Proof of the Ginzburg-Kazhdan conjecture

We prove that the affine closure of the cotangent bundle of the basic affine space of a complex semisimple group has conical symplectic singularities, which confirms a conjecture of Ginzburg and Kazhdan. We also show that this variety is $\mathbb{Q}$-factorial and has terminal singularities.


Introduction
The goal of this paper is to prove the following theorem, which in particular confirms a conjecture of Ginzburg and Kazhdan [16,Conjecture 1.3.6]: Theorem 1.1.The variety T * (G/N ) has conical symplectic singularities.
Here, G denotes a connected semisimple group over C, N = [B, B] denotes the unipotent radical of some Borel B, and T * (G/N ) is the affinization of the quasi-affine variety T * (G/N ).We also prove in Theorem 6.3 that T * (G/N ) has symplectic singularities if G is reductive, although the G m -action we construct is not conical if G is not semisimple.
Theorem 1.1 implies that T * (G/N ) conjecturally admits a symplectic dual in the sense of Braden-Licata-Proudfoot-Webster.We also prove the following theorem, which determines properties of this conjectural dual: Theorem 1.2.The variety T * (G/N ) is Q-factorial and has terminal singularities.Moreover, if G is simply connected then the ring of functions of T * (G/N ) is a unique factorization domain.
1.1.Outline of Proof.In [20], it is shown that T * (G/N ) has symplectic singularities when G = SL n .We briefly review the approach of [20] so as to compare and contrast with the approach taken in the general case here.When G = SL n , the variety T * (G/N ) admits a description as a hyperkähler reduction of a certain vector space obtained from a quiver [9].This quiver description gives a stratification of T * (G/N ) by hyperkähler varieties of even (complex) codimension, and this stratification in particular includes a dense open smooth subset denoted in [9] as Q hks .A key insight in [20] is that complement of Q hks in T * (G/N ) in fact has codimension four, which is proved by showing all other strata have codimension at least four.Results of [33], [11] then show that, to prove that T * (G/N ) has symplectic singularities, it suffices to show that T * (G/N ) is normal and that the smooth locus of T * (G/N ) admits a symplectic form, see Lemma 6.4.These facts are proved in [20] using the above stratification and the Marsden-Weinstein theorem.
A quiver type description of T * (G/N ) is not expected to exist for general G, even when G = SO 2n [8].In proving Theorem 1.1, we circumvent this issue by constructing a smooth open subscheme Q ⊆ T * (G/N ) that we expect can be identified with Q hks when G = SL n .We show that the complement of Q in T * (G/N ) has codimension at least four (Theorem 6.2), and argue directly that T * (G/N ) is normal and that the regular locus of T * (G/N ) admits a symplectic form.Using this, we show that T * (G/N ) has symplectic singularities in Section 6.2.
To show that the complement of Q has codimension at least four, we analyze the right T -action on T * (G/N ) and use the fact that the ring of functions on T * (G/N ) is a unique factorization domain when G is simply connected, see Corollary 3.5.In this case, the fact that the singular Mathematics Subject Classification 2020: 14B05, 22E57, 22E10 locus of G/N has codimension at least four allows us to show that a set strictly smaller than Q (in general) has codimension at least four, see Proposition 5.16.This of course gives the fact that the complement of Q has codimension at least four for simply connected G, which we use to derive the corresponding fact for general G in Section 6.1.The fact that T * (G/N ) has terminal singularities then follows immediately from a result of Namikawa [29].
1.2.Acknowledgments.I would like to thank Desmond Coles, Gurbir Dhillon, Louis Esser, Joakim Faergeman, Victor Ginzburg, Rok Gregoric, Boming Jia, Frances Kirwan, Joaquín Moraga, Morgan Opie, Alberto San Miguel Malaney, Kendric Schefers, and Jackson Van Dyke for many interesting and useful conversations.I would also like to thank Boming Jia and Sam Raskin for pointing out a gap in a draft version of this paper, as well as the anonymous referee, whose extremely detailed feedback greatly improved the exposition of this paper.Finally, I would also like to thank Ivan Losev and the organizers of the Quantized symplectic singularities and applications to Lie theory 2022 conference, who provided an excellent environment for me to learn the basics of the theory of symplectic singularities.

Recollections on The Affine Closure of The Basic Affine Space
We now collect the facts we will use regarding G/N and T * (G/N ) and set the notation which will be used in what follows.None of the material in Section 2 is original.More details, references, and proofs can be found in works such as [2], [17], [7], [16], and [24].
2.1.Affine Closures of Basic Affine Space and Its Cotangent Bundle.Hereafter, in every section except1 Section 5, we let G Z denote some split reductive group over Z with choice of maximal torus T Z , and let G := G k and T := T k denote the respective base changes to k := C. Denote by X • (T ) denote the lattice of characters for T , and let X • (T ) denote the lattice of cocharacters.Choose some Borel subgroup B ⊇ T , and let N denote the unipotent radical of B. Let g and t denote the Lie algebras of G and T respectively, and let g * and t * denote their respective dual Lie algebras.We will occasionally abuse notation by denoting t(Q With this notation, we have isomorphisms both induced by the differential.By the algebraic Peter-Weyl theorem the ring of functions on G/N [27,Proposition 14.26]) that the variety G/N is quasi-affine; we also reprove this fact below as an instance of a more general claim in Proposition 5.5.Therefore G/N is an open subscheme of its affine closure G/N := Spec(A).
The projection map π : T * (G/N ) → G/N is affine, and so the variety T * (G/N ) is also quasiaffine.In particular, we may identify it as an open subscheme of its affinization T * (G/N ) = Spec(R), where denotes the ring of global functions on T * (G/N ).The map π induces a canonical map induced by the inclusion A ⊆ R. The action of G × T on G/N gives a corresponding G × T action on T * (G/N ).We refer to the action of the subgroup 1 × T as 'the' T -action on T * (G/N ) and, for any λ ∈ X • (T ), we let R λ denote the graded summand of R induced from this T -action.

Moment Map Notation.
From the G × T -action on G/N , we obtain a moment map T * (G/N ) → g * × t * which lifts to a map T * (G/N ) → g * × t * W t * .Since g * × t * W t * is affine, we obtain an induced map µ : T * (G/N ) → g * × t * W t * by the universal property of affinization.

2.3.
Algebraic Gelfand-Graev Action on T * (G/N ).We recall a weaker form of one of the main theorems of [16], see also [17,Section 5.5]: 2.4.Finite Generation of Functions on T * (G/N ).We now summarize some results of [17,Section 3.6] which will be used below.For any w ∈ W , let π w denote the composite map π • w : Lemma 2.2.The ring R is finitely generated, and in particular Noetherian.Moreover, the morphism embedding, and, for any dominant λ ∈ X • (T ) and any w ∈ W , the restricted multiplication map M w,λ : Sym(g) ⊗ Zg Sym(t) ⊗ k w(A λ ) → R wλ is surjective, where w(A λ ) denotes the image of A λ in R under w.

Ring of Functions on T * (G/N )
In this section, we study the variety T * (G/N ) and its ring of functions R. We first show that the variety T * (G/N ) is Q-factorial, and moreover that R is a UFD when G is simply connected, in Section 3.1.We then construct a G m -action on T * (G/N ) in Section 3.2 and record some of its basic properties.
3.1.Factoriality and Q-Factoriality of Affine Closure of T * (G/N ).Recall that a normal variety is said to be Q-factorial if the cokernel of the map Pic(X) − → Cl(X) is torsion.We now show: Notice that G/N and T * (G/N ) are normal since the ring of functions on any smooth variety is normal, see, for example, [32,Lemma 28.7.9].To prove the remainder of Proposition 3.1, we first show the following: Lemma 3.2.The class group and the Picard group of G/N are finite.Moreover, if G is simply connected, both groups are trivial.
Proof.For any normal variety, the Picard group injects into the class group, so it suffices to show the class group of G/N is finite and, when G is simply connected, is trivial.The class group of G/N agrees with the class group of G/N since its complement has codimension 2 by the stratification (4), see, for example [18,Proposition II.6.5(b)].Since G/N is smooth, its class group and Picard group agree.However, it is known (see, for example, [27,Theorem 18.32]) that we have an exact sequence where X • (N ) := Hom AlgGp (N, G m ).Notice that X • (N ) = 0 and Pic(N ) = 0 as N is unipotent by [27,Corollary 14.18] and [27,Proposition 14.32], respectively.Therefore Pic(G/N ) ∼ − → Pic(G).Now our claim follows from the fact that Pic(G) is finite [27,Corollary 18.23] and the fact that if G is simply connected then Pic(G) is trivial [27,Corollary 18.24].□ From Lemma 3.2, we derive the following result, which in particular completes the proof of Proposition 3.1: Proposition 3.3.The class group of T * (G/N ) is isomorphic to the class group of G/N .In particular, the class group of T * (G/N ) is finite and, if G is simply connected, is trivial.
We show this after showing the following lemma: Lemma 3.4.Assume Y is an integral quasi-affine scheme and let B denote its ring of functions so that j : Y → Spec(B) is an open embedding.If B is Noetherian, then the complement of Y in Spec(B) has codimension at least two.
Proof.Choose some minimal prime p in the complement of Y , and let Z denote the integral scheme Spec(B/p).Letting U := Spec(B) \ Z and X := Spec(B), we have a containment of open subschemes This map is surjective by the induced map of functions given by (3) and injective since Y , and thus Spec(B), are integral, and thus we see that j # is an isomorphism.From this and (3), it follows that the restriction map is an isomorphism.Note that U is quasi-compact, as it is an open subset of Spec(B) for B Noetherian.However, for quasi-compact U , the main result of [28] gives that (3) is not an isomorphism if p is a divisor, so the height of p must be at least two.□ Proof of Proposition 3.3.Since the complement of T * (G/N ) has codimension at least two in T * (G/N ) by Lemma 3.4, it suffices to show the analogous claim for T * (G/N ).Since T * (G/N ) is smooth, by the Auslander-Buchsbaum theorem its Picard group and class group agree.Therefore by Lemma 3.2 it is enough to show that the map of abelian groups is an isomorphism.This map is injective, as any line bundle L for which π * (L) is trivial has the property that L ∼ = z * (π * (L)) is also trivial, where z denotes the zero section.The surjectivity follows from [10,Corollaire IV.21.4.11,Erratum], as π : T * (G/N ) → G/N is a faithfully flat morphism to a normal variety whose its fibers are vector spaces.□ Since the class group of Spec(R) = T * (G/N ) is trivial when G is simply connected, we immediately obtain: Corollary 3.5.The ring R is a unique factorization domain if G is simply connected.

3.2.
Grading on Functions on T * (G/N ).We can define a G m -action on T * (G/N ) defined as follows.Let 2ρ ∨ denote the product of the positive coroots in T .This defines a map p : where u 0 := h −1 α uh α , we see that this gives an action of G m on T * (G/N ) ∼ = G N × b and in particular equips R with a grading R = ⊕ i R i , where we use superscripts for the grading to disambiguate from the X • (T )-grading on R of Section 2.1.For a nonzero homogeneous element r ∈ R, with respect to this grading, we let c(r) denote the unique integer with r ∈ R c(r) , and refer to c(r) as the c-grading of r.We use this term since, when G is semisimple, this grading is conical, as stated in the last point of the following proposition: Proposition 3.6.The above grading on R has the following properties: (1) The maps π and µ are G m -equivariant, where G/N is equipped with a G m -action via restricting the T -action via p and G m acts on (2) For any nonzero r ∈ R λ of usual grading h r , r is homogeneous with respect to the c-grading and moreover c(r Proof.The first claim immediately follows from the fact that π and µ are G m -equivariant.By Lemma 2.2 and (1), we see that (2) holds for any λ lying in the closure of the Weyl chamber C = 1C determined by our choices of B and N .Using this as the base case, we now proceed by induction on the length of w ∈ W .For any λ ∈ wC, we may choose some simple reflection s such that sw < w.Let r ∈ R λ .By induction we see that s(r) and rs(r) are both homogeneous with respect to the c-grading, and thus r is as well since R is an integral domain.Letting h s(r) and h r denote the respective usual gradings we then obtain where both the first and last step use the inductive hypothesis.We therefore see To prove (3), it suffices to show the claim on some generating set.By Lemma 2.2, we may choose this generating set whose elements are a basis of g ⊕ t as well as the elements w(a) for all a ∈ A λ for λ dominant and w ∈ W .The former case follows from (1) and the W -equivariance of µ, so we may assume r = w(a) for some a ∈ A λ .However, for such r, it is known [17, Remark 3.2.2(1)],building on the analogous claim for differential operators [24, Proposition 2.9], [2], that the usual grading of w(a) is ⟨λ − w(λ), ρ ∨ ⟩.In particular, we see Furthermore, when G is semisimple, we have that ⟨λ, 2ρ ∨ ⟩ > 0 for all nonzero dominant λ.Therefore for such G each element in the set has positive c-grading, where λ varies over the dominant nonzero weights.Since this set generates R by Lemma 2.2, we obtain (4).
Proof.Fixing x, y as above, by Proposition 3.6(2) we may assume that there exists λ, λ ′ ∈ X • (T ) and h, h ′ ∈ Z ≥0 such that i = ⟨λ, 2ρ ∨ ⟩ + 2h, j = ⟨λ ′ , 2ρ ∨ ⟩ + 2h ′ , and, with respect to the G m -action given by fiber dilation, the grading on x, respectively y, is h, respectively h ′ .The Poisson bracket on R preserves the T -grading and lowers the (usual) degree of a vector field by one (which can be checked locally on T * (G/N )).Therefore we see that {x, y} ∈ R λ+λ ′ and its grading from the G m -action given by fiber dilation is h + h ′ − 1, and so its c-grading is Section 5].In particular, it is shown that this grading is compatible with a natural G m -action given by identifying T * (SL n /N ) with a hyperkähler reduction of a certain vector space [9]-see [20,Proposition 5.5] for the precise statement.

Preliminary Results in Algebraic Geometry
4.1.GIT Quotients of Integral Varieties.We now record two properties of GIT quotients of integral affine varieties with G m -actions which will be used below.
Lemma 4.1.Assume S is a graded integral finitely generated k-algebra such that there is a positively graded or negatively graded homogeneous element x.Then dim(Spec(S) G m ) ≤ dim(Spec(S)) − 1, where we equip Spec(S) with the G m -action corresponding to the grading on S.
Proof.The fact that S is an integral domain implies that the multiplication map S 0 ⊗ k k[x] → S is injective, since we may check if an element of S is nonzero by checking if each projection onto each homogeneous summand of S is nonzero.We thus see that the morphism where the second equality uses the fact that S is finitely generated and therefore in particular Noetherian.□ Recall that, for any affine variety Spec(S) with an action of G m , this action is determined by a Z-grading on the ring S.Moreover, the fixed point subscheme Spec(S) Gm is a closed subscheme cut out by the ideal I generated by the S i for i ̸ = 0. We have an induced map Spec(S/I) = Spec(S) Gm → Spec(S) G m = Spec(S 0 ) by composing the inclusion and the quotient map.Moreover, the induced map of rings S 0 → S/I is surjective since f = i f i ∈ S is congruent to f 0 in S/I.This proves the following observation: Proposition 4.2.For any affine variety Spec(S) with an action of G m , the induced map Spec(S) Gm → Spec(S) G m is a closed embedding.Proof.For any variety Y , we let T Y denote its tangent sheaf.Notice that we have a canonical map over G/N given by the universal property of affinization and the fact that Spec(Sym • G/N (T G/N )) is affine.Therefore, we obtain a canonical map ϕ given by the composite where this isomorphism is given by the fact that S is smooth, for which the composite is the identity.Since this map is a section and ϕ is separated we see that σ is a closed embedding of equidimensional integral schemes, and therefore is an isomorphism.□

The Singular Locus for Simply Connected G
In this subsection, we study the singular locus of T * (G/N ) in the case where G is simply connected.For such G we also introduce the subset Q and study its basic properties.To avoid excessive repetition, in all of Section 5 we assume that G is simply connected.The first sentence of Proposition 5.1, whose proof heavily uses ideas of [23], also holds by similar methods in a more general context where N = [B, B] is replaced by the commutator of an arbitrary parabolic subgroup, see Proposition 5.5.We use this generalization to prove the second sentence of Proposition 5.1.

5.1.1.
Commutator of Parabolic.Choose a subset I of the set of simple coroots ∆ ∨ and let J denote its complement.For each simple coroot c, we let ω c ∈ X • (T ) denote the fundamental weight dual to c and choose a nonzero highest weight vector ⃗ v c in the irreducible representation V c of highest weight ω c .(Each ⃗ v i is of course unique up to nonzero scalar multiple and our constructions will be independent of our choices of ⃗ v i .)If we let L j denote the line spanned by ⃗ v j for every j ∈ J, then P ∆ ∨ \{j} is stabilizer of L j , and moreover P I = ∩ j∈J P ∆ ∨ \{j} , both of which can be seen from for example the fact that any subgroup of G containing our choice of Borel subgroup is a standard parabolic subgroup [19,Theorem 29.3(a)] by computing the simple root vectors in the Lie algebra of the group.(In particular, P ∅ = B.) When I contains a single element i we will also let P i := P {i} and, when there is no danger of confusion, we will also denote P i by P α where α is the simple root associated to i.
We also define the G-representation V J := ⊕ j∈J V j , and let E J := span V J {⃗ v j : j ∈ J} which is a subspace of V J which is closed under the action of P I and has dimension |J|.We naturally obtain a representation (5) ρ I : where Q I denotes the kernel of the map of ( 5), or equivalently the intersection of stabilizers of the ⃗ v j for j ∈ J.Moreover, since G Proof.By (7) we have an isomorphism of varieties We can use similar methods to the proof of Proposition 5.2 to obtain an explicit description of the G-stabilizer of any vector ⃗ v ∈ E J .By the semidirect product decomposition of (7), to compute the stabilizer of any vector it suffices to compute the stabilizer of ⃗ v J := j∈ J ⃗ v j for any fixed subset J ⊆ J.We describe this in terms of Ĩ := ∆ ∨ \ J: Proof.Letting S ⃗ v j denote the stabilizer of ⃗ v j , we have Since the stabilizer of a vector is contained in the stabilizer of the line it spans, the stabilizer of ⃗ v j in G is Q ∆ ∨ \{j} for any j ∈ ∆ ∨ ; in other words, and so in particular we see that However, the kernel Q Ĩ of the representation ρ Ĩ is evidently the intersection ∩ j∈ J (S ⃗ v j ∩ P I ) = P I ∩ j∈ J S ⃗ v j so our claim follows from combining these above equalities.□ Remark 5.4.We thank the referee for suggesting the above proof of Corollary 5.3, which is more simple than the proof originally given by the author.
Proof.Notice that the representation morphism ρ : G × V J − → V J induces a map ρ : G P I × V J → V J and so we obtain an induced map ρ| E J : G We first claim that ρ| E J is proper.To see this, notice that, if ϕ denotes the isomorphism G given by ϕ(g, v) := (gP, gv), we have proj V J ϕ = ρ.This shows that ρ is proper, and, since ρ| E J is the restriction to the closed subvariety G P I × E J , we see ρ| E J is proper as well.In particular, the image X I of ρ| E J is closed.Therefore, X I is affine and every closed point of X I can be written as g⃗ v for some closed point ⃗ v ∈ E J (k).Now, notice the P I -orbits of E J are equivalently given by the G J m -orbits on E J .Since we have a G J m -equivariant isomorphism E J ∼ = j∈J L j we may explicitly compute the G J m -orbits on this space and we see that the G J m -orbits on E J are precisely of the form for some K ′ ⊆ J, such that O K ′ lies in the closure of O K ′′ if and only if K ′ ⊇ K ′′ .Now, using (7), we see that GO K ′ is equivalently the G-orbit of j∈J\K ′ ⃗ v j and therefore GO K ′ ∼ = G/Q I∪K ′ by Corollary 5.3.We therefore have GO K ′ = G/[P I∪K ′ , P I∪K ′ ] by Proposition 5.2, which gives the stratification (9) with the closure relations as in Proposition 5.5.
The variety X I is normal by [23, Proposition 1].We claim that the complement of G/Q I in X I is a codimension two subset.Indeed, this follows from the fact that for any coroot j ∈ J the Lie algebra of any Q I∪{j} contains a Levi factor SL j 2 of P j and so, comparing the T -weight spaces of the Lie algebras, we see that dim(Q Therefore since X I is a normal affine variety (it is a closed subscheme of E), we have that X I is the affine closure of G/Q I and so our stratification (9) gives our claim.□ 5.1.3.The Symplectic Vector Bundle.Recall that in [22, Section 2.1], the authors choose an SL 2triple associated to a fixed simple root α and construct a certain rank two symplectic vector bundle f α : V α → G/Q α such that V α admits an action of G × T for which the map f α is equivariant for this action and such that the complement of the zero section of V α can be identified with G/N .In particular, the variety T * (V α ) has an endomorphism given by the symplectic Fourier transform see for example [17,Appendix B] where, in the notation of [17], we specialize to ℏ = 0.This notation is justified as the composite endomorphism (read left to right) induced by the restriction maps (which are equivalences since the complement of T * (G/N ) in T * (V α ) has codimension two) and pulling back by S α is, by definition, the automorphism s α of R given by the Gelfand-Graev action, see [17].
Proof.This is a standard argument (see, for example, the proof of [13, Lemma 3.21]) but since the argument is short we repeat it for the convenience of the reader.The morphism f α is affine as V α is a vector bundle, and therefore quasi-affine.Since the terminal map from G/[P α , P α ] is quasi-affine by Proposition 5.5, our claim follows from the fact that composition of quasi-affine morphisms is quasi-affine [ Proof.Recall that the complement of G/N ⊆ V α has codimension two, since at the reduced level it can be identified with the scheme theoretic image of the zero section G/Q α → V α and V α is a rank two vector bundle.Therefore since V α is smooth and in particular normal we therefore see that the restriction map gives an equivalence O(V α ) ∼ − → A. Since V α is quasi-affine by Corollary 5.6, the affinization map for V α is an open embedding, and so we have an open embedding V α ⊆ G/N given by the composite (read left to right)

Irreducible Elements of Functions on T * (G/N ).
In this section, we use the Gelfand-Graev action to compute some irreducible elements of R: Lemma 5.9.For all fundamental weights ω i , w ∈ W , and nonzero z ∈ A ω i , the element w(z) ∈ R is irreducible.
Proof.It suffices to show this in the case when w = 1 since any w ∈ W gives a ring automorphism which in particular preserves irreducibility.Now, if z = ab for some a, b ∈ R, then by the Z ≥0grading given by the usual G m -action on the cotangent bundle, we see that a, b ∈ A. However, in A, we may identify the X • (T )-grading with a (Z ≥0 ) r grading using the fundamental weights.Then the irreducibility of z then follows as A 0 = k, which implies that any nonzero element of A v for v ∈ (Z ≥0 ) r of length one must be irreducible.□

Torus Stabilizers and Projections.
For any global function f on some scheme Y , we let D(f ) denote the complement of the vanishing locus of f , and, for any subset of global functions F , we let D(F ) := ∪ f ∈F D(f ).We now prove the following proposition, which informally says that for any p ∈ D(R λ ), there exists some w ∈ W such that π w (p) lies in an open locus of G/N determined by root hyperplanes which do not contain λ.
Proposition 5.10.Assume λ ∈ X • (T ) and w ∈ W such that wλ lies in the closure of the dominant Weyl chamber.Write wλ = i n i ω i with n i ∈ Z ≥0 , and let S wλ denote the subset of fundamental weights ω i for which n i ̸ = 0. Then π w (D(R λ )) maps into the open subscheme ∩ ω i ∈S wλ D(A ω i ).In particular, if λ is regular then π w (D(R λ )) maps into G/N .
Proof of Proposition 5.10.By Lemma 2.2, we see that the multiplication map Sym(g) ⊗ Zg Sym(t) ⊗ A wλ → R wλ is surjective.As A is generated by the union of the A ω where ω varies over the fundamental weights, we therefore in particular obtain the multiplication map ( 10) Let I denote the ideal of R generated by R wλ and, for a fixed ω i ∈ S wλ , let J i denote the ideal of R generated by A ω i .The fact that the multiplication map of ( 10) is surjective implies that R wλ ⊆ J i and so I ⊆ J i .In particular, we see that A ω i ̸ ⊆ wp.Thus π w (p) = A ∩ wp does not contain all of A ω i for any ω i ∈ S wλ , as desired. □ From this, we derive the following: Corollary 5.11.Assume λ 1 , ..., λ q ∈ X • (T ) span the vector space t * (Q).Then there exists some w ∈ W such that π w maps the set ∩ i D(R λ i ) into G/N .In particular T acts with no stabilizer on any point in We show Corollary 5.11 after proving the following Lemma: Lemma 5.12.Assume L ⊆ Q d is some full rank lattice and choose some basis Denote by C the R >0 -span of S, i.e.
and assume Z is some closed subset of R d which does not contain C and which is closed under scaling by any positive real number.Then the Z >0 -span of S contains a point of L not in Z.
is open, and so in particular C ∩ Z c contains an open ball of some positive radius since by assumption it is nonempty.Since C ∩ Z c is closed under the scaling of any positive real number, for any r ∈ R >0 , C ∩ Z c contains an open ball of radius r.However, for any full rank lattice, there exists some M such that all points in R d are distance at most M from a point on that lattice.Choosing r > M we see that there is an element of x ∈ L ∩ Z c in the R >0 -span of S. As S is a basis and x in particular lies in Q d , we see that x lies in the Q >0 -span of S.There exists some positive integer N such that N x therefore is a Z >0 -linear combination of the ⃗ v i , as desired.□ Proof of Corollary 5.11.Assume p ∈ ∩ i D(R λ i ).Let L := X • (T ) and choose some subset S of the λ i such that S is a basis of X • (T ) ⊗ Z Q.If Z denotes the union of root hyperplanes, we may apply Lemma 5.12 to show there is some λ ∈ X • (T ) which lies in the interior of some Weyl chamber and which is a Z >0 -linear combination of the elements of S. Since p ∈ D(R λ i ) for every i, we see p ∈ D(R λ ).Choose the (unique) w ∈ W which takes λ to the dominant Weyl chamber.Then since wλ is regular, by Proposition 5.10 we see that π(wp) maps to G/N .In particular, π(wp) has no T -stabilizer, and thus neither does p itself, since π is T -equivariant.□ From this we may also derive a similar result for the locus of points whose T -stabilizer has dimension one: Corollary 5.13.Assume that the T -stabilizer of some point p ∈ T * (G/N ) has dimension one.Then there exists some w ∈ W and some simple root α such that π(wp) ∈ G/[P α , P α ].In particular, the T -stabilizer of p is wG α m w −1 .Proof.For such a point p, we define the set D := {λ ∈ X • (T ) : p ∈ D(R λ )}.Using (1), we may view D as a subset of t * (Q) and let V Q denote the span of D in t * (Q).
We claim that the dimension of V Q is at least dim(t * ) − 1.To see this, assume the dimension of V Q was less than dim(t * ) − 1.In this case, there would be two linearly independent elements ν 1 , ν 2 ∈ t(Q) such that ν 1 (s) = 0 = ν 2 (s) for any s ∈ D. Here, we use the finite dimensionality of t(Q) to canonically identify it with the vector space dual of t * (Q).Now using the isomorphism (2) we see that we may replace ν 1 and ν 2 by a positive integer multiple if necessary and additionally assume that both ν 1 and ν 2 are in X • (T ).By the definition of D, we have that the image of each map ν i : G m → T stabilizes p.However, the fact that ν 1 and ν 2 form a linearly independent set implies that the subgroup generated by the images of these ν i has dimension two, violating our assumption on the dimension of the stabilizer of p.
Let V R denote the R-span of the elements of D, let Z ′ denote the union of every intersection of two distinct root hyperplanes, and let and let L denote the lattice generated by S. Since the dimension of V Q is at least dim(t * ) − 1, we see that Z does not contain the R ≥0 -span of S, and so we may apply Lemma 5.12 to see that the Z ≥0 -span of the elements of S contains some λ ∈ X • (T ) such that λ lies on at most one root hyperplane.Since the Z ≥0 -span of S is contained in D as D is its own Z ≥0 -span, we see that p ∈ D(R λ ).
Choose some w ∈ W such that wλ lies on a root hyperplane cut out by at most one simple coroot.By Proposition 5.10, we see that if wλ is contained in no root hyperplanes then π(wp) projects into G/N and so in particular the T -action on p is free.Thus wλ is contained in exactly one root hyperplane cut out by the vanishing of a simple coroot α ∨ .Therefore by Proposition Proof.If the T -stabilizer of p ∈ T * (G/N ) has dimension zero, then we in particular see that the set of wω i for which p ∈ D(R wω i ) spans the rational points of t * , since otherwise they would be contained in some hyperplane cut out by some α ∈ t(Q) and thus fixed by some subtorus.Therefore by Corollary 5.11 some element of the W -orbit of p projects to G/N under π.If the T -stabilizer of p has dimension one, by Corollary 5.13 some point in its W -orbit projects to some point of G/[P α , P α ] under π.In either case, π w (p) lies in G/N ∪ G/[P α , P α ] for some w.Since G/N ∪ G/[P α , P α ] ⊆ S by our simply connectedness assumption, we see that p is smooth by Proposition 4. With the exception of Remark 5.18, the proof of Proposition 5.16 will occupy the remainder of Section 5.4.We first construct a stratification of T * (G/N ) by T -invariant locally closed subschemes which will also be used in Section 5.5.Let F denote the set of fundamental weights.For any (w, ω) Notice that A ω = A 1,ω for any ω ∈ F .By Lemma 2.2, the sets give a stratification of T * (G/N ) by locally closed T -invariant subschemes, where S varies over subsets of W × F .Moreover, for a fixed S, any two closed points in S S have the same (right) T -stabilizer T S .Since there are finitely many such S S , Proposition 5.16 follows from the following proposition: Proposition 5.17.Assume that S ⊆ W × F such that T S has dimension at least two.Then S S has codimension at least four in T * (G/N ).
Proof.Fix a subset S ⊆ W × F such that T S has dimension at least two.Letting S 0 := R, we recursively construct k-algebras S 1 , S 2 , S 3 , S 4 such that for every i ∈ {0, 1, 2, 3}, and such that by construction S S is a locally closed subscheme of Spec(S 4 ).The existence of this construction automatically implies that S S has codimension at least four.
Choose some fundamental weight ω such that (1, ω) ∈ S. Such an ω exists since, if not, where the intersections are taken over all fundamental weights, and therefore any point of S S has no T -stabilizer by, for example, Corollary 5.11.Choose some nonzero a 1 ∈ A ω and let and (1, ω) ∈ S, S S is a locally closed subscheme of Spec(S 1 ).Moreover, since R is a unique factorization domain and a 1 is an irreducible element of R by Lemma 5.9, S 1 is an integral domain.
Since a 1 is a nonzero element in an integral domain, we also see that (11) holds when i = 0. We also have that, since a 1 is homogeneous with respect to the X • (T )-grading, the ring S 1 admits a grading by X • (T ).The grading on S 1 has the property that (12) S 1 λ ̸ = 0 for any λ ∈ X • (T ) by Lemma 2.2 and the unique factorization of R given by Corollary 3.5.
As T S has dimension at least two, we may choose two elements γ 1 , γ 2 ∈ X • (T S ) which are linearly independent in Lie(T S )(Q), and we denote by T 1 , respectively T 2 the rank one subtori generated by γ 1 , respectively γ 2 .Define S 2 := S 1 γ 1 =0 so that, by definition, S 2 is the subring of S 1 generated by those S 1 λ such that ⟨γ 1 , λ⟩ = 0. Then since ( 12) holds, we may apply Lemma 4.1 to see that (11) holds when i = 1.Moreover, since S S is a locally closed subscheme of Spec(S 1 ) T 1 and the quotient map identifies Spec(S 1 ) T 1 as a closed subscheme of Spec(S 2 ) = Spec(S 1 ) T 1 by Proposition 4.2, we see that we may view S S as a locally closed subscheme of Spec(S 2 ).Similarly, define S 3 = S 2 γ 2 =0 .Since (12) holds for λ which satisfy ⟨γ 1 , λ⟩ = 0 ̸ = ⟨γ 2 , λ⟩ we see that we may similarly apply Lemma 4.1 to see that (11) holds when i = 2, and, exactly as above, we may view S S as a locally closed subscheme of Spec(S 3 ) = Spec(S 2 ) T 2 .Moreover, S 3 is an integral domain as it is a subring of the integral domain S 1 .
Finally, choose some a 2 ∈ A ω such that {a 1 , a 2 } is linearly independent and choose some nonzero b ∈ A w 0 ,w 0 (−ω) .If we set S 4 := S 3 /(a 2 b) then, since ω ∈ S, S S (viewed as a locally closed subscheme of Spec(S 3 ) via the quotient map as above) is contained Spec(S 4 ).We now claim that a 2 b is not zero in S 3 .To see this, notice that if a 2 b = 0 in S 3 ⊆ S 1 then there exists some f ∈ R such that a 1 f = a 2 b in R.However, since a 1 , a 2 , b are irreducible in R by Lemma 5.9, this would violate the fact that R is a unique factorization domain, i.e.Corollary 3.5.Since a 2 b is a nonzero element in the integral domain S 3 , any irreducible component of Spec(S 4 ) has codimension 1 in Spec(S 3 ).□ Remark 5.18.The reader who is only interested in the proof of the main theorems when G is simply connected may proceed directly to Section 6.2, replacing the usage of Theorem 6.2 in final sentence of the proof of Theorem 6.3 with the usage of Theorem 5.15.
5.5.Free Locus Has Codimension Four.Using the notation introduced in Section 5.4, let Z i denote the closed subscheme V (A ω i ) ∩ V (A w 0 ,w 0 (−ω i ) ), and let U i denote its open complement.In this section, we study properties of the open subscheme Q := ∪ w w(∩ r i=1 U i ).We first give a more explicit description for Q, which can be compared to [14, Proposition 5.1.4]:Proposition 5.19.We have an equality (13) Q = ∪ w π −1 w (G/N ).Moreover, Q is the set of points of T * (G/N ) for which the (right) T action is free and is the set of points of T * (G/N ) for which the T -stabilizer has dimension zero.In particular, any point of T * (G/N ) whose T -stabilizer has dimension zero has trivial stabilizer.
Proof.To show ⊆ in (13), by W -equivariance it suffices to show ∩ r i=1 U i ⊆ ∪ w π −1 w (G/N ).Choose some homogeneous function f i ∈ A ω i ∪ A w 0 ,w 0 (−ω i ) and let λ i ∈ {±ω i } denote the degree of f i .It further suffices to show that ∩ r i=1 D(f i ) ⊆ ∪ w π −1 w (G/N ) which follows from Corollary 5.11.Conversely, by W -equivariance we may show the containment π −1 (G/N ) ⊆ ∪ w w(∩ r i=1 U i ), but this follows from the fact that π −1 (G/N ) = ∩ r i=1 D(A ω i ) ⊆ ∩ r i=1 U i .By the T -equivariance of π, we see that any point in T * (G/N ) which maps to G/N under π must itself have trivial (right) T -stabilizer.Now assume that the right T -action on T * (G/N ) for some point p has dimension zero.Then for any γ ∈ X • (T ) there exists some λ ∈ X • (T ) and f ∈ R λ such that ⟨λ, γ⟩ ̸ = 0 and f does not vanish at p, as otherwise it would be stabilized by the one parameter subgroup γ : G m − → T .In particular, if we let S denote the set of all λ ∈ X • (T ) such that R λ contains a function which does not vanish at p, then we see that the elements of S span the real points of t * .Thus we see that p ∈ π −1 w (G/N ) for some w ∈ W by Corollary 5.11 and, therefore by the above must also have trivial T -stabilizer. □ From this, we can derive the codimension result on the complement of Q (for G simply connected) stated in the introduction, whose proof occupies the remainder of this entire subsection: Corollary 5.20.The complement of Q has codimension at least four.
For this remainder of this section, we fix a simple root α, choose an SL 2 -triple for α and use the notation of Section 5.1.3.We also let p α : T * (V α ) → V α denote the projection map and set p α,s := p α • S α where S α : T * (V α ) → T * (V α ) is the symplectic Fourier transform.We also define Y α := G/Q α and view Y α as a closed subscheme of V α via the zero section map.
Lemma 5.21.The scheme theoretic intersection Proof.First, the symplectic Fourier transform on V α is, by construction, an automorphism over Y α .Therefore the map (p α , p α,s ) : factors through the closed subscheme V α × Yα V α .We wish to compute the dimension of the closed subscheme gives the symplectic form In particular, we obtain isomorphisms ( 14) where the final isomorphism is induced by the symplectic form.Under the composite identification obtained from reading ( 14) left to right one can directly follow the construction of the symplectic Fourier transform as in, for example [17,Appendix B], to see that it is given by the automorphism of T * ( C) × A 2 × A 2 .Therefore, using the above trivialization, we may identify p α | U with the map where z is the image of p under the map T * (C) → C. We may similarly identify the map p α,s | U with (z, again using the identification induced by the composite of the isomorphisms of ( 14).Therefore, via the isomorphisms of ( 14), we can identify which evidently has codimension four in U .□ Proof of Corollary 5.20.We temporarily denote by Z Q the complement of Q in T * (G/N ).Since we can identify Q with the locus where T acts with dimension zero stabilizers Proposition 5.19, we can write

those points whose right T -stabilizer has dimension one and Z ≥2
Q is the closed subscheme consisting of those points whose right T -stabilizer has dimension at least two.By Proposition 5.16, the codimension of Z ≥2 Q is at least four, so it suffices to show that Z 1 Q has codimension at least four.By Corollary 5.13, we may write Z 1 Q as the union , where w varies over W , α varies over the simple roots, and Z 1 Q,w,α denotes the subset of points of We recall the following standard lemma on extendability of differential forms as applied to the theory of symplectic singularities: Lemma 6.4.Assume X is some irreducible variety.
(1) Assume Z is some closed subscheme of X reg whose codimension is larger than 1.Then if ω ∈ Ω 2 (X reg \ Z) is some symplectic form, then ω extends to a symplectic form on X reg .(2) If in addition X is normal and codim X (X \ X reg ) ≥ 4, X has symplectic singularities.
Proof.A standard Hartog's lemma argument gives that ω extends to a symplectic form on X reg , see, for example, [1, Section 4, Remarque (3)].Now, any normal irreducible variety with a nondegenerate 2-form on the regular locus has symplectic singularities if and only if, for any resolution p : Y → X of singularities, the induced 2-form p * (ω) extends to a 2-form on Y .However, since the codimension of the singular locus is larger than 3, this extension follows directly from the main theorem of [11].□ Proof of Theorem 6.3.We check the hypotheses of Lemma 6.4.The fact that T * (G/N ) is normal follows from Proposition 3.1.The variety T * (G/N ) is an open subscheme of T * (G/N ) whose complement has codimension at least two by Lemma 3.4, so the symplectic form on T * (G/N ) necessarily extends to a symplectic form on the smooth locus of T * (G/N ).Finally, the fact that T * (G/N ) has a singular locus of codimension ≥ 4 follows from Theorem 6.2.□ 6.3.Consequences of Main Theorem.We have seen that T * (G/N ) has symplectic singularities in Theorem 6.3 and that its singular locus has codimension at least four in Theorem 6.2.Recall that all singular symplectic varieties whose singular locus has codimension at least four are terminal by the main result of [29].Therefore we immediately obtain the following result which, combined with Proposition 3.1, completes the proof of Theorem 1.2: Corollary 6.5.The variety T * (G/N ) has terminal singularities.
We also claim that, if G is semisimple and not a product of copies of SL 2 , that T * (G/N ) is singular: Proposition 6.6.When G is semisimple and not a product of copies of SL 2 , the cone point 0 ∈ T * (G/N ) is singular.
Proof.Notice that the ideal cutting out the image of the closed embedding given by the zero section z : G/N − → T * (G/N ) is homogeneous for both the torus action and the usual G m -action induced by scaling fibers on T * (G/N ).Therefore, by Proposition 3.6(2), this ideal is also homogeneous for the conical G maction.In particular, the cone point is contained in this closed subscheme.
It is well known that G/N is singular when G is semisimple and not a product of copies of SL 2 -for example, this follows from the fact that the ring of differential operators on any smooth affine variety is generated by derivations [26,Corollary 15.6] but the results of [24] show that this is not the case for such G.Since the singular locus of a scheme is closed and the singular locus of a scheme with a group action is closed under the action of that group, we see that the singular locus of G/N contains the cone point for such G since any nonempty closed G m -invariant subscheme of G/N contains the cone point.Therefore dim(T 0 (G/N )) > d := dim(G/N ), and, since z • π = id, π induces a surjective map (16) T 0 (T * (G/N )) → T 0 (G/N ) on tangent spaces.Note also that, by for example Proposition 4.3, the generic fiber of π has dimension exactly d.Therefore, by upper semicontinuity of the fiber dimension, we see that the fiber F of π at the cone point of T * (G/N ) has dimension at least d.In particular, T 0 (F ) has dimension at least d and lies in the kernel of the map (16).Therefore by rank-nullity the dimension of T 0 (T * (G/N )) is larger than d + d = dim(T * (G/N )), and so the cone point is singular.□ On the other hand, the codimension of the singular locus of T * (G/N ) is at least four by Theorem 6.2.Thus the Q-factoriality of T * (G/N ) in Proposition 3.1 allows us to use [12,Corollary 1.3] to show the following, which generalizes a remark of [16,Section 1.3] for G = SL 3 to all types: Corollary 6.7.If G is semisimple and not a product of copies of SL 2 then the variety T * (G/N ) does not admit a symplectic resolution.
The fact that T * (G/N ) admits conical symplectic singularities for semisimple G implies that is a natural object in the study of symplectic duality.The following result determines properties of the conjectural symplectic dual to T * (G/N ) and should be compared to the expectations of [8, Section 8]: Corollary 6.8.There are no nontrivial flat Poisson deformations of T * (G/N ).
Proof.It suffices to show that the vector space HP 2 (T * (G/N )) is zero, see, for example, [15], [31], [30].In fact, HP 2 (Y) vanishes for any normal affine variety Y with terminal symplectic singularities and finite class group.We give the proof of this well known result here for completeness.
Since Y has terminal symplectic singularities, we have that HP 2 (Y) ∼ = H 2 (Y, C), where Y denotes the smooth locus of Y [31].Now [25,Lemma 4.4.6]shows that the first Chern class gives an isomorphism Pic(Y )

4. 2 .
Fiber of Projection Over Smooth Locus.Next, we compute the fiber of π over its smooth locus S. Notice that the complement of T * (G/N ) in T * (S) has codimension ≥ 2 by the stratification(4).From this, we see that O(T * (S)) ∼ − → O(T * (G/N )), and therefore we obtain a canonical map σ : T * (S) → T * (G/N ) × G/N S. Proposition 4.3.The map σ is an isomorphism.

5. 1 .
Stratification of Affine Closure of G/N .For any subset of simple coroots or subset of simple roots θ, we let P θ denote the standard parabolic subgroup containing B and labeled by θ.The goal of Section 5.1 is to prove the following well known proposition whose proof we are unable to find in the literature: Proposition 5.1.The variety G/N is quasi-affine and its affine closure admits a stratification (4) G/N = θ G/[P θ , P θ ] into the orbits of the action of G × T , where θ varies over all subsets of simple roots.Moreover, the smooth locus of G/N contains all locally closed subschemes G/[P θ , P θ ] where |θ| = 1.

|J|m
is abelian, Q I contains [P I , P I ] and, since [P I , P I ] and Q I are affine (as they are closed subgroup schemes of G), the inclusion map [P I , P I ] − → Q I is a closed embedding.Proposition 5.2.The closed embedding [P I , P I ] − → Q I is an isomorphism.
Cartier gives any affine algebraic group in characteristic zero is smooth (see for example[27,  Theorem 3.23]) and so any connected affine algebraic group over our characteristic zero field is in particular irreducible; it thus suffices to show that this closed embedding induces an equality of the Lie algebras of [P I , P I ] and Q I .Moreover, both [P I , P I ] and Q I are normalized by P I so their Lie algebras in particular admit T -representations.However, the Lie algebra of [P I , P I ] contains all of n = [b, b] and each h β := [e β , f β ] and each f β = −2[h β , f β ] for every simple coroot β ∨ ∈ I where e β ∈ n β and f β ∈ n − β yield an sl 2 -triple.Therefore the tangent spaces of both groups of Proposition 5.2 are equal, as desired.□

5. 1 . 2 . 1 : 5 . 5 .
Stratification of Affine Closure of G/[P I , P I ].From Proposition 5.2, we can now show the following claim, which when I = ∅ recovers the first sentence of Proposition 5.Proposition The variety G/[P I , P I ] is quasi-affine.Moreover, we have a G × T -equivariant stratification of its affine closure (8) G/[P I , P I ] = K⊇I G/[P K , P K ]

1 : 5 . 8 .
whose restriction to the open G-orbit G/N the open embedding G/N ⊆ G/N .Therefore the claim follows by comparing the stratification of V α = G/N ∪ G/[P α , P α ] into the orbits of the action of G × T to the stratification of Proposition 5.5 with I = ∅.□Since V α is smooth, we in particular see the following, which completes the proof of Proposition 5.Corollary The smooth locus of G/N contains G/N and G/[P α , P α ] for any simple root α.

Corollary 5 . 14 .
5.10 π(wp) ∈ G/[P α , P α ].The latter claim follows from the fact that π is compatible with the T -action, which shows that the T -stabilizer of wp is a closed subgroup scheme of G α m .□ If the right T -stabilizer of some point p ∈ T * (G/N ) has dimension zero or one, then p is a smooth point.

3 . □ 5 . 4 . 1 : 5 . 15 .Proposition 5 . 16 .
Codimension of Singular Locus of Affine Closure of T * (G/N ).We now use the results of Section 5.3 to prove the following, which is a key ingredient in our proof of Theorem 1.Theorem The singular locus of T * (G/N ) has codimension at least four.By Corollary 5.14, Theorem 5.15 follows directly from the following proposition: The locus of points of T * (G/N ) whose T -stabilizer has dimension ≥ 2 has codimension at least four.
where we regard T * (V α ) (and thus Z α ) as a scheme over Y α in the natural way.It suffices to do this on an open cover of V α .Let C denote the open B-orbit of G/P α , and let C denote its inverse image under the quotient mapG/Q α → G/P α .Defining U 0 := f −1 α ( C), the open subset U := p −1 α (U 0 )gives a nonempty open subscheme of T * (V α ).Moreover, since f α and p α are G-equivariant and the action of G on G/Q α is transitive, we see that U and its G(k)-translates cover T * (V α ).It therefore suffices to show U ∩ Z α has codimension four in U by G(k)-equivariance.The construction of V α [22, Section 2.1] gives a trivialization U 0 Therefore, since Pic(Y ) is finite (since it is a subgroup of the class group of Y) we see that H 2 (Y, C) ∼ = H 2 (Y, Q) ⊗ Q C ∼ = 0 as desired.□ embedding, and moreover if we denote the image of α ∨ 32, Lemma 29.13.4].