in particular X0 Finally Propositions 4.2 (2) and 4.1(d) show that the image is U − × U. 2  Let t−αi be the regular function on Tad = T  /Z(G) given by the character −αi . Let    Tad ⊂ X af f be the closure and Tad,0 = Tad ∩ X0af f .   ∼ Proposition 5.1. Tad,0 is smooth and its = C[t−α0 , . . . , t−αr ] ∼ = Ar+1 . In particular Tad,0  fan is given by the cone on the negative Weyl alcove −Al0 ; the fan Tad is given by the   cone on the Weyl aclove decomposition of tR = Lie(Tad )R .  Proof. Let S be the set of nonzero weight spaces of V = V (0, λ, l). The image of Tad in af f μ−(0,λ,l) −αi X0 is μ∈S t . By Proposition 4.1(b) we see that t appears in this product

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and moreover by 4.1(a) all other terms in the product are monomials in t−αi . The first statement follows. The second statement follows because 

 Tad =

 wTad,0 w−1

w∈W af f

and −Al0 is a fundamental domain for the action of W af f on t R.

2

The proof of Proposition 5.3 is adapted from [6, 6.1.7]. Proposition 5.3. There is a U − × U equivariant isomorphism  U − × U × Tad,0 − → X0af f a

(l, u, t) → l · t · u  Proof. First note that the restriction to U − × U × Tad is just the multiplication map and this is open by 4.2(2); consequently the morphism is birational. By Lemma 5.2, we have (πv ,πv∗ )

a morphism X0af f −−−−−−→ U − × U which is moreover U − × U equivariant.  The composition U − × U × Tad,0 → X0af f → U − × U is given by (l, t, u) → (l, u). − By U × U equivariance we reduce to the finite dimensional question of showing  (πv , πv∗ )−1 (1, 1) = Tad,0 .   As (πv , πv∗ )−1 (1, 1) is closed and contains Tad we have Tad,0 ⊂ (πv , πv∗ )−1 (1, 1). Let  −1 − p ∈ (πv , πv∗ ) (1, 1) then there exists a morphism f ∈ U Tad U(C((t)) that extends to f ∈ X0af f (C[[t]]) such that f (0) = p. Let γ = (πv , πv∗ ) ◦ f ∈ U − (C[[t]]) × U(C[[t]]). Then   γ −1 f (0) = p and (γ −1 f )|Spec C((t)) ∈ Tad (C((t))) hence p ∈ Tad,0 . 2 f Proposition 5.4. X af f is an ind scheme containing Gaf ad as a dense open sub ind-scheme. f af f Proof. By construction X af f is an ind-scheme and Gaf is ad is dense. Moreover X0 af f open in X . We show that

X af f =



gX0af f

(9)

g∈L G×L G f f af f af f then the fact that Gaf reduces to showing that Gaf is open ad is open in X ad ∩ X0 af f in X0 . ! First let Z = X af f − g∈L G×L G gX0af f ; it is L G × L G stable and closed. We will show if Z is nonempty then [v ⊗ v ∗ ] ∈ Z(C) which is a contradiction hence Z will be empty. As a preliminary step suppose W is a highest weight representation under a semisimple group G with highest weight w. Let Uα (t) ∼ = Ga = Spec C[t] be the root subgroup

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associated to a root α of G. For any p ∈ PW the specialization of p under Uα (1/t) is the point p0 = limt→0 Uα (1/t)p ∈ PW . By repeatedly specializing any p ∈ PW by all possible Uα (1/t) for positive roots α we eventually specialize to the highest weight as this is the only point stable under all Uα for α positive. We apply this observation to Z ⊂ PEndind (V ). Let Zf in ⊂ Z be the closed sub-indscheme consisting of points that lie in P(W1 ⊗ W2 ) where W1 ⊂ V and W2 ⊂ V ∗ are  finite dimensional. Then Zf in is preserved by the action of L poly G × Lpoly G. Moreover the specialization pf in under a generic 1 parameter subgroup of T  × T  of any p ∈ Z lands in Zf in . Finally by specializing pf in under root subgroups we specialize to [v ⊗ v ∗ ]. As Z is L G × L G stable and closed we conclude [v ⊗ v ∗ ] ∈ Z which is a contradiction. Therefore X af f = ∪γ1 ,γ2 ∈L G γ1 X0af f γ2 . Thus to prove the proposition it suffices to f af f f af f  show that Gaf is open in X0af f . By Proposition 5.3, Gaf = U − Tad U ad ∩ X0 ad ∩ X0 which is open. 2 Let Y = ∪i Yi be an ind-scheme and let Z = ∪i (Z ∩ Yi ) = ∪i Zi be a closed sub ind-scheme. We say Z is a Cartier divisor if Zi ⊂ Yi is a Cartier divisor for each i. f af f − X0af f . Then the Zi are Cartier Proposition 5.5. Set Z1 = X af f − Gaf ad and Z2 = X divisors. Both have r + 1 irreducible components.

Proof. For Z1 using (9) it is enough to show that Z1 ∩X0af f is Cartier. By Proposition 5.3 there is a projection  ∼ X0af f → Tad,0 = Ar+1

and Z1 ∩ X0af f is the pull back of the union of hyperplanes hence Cartier and has r + 1 components. To show Z2 is Cartier we use the embedding in some PEndind (V ). Let v ∈ V be the highest weight. Then [v ⊗ v ∗ ] is a section of O(1) on PEndind (V ) and Z2 = {[v ⊗ v ∗ ] = 0} ∩ X af f hence Z2 is Cartier. We can determine the number of components by looking at field valued points. Because Z2 is B − × B stable we can use the Birkhoff factorization (5) to identify Z2 (k)|Gaf f : ad

Z2 (k)|Gaf f =

r 

ad

U − si B

i=0

hence Z2 has r + 1 components. 2 f A caveat about Theorem 5.1. The ind-scheme Gaf ad is not the union of smooth schemes af f [10, 5.4]. However Gad is formally smooth: if R is a C algebra and I is a nilpotent ideal f then any R/I point of Gaf ad lifts to an R point. As such we can at most ask for embeddings that are formally smooth.

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Theorem 5.1. Let G be a simple, connected and simply connected group over C with f maximal torus T and set r = dim T . The ind-scheme X af f contains Gaf ad as a dense open sub-ind scheme and further (a) X af f is formally smooth and independent of the choice of regular dominant weight λ = (0, λ, l). f (b) The boundary X af f − Gaf ad is a Cartier divisor with r + 1 components D0 , . . . , Dr . (c) Each Di is formally smooth and ∪i Di is locally isomorphic to a product S × Z where Z is the union of the hyper planes in Ar+1 . f  af f (d) The torus Tad := C× ×T /Z(G) ⊂ Gaf which ad has a partial compactification in X  ∼ r+1   is an affine toric variety Tad,0 = A and there is a bijection between Tad ×Tad orbits in Ar+1 and L G ×L G orbits in X af f . In particular, the L G ×L G orbit closures are given by subsets I ∈ {0, . . . , r} such that to I we associate ∩i∈I Di . Proof. Let λ = (0, λ, l), μ = (0, μ, l ) be two regular dominant weights and deaf f f be the closure of Δ(Gaf note Xλaf f , Xμaf f the respective embeddings. Let XΔ ad ) in af f Xλaf f × Xμaf f . The projection pλ : XΔ → Xλaf f is equivariant and Proposition 5.3 imaf f af f ∼ af f  − plies XΔ,0 := p−1 λ (Xλ,0 ) = U · Tad,0 · U; that is, the restriction of pλ to XΔ,0 is an isomorphism and therefore induces a L G × L G equivariant isomorphism on





af f af f g.XΔ,0 = XΔ → Xλaf f =

g∈L G×L G

af f g.Xλ,0 .

g∈L G×L G

 This shows X af f is independent of the choice of a regular dominant λ. Further Tad,0

is smooth hence by Proposition 5.3, X0af f is formally smooth hence (a). Part (b) is Proposition 5.5. f af f As the boundary X af f − Gaf × Gaf f stable it is enough to verify (c) on ad is G af f X0 . We have by Proposition 5.3 X af f − Gaf f ∩ X0af f = U − × Hy × U  ∼ where Hy is the union of hyperplanes in Tad,0 = Ar+1 . This proves (c).   For (d) notice any L G × L G orbit Orb in X af f intersects X0af f . By Proposition 5.3   we see Orb intersects Tad,0 . We readily see that the stabilizer of Tad,0 in Gaf f × Gaf f is    Tad × Tad . In particular distinct T  × T  orbits in Tad,0 give rise to distinct Gaf f × Gaf f af f orbits in X .  To finish let Hi denote the ith hyperplane in Tad,0 = Ar+1 . The final statement of (d) follows because the T  × T  orbit closures are given by intersecting the Hi . 2

From (d) we see that there are 2r+1 orbits in X af f : X af f = I∈{0,...,r} Orb(I).

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A priori this is just a set theoretic statement about R points. However because the orbits form a stratification of X af f in the sense that Orb(I) = J⊃I Orb(J) one can show each orbit is locally closed and hence a sub-ind scheme of X af f . After Proposition 5.8 we take a detailed look at the orbits. The analysis will be used to give the orbits modular interpretations. We now turn to computing the Picard group of X af f . A crucial step is Proposition 5.6; the proof is due to Sharwan Kumar. Proposition 5.6. P ic(U − ) = 0. Proof. For any w ∈ W af f we have a Schubert variety BwB/B ⊂ LG/B; set Uw− = ! U − ∩ BwB/B. In fact U − ⊂ LG/B and we get an ind-structure on U − = n Un− where ! Un− = l(w)≤n Uw− . This ind-scheme structure is compatible with ind-scheme structure previously defined on LG by [14, prop. 4.2]. We show P ic(Uw ) = 0 for all w. Fix w and abbreviate Y = Uw− . For any k ∈ N we have a short exact sequence f →f k Z/k → O× −−−−→ O× ; using that H 1 (Y, O× ) ∼ = P ic(Y ) and looking at the long exact Y

et

Y

Y

sequence in étale cohomology we get 1 2 · · · → Het (Y, Z/k) → P ic(Y ) → P ic(Y ) → Het (Y, Z/k) → · · · ∗ ∗ By the proof of [13, 7.4.17], Y is contractible and because Het (−, Z/k) = Hsingular (−, Z/k) the outer terms vanish. Alternatively, [9, pg. 48] shows that Y is algebraically conL →L⊗k

tractible. It follows that P ic(Y ) −−−−−→ P ic(Y ) is an isomorphism for any k. We now show P ic(Y ) is finitely generated and together with the previous statement it will follow that P ic(Y ) = 0. Y is a normal variety with dim Y = l(w) so by [11, 2.1.1] P ic(Y ) embeds in the Chow group P ic(Y ) ⊂ Al(w)−1 (Y ). So we reduce to showing Al(w)−1 (Y ) is finitely generated. By [11, 1.8] there is a surjection Al(w)−1 (BwB/B) → Al(w)−1 (Y ). By [11, 19.1.11b] Al(w)−1 (Y ) = H2(l(w)−1) (BwB/B, Z) and finally the Bruhat decomposition implies the latter group is finitely generated. 2  Corollary 5.7. P ic(U) = P ic(Tad,0 U) = P ic(X0af f ) = 0.

Proof. The group U is a pro-unipotent pro group [13, 4.4]. In particular there is a family π F of normal subgroups N ⊂ U such that for N ∈ F the quotient U −−N→ U/N is a morphism and U/N is a finite dimensional unipotent group. ! Write U/N = Spec AN then U = Spec N ∈F AN . Moreover each AN is a polynomial ring so U ∼ = Spec C[x1 , x2 , . . . ] = Spec S. Let L be any line bundle. Then as U is quasi compact L is trivialized on a finite number of open subsets which we can take to be Spec Sfi . Let f = fi then f lies is some R = C[x1 , . . . , xm ] and evidently L is the pull back of a line bundle on Spec R  U) = 0 as hence L is trivial and P ic(U) = 0. This immediately implies that P ic(Tad,0  ∼ r+1 T =A . ad,0

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By the previous proposition, P ic(Un− ) = 0. Writing Un− = Spec Bn and replacing S, R with S ⊗ Bn and R ⊗ Bn in the argument above allows us to conclude that  P ic(Un− Tad,0 U) = 0 for all n from which it follows that P ic(X0af f ) = 0. 2 Proposition 5.8. The group P ic(X af f ) is free of rank r+1. The components of the Cartier divisor X af f − X0af f form a basis. af f af f Proof. For each n the components of Xnaf f − X0,n are Cartier and P ic(X0,n ) = af f  U) = 0. We get that the components of Xnaf f − X0,n generate P ic(Xnaf f ). P ic(Un− Tad,0 A relation among these generators is a principal divisor div(f ) which is invertible on X0af f . Since X af f − X0af f is B − × B stable we obtain, for every p ∈ X0af f , a map B − × B → Gm given by b → f (bp)/f (p). This map must be trivial on the unipotent factors hence is a character and f is an eigenfunction. Consequently the values of f on  X0af f are determined by its restriction to Tad,0 . It follows that f must be a constant c and f − c is zero on a dense open set hence there are no relations. 2

5.2. Orbits From 5.1 the Gaf f ×Gaf f orbits of X af f are indexed by subsets I ⊂ {0, . . . , r}. We first want to show these orbits fiber over a product of flag varieties for Gaf f (Proposition 5.2). To this end we set up some notation.  ∼ Recall Tad,0 = Ar+1 ; for I ⊂ {0, . . . , r} let eI be the point with 0s in the coordinates in I and 1s in the coordinates in the complement of I. Then the eI are base  points for the torus orbits in Tad,0 . This convention is set up to match up with our convention on indexing LG × LG orbits in X af f . If we consider Ar+1 as embedded in diagonal (r + 1) × (r + 1) matrices then eI define idempotent endomorphisms of rank r + 1 − #I.  We exploit the embedding Tad,0 ⊂ Pind [v⊗v ∗ ] End(V ) and denote by [eI ] the endomorphism of V . Similarly to in Lemma 4.5 we consider elements of PEndind (V ) as N × N matrices in such a way that the [eI ] embed as diagonal matrices, i.e. [eI ] is a projection onto subspace given by a finite subset [I] ⊂ N. In general the rank of eI and [eI ] are different. Example 3. Take G = SL2 (C). The maximal torus of Gaf f is T af f = C× × T × C× c = t 0  ± ± af f −2 Spec C[d ] × 0 t−1 × Spec C[c ]. The simple roots are given by α0 (T ) = t d and α1 (T af f ) = t2 . Take the regular dominant weight λ = 2ω 0 + 3ω 1 = (0, 3, 5). Let V = V (λ) and let v −1 2 be a highest weight of V . Then Tad,0 = Tad ∩ P[v⊗v∗ ] End(V ) ∼ = Spec C[α−1 0 , α1 ] = A . ± 3 4 Consider the operators X(−1), Y ∈ sl2 (C[z ]). Then X(−1) v = 0 = Y v = 0 but v, Y v, Y 2 v, Y 3 v, X(−1)v, X(−1)2 v are independent weight vectors for T af f . Extend this  to a basis of weight vectors for V . Then Tad,0 embeds in P[v⊗v∗ ] End(V ) as

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⎛

1 ⎜0 ⎜ ⎜0 ⎜ ⎜ −1 −1 0 (α0 , α1 ) → ⎜ ⎜ ⎜0 ⎜ ⎜0 ⎝ .. .

0 α−1 0 0 0 0 0 .. .

0 0 α−2 0 0 0 0 .. .

0 0 0 α−1 1 0 0 .. .

0 0 0 0 α−2 1 0 .. .

0 0 0 0 0 α−3 1 .. .

711

⎞ ··· ···⎟ ⎟ ···⎟ ⎟ ⎟ ···⎟ ⎟ ···⎟ ⎟ ···⎟ ⎠ .. .

−1 −1 −1 where the rest of the diagonal entries are elements in α−1 0 α1 C[α0 , α1 ]. In this case we have e1 = (1, 0), [e1 ] is a rank 3 idempotent and [{1}] = {1, 2, 3}. Similarly e0 = (0, 1), [e0 ] is a rank 4 idempotent and [{0}] = {1, 4, 5, 6}.

Let PI = LI UI and PI− = LI UI− be the parahoric subgroups with Levi decomposition as in (3), (4). Let Z(LI ) ⊂ LI denote the center and LI,ad = LI /Z(LI ) the adjoint group. Proposition 5.2. For I ⊂ {0, . . . , r} let Orb(I) be the L G ×L G orbit in X af f according to Theorem 5.1(d). Then there is a surjective map π : Orb(I) → Gaf f /PI × Gaf f /PI− and all the fibers are isomorphic to LI,ad . Proof. Let gi ∈ Gaf f (R). We denote by [gi ] the associated N × N matrix. The matrix [eI ] is idempotent so [g1 ][eI ][g2−1 ] = [g1 ][eI ][eI ][g2−1 ]

(10)

For any N ×N matrix [m] let [m]N×[I] be the N ×#[I] matrix formed from the columns of [m] indexed by [I]. Define [m][I]×N similarly. The only nonzero entries of [g1 ][eI ] are [g1 ]N×[I] and similarly for [eI ][g2 ]. Thus the factorization (10) yields a map [g1 ]eI [g2−1 ] → [g1 ]N×[I] × [g2 ][I]×N into a product of Grassmannians: "

#[I]

π : Orb(I) → P

"

#[I]

V ×P

V ∗.

Let W ⊂ V be the subspace that [eI ] projects to. Then evidently the above map is ##[I] ∗ ##[I] the Gaf f × Gaf f orbit of W× W which, via 4.1(e) and 4.3 is identified with Gaf f /PI × Gaf f /PI− . Notice W is a highest weight representation for a regular dominant highest weight of ##[I] ##[I] ∗ LI and the fiber π over ( W× W ) consists of the orbit of [id] in PEnd(W ) which is LI,ad . 2

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Corollary 5.9. Let I ⊂ {0, . . . , r} and consider the Levi decomposition PI = LI UI . Then af f ×Gaf f Orb(I) = G Stab(e and I) Stab(eI ) = {(l1 u1 , l2 u2 ) ∈ LI UI × LI UI− |l1 l2−1 ∈ Z(LI )} = Z(LI ) × Z(LI )Δ(LI )  UI × UI− . Proof. By Proposition 5.2 we have Stab(eI ) ⊂ PI × PI− . Further UI × UI− ⊂ Stab(eI ) so PI ×PI− Stab(eI )

×LI = Stab(eLII)∩(L . Finally Proposition 5.2 also implies I ×LI ) Hence Stab(eI ) ∩ (LI × LI ) = Z(LI ) × Z(LI )Δ(LI ). 2

LI ×LI Stab(eI )∩(LI ×LI )

= LI,ad .

The following is not used in the modular interpretation of Orb(J) but we think the result is interesting in its own right. Proposition 5.10. Let C → R be a field extension and J ⊂ {0, . . . , r} and Orb(J) ⊂ X af f be as in Corollary 5.9. Let WJ denote the Weyl group of the Levi factor LJ ⊂ PJ . Then Orb(J)(R) =



B − (R)w1 .eJ .WJ w2 B(R)

w1 ∈W af f /WJ w2 ∈WJ \W af f

The restriction to field valued points is because we use the Bruhat decomposition and properties of unipotent groups that in general do not hold over rings. Note when J = ∅ the disjoint union is the usual Birkhoff decomposition. When J = {0, . . . , r} then the disjoint union is the stratification by Schubert cells of L G/B × L G/B − . Proof. We look at field valued points through out so we drop it from the notation. The given expression is stable under the action of B − × B and disjointness of the expression is implied by the fact that B − × B ∩ W af f = 1 so it’s enough to prove the proposition with  replaced with ∪. Let C(J) denote B − \Orb(J)/B. For subsets A, B ⊂ L G let C(J, A, B) ⊂ C(J) be the subset of cosets that have a representative of the form AeJ B. The proposition amounts to showing C(J, W af f , W af f ) = C(J). The assertion of passing to WJ cosets in W af f follows because WJ ⊂ LJ and Δ(LJ ) ⊂ Stab(eJ ). Trivially we have C(J) = C(J, L G, L G). The Birkhoff factorization (5) together with Lemma 5.9 implies C(J, L G, L G) = C(J, W af f LJ , W af f ). Let BLJ ⊂ LJ be a Borel subgroup, ULJ its unipotent radical and UL−J the opposite radical. Applying the Bruhat decomposition LJ = UL−J WJ BLJ we further see C(J, W af f LJ , W af f ) = C(J, W af f UL−J , WJ ULJ W af f ). A general element of C(J, W af f UL−J , WJ ULJ W af f ) is w1 u− eJ w3 vw2 . We have w1 u− eJ w3 vw2 =

   w1 u− w1−1 w1 eJ w3 w2 w2−1 vw2



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The result follows if we can arrange that (*) w1 u− w1−1 ∈ U − and (**) w2−1 vw2 ∈ U. We show this indeed the case. Under the natural involution the proof for (*) and (**) is the same; we prove (*). Further, over fields ULJ is generated by some number of simple root subgroups Uαi ∼ = Ga . Therefore we can reduce to proving (*) for U−αi ∈ UL−J . We prove (***): if w ∈ W af f and −αi is a negative root then w(−αi )w−1 is a positive root ↔ there exists an reduced expression for w ending in si . When we apply (***) in (*) we have si ∈ WJ so we can replace w with wsi repeating we must eventually arrive at an element that cannot have a reduced expression ending in si . Thus we can arrange for (*) once we prove (***). In [6, pg. 61] it is shown that if w has a reduced expression ending in si then w(−αi )w−1 is a positive root. Now suppose w(−αi )w−1 is a positive root. By abuse of notation let w denote a reduced word for w. If w = wsi is not reduced then by the exchange property for reflection groups we can find a reduced expression for w ending in si hence we must show w is not reduced. If it is reduced then w (−αi )(w )−1 is a positive root by [6, pg. 61]. But w (−αi )(w )−1 = −w αi (w )−1 = −wsi αi si w−1 = −w(−αi )w−1 and w(−αi )w−1 was assumed to be positive, a contradiction. 2 5.3. Polynomial loop group We can establish a result analogous to Theorem 5.1 for the polynomial loop group. In this setting everything is easier. Fix a highest representation V = V (λ) = ∪k Vk = ∗ ⊕k grVk . Define the restricted dual Vres := ∪k (Vk )∗ = ⊕(grVk )∗ . af f af f  Set Gpoly to be the pullback of G under the inclusion L poly G → L G: 1

C× c

Gaf f

L G

1

f Gaf poly

L poly G

1

=

1

C× c

f Recall for an integer k we use n(k) to indicate that (Gaf poly )k maps Vk into Vn(k) . f ∗ ∗ Similarly we have that (Gaf poly )k maps Vk into Vm(k) .

Let R be a C-algebra. Define an ind scheme Endpoly (V ) = ∪i Endpoly (V ) where i ∗ Endpoly (V )(R) = {φ ∈ End(V )(R)|φ(Vk (R)) ⊂ Vn(i+k) and φ(Vj∗ (R)) ⊂ Vm(i+j) (R)} i

Proceeding exactly as in Lemma 4.5 shows (V ) is represented by an affine scheme and Endpoly (V ) Lemma 5.11. The C space Endpoly i is an affine ind-scheme.

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Comparing, Endind i (V ) is an infinite product of infinite dimensional scheme whereas poly Endi (V ) is an infinite product of finite dimensional schemes. Further Endpoly (V ) is i affine and its coordinate ring has a natural grading so we can form PEndpoly (V ). Let λ = (0, λ, l) be a regular dominant weight of Lpoly G and let V = V (λ). For every poly k ≥ 0 we have a locally closed embedding (L (V )n(k) . Define poly G/Z(G))k ⊂ PEnd poly Xkpoly = (L (V )n(k) poly G/Z(G))k ⊂ PEnd

X poly = ∪k Xkpoly

(11)

Similarly we have an affine ind-scheme P[v⊗v∗ ] Endpoly (V ). Define poly X0,k = Xkpoly ∩ P[v⊗v∗ ] Endpoly (V )n(k) poly X0poly = ∪k X0,k

(12)

Corollary 5.12. Let G be a simple, connected and simply connected group over C with maximal torus T and set r = dim(T ). The ind-scheme X poly contains L poly G/Z(G) as a dense open sub-ind scheme and further (a) X poly is formally smooth and independent of the choice of regular dominant weight λ. (b) The boundary X poly − L poly G/Z(G) is a Cartier divisor with r + 1 components D0 , . . . , Dr . The L G × L poly poly G orbits closures are in bijection with subsets I ⊂ {0, . . . , r} in such a way that to I we associate ∩i∈I Di . (c) Each Di is formally smooth and ∪ri=0 Di is locally a product S × Z where S is an ind-scheme and Z is the union of hyperplanes in Ar+1 . (d) X poly − X0poly is a Cartier divisor and with r + 1 components which freely generate P ic(X poly ). (e) Let Orb(I) ⊂ X poly be the sub ind-scheme which is the orbit of eI . Then X poly = I⊂{0,...,r} Orb(I). Moreover Orb(I) = J⊃I Orb(J) and we have a map  − π : Orb(I) → L poly G/PI × Lpoly G/PI with fiber LI,ad . Proof. As Lpoly G ⊂ LG all the proceeding arguments for LG apply to Lpoly G mutatis mutandis. The most significant difference is that Lpoly G ∩ U is abstractly isomorphic to U − . So for example Corollary 5.7 can be replaced with a simple extension of Proposition 5.6. 2 An improvement of X poly is that its boundary is a proper: Proposition 5.13. The boundary X poly − Lpoly G is a projective ind-variety. Proof. Embed X poly ⊂ PEndpoly (V ). Then X poly − Lpoly G is the union of L poly G × L G orbits of [e ] for nonempty subsets I ⊂ {0, . . . , r}. Each [e ] lies in the ind variety I I poly

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 PEndf in (V ) = ∪k PEnd(Vk ) and moreover PEndf in (V ) is preserved by L poly G ×Lpoly G. Therefore X poly − Lpoly G is closed ind-subvariety of a projective ind-variety. 2

5.4. Modular interpretation Let C be a nodal curves with a unique node p ∈ C such that C − p is affine. Consider the following moduli functor TGad ,C : Aff C → Set R → {P, t} where P is a principal Gad bundle on C ×Spec R and t : P |(C−p)×Spec R → Gad ×(C −p) × Spec R is a trivialization. Proposition 5.14. The functor TGad ,C is represented by an ind-variety also denoted by TGad ,C . Further π0 (TGad ,C ) = π1 (Gad ) = Z(G) and each connected component is isomorphic to Orb({0}) ⊂ X poly . Proof. Given a Gad bundle P on (C−p)×Spec R there is, by [8], an étale base change R → R such that P is trivial on (C − p) × Spec R . Also P is trivial on a formal neighborhood Spec R [[x, y]]/(xy) of the node p. Hence P is determined by transition function γ ∈ Gad (R ((x)) × R ((y))) = Lx Gad (R ) × Ly Gad (R ). The resulting bundle over Spec R ⊗R R is the same under the two maps R → R ⊗R R . By a standard descent argument we obtain a surjective map Lx Gad × Ly Gad → TGad ,C . Two transition functions γ, γ are identified in TGad ,C if they are related by an isomorphism over Spec R[[x, y]]/(xy). Thus we see TGad ,C =

Lx Gad × Ly Gad Gad [[x,y]] (xy)

[[x,y]] where Gad(xy) = Δ(Gad )  (N × N ) and N = Ker(L+ G → G). Let e{0} ∈ X poly (C). We observe that Δ(Gad )  (N × N ) is closely related to Stab(e{0} ) = Z(G) × Z(G)Δ(G)  (Npoly × N − ) where Npoly := N ∩ Lpoly G. To connect them we note 1) the connected component of the identity of LGad is (LGad )0 = LG/Z(G), 2) N − ∼ = Npoly and 3) Lpoly G/N − ∼ = Lpoly G/Npoly ∼ = LG/N . Therefore

(TGad ,C )0 = ∼ =

Lpoly G/Z(G) × Lpoly G/Z(G) Δ(Gad )  (Npoly × Npoly ) Lpoly G × Lpoly G Z(G) × Z(G)Δ(G)  (Npoly × N − )

= Orb({0}) ∈ X poly .

2

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The moduli stack MGad (C) of Gad bundles on C is a quotient of TGad ,C and is in general not complete. At this point we would like to say we can complete MGad (C) by adding modular objects corresponding to the closure Orb({0}) ⊂ X poly . This will indeed be the case. Notice that Orb({0}) = ∪0∈I Orb(I) does not run through all of the orbits in X poly . In fact the remaining orbits also have a modular interpretation and these are necessary to study not just a fixed smooth curve C but families of curves Ct that have a nodal special fiber C0 . The modular interpretation behind the rest of the orbits is in [18]. Acknowledgments This research was supported by the University of California Berkeley, an NSF graduate research fellowship DGE 1106400, and by Caltech. I thank my advisor Constantin Teleman for suggesting this project. I also thank the referee for helpful comments and corrections. References [1] V. Baranovsky, V. Ginzburg, Conjugacy classes in loop groups and G-bundles on elliptic curves, Int. Math. Res. Not. IMRN 15 (1996) 733–751, http://dx.doi.org/10.1155/S1073792896000463. [2] A. Beauville, Y. Laszlo, Conformal blocks and generalized theta functions, Comm. Math. Phys. 164 (2) (1994) 385–419, http://projecteuclid.org/getRecord?id=euclid.cmp/1104270837. [3] D.N.D. Ben-Zvi, Elliptic Springer theory, arXiv:1302.7053. [4] R. Bezrukavnikov, D. Kazhdan, Geometry of second adjointness for p-adic groups, Represent. Theory 19 (2015) 299–332, http://dx.doi.org/10.1090/ert/471, with an appendix by Yakov Varshavsky, Bezrukavnikov and Kazhdan. [5] Y.V.R. Bezrukavnikov, D. Kazhdan, A categorical approach to the stable center conjecture, arXiv: 1307.4669. [6] K. Brion, The Frobenius Splitting Method in Algebraic Geometry, Progr. Math., vol. 231, Birkhäuser, Boston, MA, 2005. [7] C. De Concini, C. Procesi, Complete symmetric varieties, in: Invariant Theory, Montecatini, 1982, in: Lecture Notes in Math., vol. 996, Springer, Berlin, 1983, pp. 1–44. [8] V.G. Drinfel’d, C. Simpson, B-structures on G-bundles and local triviality, Math. Res. Lett. 2 (6) (1995) 823–829, http://dx.doi.org/10.4310/MRL.1995.v2.n6.a13. [9] G. Faltings, Algebraic loop groups and moduli spaces of bundles, J. Eur. Math. Soc. (JEMS) 5 (1) (2003) 41–68, http://dx.doi.org/10.1007/s10097-002-0045-x. [10] S. Fishel, I. Grojnowski, C. Teleman, The strong Macdonald conjecture and Hodge theory on the loop Grassmannian, Ann. of Math. (2) 168 (1) (2008) 175–220, http://dx.doi.org/10.4007/annals. 2008.168.175. [11] W. Fulton, Intersection Theory, 2nd edition, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics (Results in Mathematics and Related Areas. 3rd Series. A Series of Modern Surveys in Mathematics), vol. 2, Springer-Verlag, Berlin, 1998. [12] I. Kausz, A Gieseker type degeneration of moduli stacks of vector bundles on curves, Trans. Amer. Math. Soc. 357 (12) (2005) 4897–4955, http://dx.doi.org/10.1090/S0002-9947-04-03618-9 (electronic). [13] S. Kumar, Kac–Moody Groups, Their Flag Varieties and Representation Theory, Progr. Math., vol. 204, Birkhäuser Boston, Inc., Boston, MA, 2002. [14] Y. Laszlo, C. Sorger, The line bundles on the moduli of parabolic G-bundles over curves and their sections, Ann. Sci. Éc. Norm. Supér. (4) 30 (4) (1997) 499–525, http://dx.doi.org/10.1016/ S0012-9593(97)89929-6.

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717

[15] J. Martens, M. Thaddeus, Compactifications of reductive groups as moduli stacks of bundles, arXiv:1105.4830. [16] D.S. Nagaraj, C.S. Seshadri, Degenerations of the moduli spaces of vector bundles on curves. II. Generalized Gieseker moduli spaces, Proc. Indian Acad. Sci. Math. Sci. 109 (2) (1999) 165–201, http://dx.doi.org/10.1007/BF02841533. [17] S. Schieder, The Drinfeld–Lafforgue–Vinberg degeneration I: Picard–Lefschetz oscillators, arXiv: 1411.4206. [18] P. Solis, A complete degeneration of the moduli of g-bundles on a curve, arXiv:1311.6847. [19] P. Solis, Infinite type toric varieties and Voronoi tilings, arXiv:1608.08581. [20] C. Teleman, C.T. Woodward, The index formula for the moduli of G-bundles on a curve, Ann. of Math. (2) 170 (2) (2009) 495–527, http://dx.doi.org/10.4007/annals.2009.170.495. [21] J. Tits, Ensembles ordonnés, immeubles et sommes amalgamées, Bull. Soc. Math. Belg. Sér. A 38 (1986) 367–387 (1987). [22] J. Tits, Uniqueness and presentation of Kac–Moody groups over fields, J. Algebra 105 (2) (1987) 542–573, http://dx.doi.org/10.1016/0021-8693(87)90214-6. [23] A. Tolland, E. Frenkel, C. Teleman, Gromov–Witten gauge theory I, arXiv:0904.4834. [24] E.B. Vinberg, On reductive algebraic semigroups, in: Lie Groups and Lie Algebras: E.B. Dynkin’s Seminar, in: Amer. Math. Soc. Transl. Ser. 2, vol. 169, Amer. Math. Soc., Providence, RI, 1995, pp. 145–182.