Springer Theory: Lecture Notes from a Reading Course George H. Seelinger

Springer Theory: Lecture notes from a reading course

George H. Seelinger

1. Introduction (presented by Weiqiang Wang) Yarrrrr pirates! (8/28/2017) Lecture 1: Semisimple Lie Algebras and Flag Va- rieties (presented by Mike Reeks).

2. Semisimple Lie Algebras and Flag Varieties In this section, we will review essential facts for our study going forward. Fix G to be a complex semisimple Lie group with Lie algebra g. 2.1. Proposition. g is a G-module via the adjoint action. That is, for g ∈ G, x ∈ g g.x = Add(φg)I (x) −1 where φg : G → G maps h 7→ ghg for h ∈ G.

Proof. Add in a proof here 2.2. Definition. A maximal solvable subgroup of G is called a Borel subgroup. 2.3. Definition. A torus of a compact Lie group G is a compact, connected, abelian Lie subgroup of G. 2.4. Proposition. Given a Borel subgroup B ≤ G of a Lie group, B has a maximal torus T ≤ G and a unipotent radical U ≤ G such that T · U = B.

Proof. Add in a proof here 2.5. Example. Let G = SLn(C). Then,       ∗ ∗ ∗ 0 1 ∗       B =   ,T =   ,U =         0 ∗ 0 ∗ 0 1 In particular, B = TU. 2.6. Definition. When considering the tangent spaces at the identity, we get a Borel subalgebra b = h ⊕ n where h is a Cartan subalgebra and n is a nilradical. 2.7. Remark. Note that n = [b, b] and b/[b, b] =∼ h. 2.8. Definition. We deﬁne the rank of a Lie algebra g to be rank g := dim h

2.9. Lemma. (a) Let B ≤ G be Borel. Then, B = NG(B) (that is, B is self-normalizing). (b) All Borel subgroups are G-conjugate

Proof. Add a proof here

3 2.10. Definition. For x ∈ g, we deﬁne

Zg(x) = {y ∈ g | [x, y] = 0},ZG(x) = {g ∈ G | g.x = x}

2.11. Example. Let g ⊆ gl2(C) be all 2 by 2 upper-triangular matrices. Then,

2.12. Proposition. dim(Zg(x)) ≥ rank g

Proof. Let x be semisimple. Then, x is contained in some Cartan subalgebra h. However, h ⊆ Zg(x). Thus, the proposition is true for x semisimple. However, semisimple elements are Zariski dense, so the result applies for the closure. Check this last thing. 2.13. Definition. We say x ∈ g is regular if dim Zg(x) = rank g. 2.14. Definition. We say x ∈ g is semisimple (resp. nilpotent) if ad(x) is semisimple (resp nilpotent). 2.15. Remark. This result is actually somewhat non-trivial, but is cov- ered by Engel’s theorem and related results.

Now, recall that any x ∈ g has a Jordan decomposition x = s + n where s ∈ g is semisimple and n ∈ g is nilpotent and s, n commute. 2.16. Definition. Let gsr be the set of semisimple regular elements in g. 2.17. Example. Note that not all regular elements are semisimple. Take   0 1         x =  1        0

Such an element is regular in sln because Zg(x) is spanned by the basis 2 n−1 n−1 {x, x , . . . , x }, which is the same size as the basis {eii − ei+1,i+1}i=1 that spans the standard Cartan subalgebrah in sln. However, since it is nilpotent, it cannot be semisimple. 2.18. Proposition. Fix h ⊆ b ⊆ g. Then, (a) Any element of h is semisimple and any semisimple element is G- conjugate to an element of h. sr (b) If x ∈ g , then Zg(x) is a Cartan subalgebra. (c) gsr is a G-stable dense subset of g (in the Zariski topology?)

“Proof”. Any element pf h is semisimple by deﬁnition. Also, in Humphreys chapter 16, he gives an argument that all Cartan subalgebras are G-conjugate

4 using the conjugacy of Borel subalgebras.

For the second claim, clearly Zg(x) is a subalgebra. Varadarajan gives an argument for why it is a Cartan subalgebra. reproduce this here. 2.19. Example. A typical example of part (b) in sln is given by   λ 0  1    x =     0 λn with λi distinct. Then, Zg(x) is given by matrices (aij) such that X X 0 = [(aij), x] = [aijeij, x] = (aijλjeij − λiaijeij) i,j ij

So, if i 6= j, then aij = 0 since λi 6= λj. That is to say, Zg(x) is all diagonal matrices in sln 2.20. Lemma. There exists a G-invariant polynomial P with coeﬃcients in g such that x ∈ gsr ⇐⇒ P (x) 6= 0

Proof. Take x ∈ g. Since dim ker(adx) ≥ rank g, the characteristic polynomial of adx is given by r r+1 det(tI − adx) = t Pr(x) + t Pr+1(x) + ··· where Pi are G-invariant polynomials on g and r = rank g. So, we have that x is regular if and only if t = 0 is a zero of order r of the characteristic polynomial. However, this is possible if and only if Pr(x) 6= 0. Thus, for x is semisimple and regular, we have found a desired polynomial. Since this polynomial is G-invariant and all semisimple elements are conjugate to an element in h, Figure out why this polynomial Flag Variety. applies to all regular semisim- 2.21. Definition. Deﬁne B to be the set of all Borel subalgebras of g. ple elements. 2.22. Proposition. B is a projective variety. 2.23. Definition. Let Gr be the Grassmanian of dim b-dimensional subspaces of g. 2.24. Proposition. B ⊆ Gr as a closed subvariety of solvable subalgebras. 2.25. Proposition. The stabilizer of b under the G-adjoint action is B, the Borel subgroup of G.

5 2.26. Proposition. For g ∈ G, the map g 7→ g·b·g−1 induces a bijection G/B → B Moreover, this is a G-equivariant isomorphism of varieties. n 2.27. Example. In type A, B = {F = (0 ⊆ F1 ⊆ F2 ⊆ · · · ⊆ Fn = C )} where dim Fi = i, the variety of complete ﬂags.

2.28. Proposition. Fix B ≤ G with Lie algebra b0. Then, we have the following bijections. 3 1 2 NG(T )/T = WT BnG/B {B-orbits on B} {G-diagonal orbits on B × B} where 1 is given by B · g · B 7→ B(g · B/B), 2 is given by B.b 7→ G.(b0, b), and 3 is given by W 7→ B · w · B where w is a representative of W in N (T ). G Get some handle on the proofs of 2.29. Corollary. As a result, B = F B , where B is a Bruhat w∈W w w these proposi- cell. tions! So W is a ﬁxed point set of the ∗ action on B. w ∈ W is w · B/B, C Understand this w ∈ W . So, B = B · (wB/B) gives us T W better. Why G G G/B = B = BW = BwB/B does this follow w∈W and why do we care? 2.30. Corollary. Each B-orbit in B contains exactly one point of the form wB/B, w ∈ WT . Make sense of this statement. 2.31. Proposition (Plucker embedding). Skipped for now.

2.0.1. Extended sln example. Let G = SLn and g = sln. Then, we have the following result. 2.32. Lemma. B is naturally identiﬁed with the variety of complete ﬂags.

Proof. Let F be a complete ﬂag. Then,

bF = {x ∈ sln | x(Fi) ⊆ Fi, ∀i} 1 2 For example, if F is the coordinate ﬂag, that is F = 0 ⊆ C ⊆ C ⊆ · · · ⊆ n C , then   ∗ b b   F = =  

Note that any 2 ﬂags are conjugate by the SLn action, so bF is Borel for every ﬂag F . Thus, our map is surjective. Hence, it is bijective and an isomorphism of algebraic varieties why does this last statement follow?

6 n 2.33. Lemma. Consider C /Sn. It is isomorphic as a variety to C[λ]n−1, polynomials in λ of degree less than or equal to n − 1.

n Proof. Consider the map ψ : C → C[λ]n−1 given by n Y (x1, . . . , xn) 7→ λ − (λ − xi)

The result of this map does not depend on the order of the xi’s, so we can n mod out by the action of S on this map.

n n Now, given a linear map x: C → C , we have an unordered n-tuple of P eigenvalues {xi}, but if x ∈ sln, we know that xi = 0 since tr x = 0. So, take n−1 ∼ X C = {(x1, . . . , xn) | xi = 0} n n as the n − 1-dimensional hyperplane in C . It is still stable under the S - n−1 action, giving us a map φ: sln → C /Sn given by

x 7→ (x1, . . . , xn), the eigenvalues of x

2.34. Definition. Given the information above, for g = sln, we deﬁne the incidence variety of g, denoted g˜, to be

g˜ := {(x, F ) ∈ sln × B | x(Fi) ⊆ Fi} 2.35. Proposition. The following diagram commutes

g˜ µ ν

n−1 n g C ⊆ C

φ ψ n−1 C /Sn

n−1 where ν : g˜ → C sends (x, F ) to the ordered list of its eigenvalues λi where λi is the eigenvalue of the induced map Fi/Fi−1 → Fi/Fi−1 and µ: g˜ → g is the standard projection onto the ﬁrst coordinate (x, F ) 7→ x.

2.36. Definition. Let Bx := {F ∈ B | x(Fi) ⊆ Fi, ∀i}. sr 2.37. Proposition. For all x ∈ g , the set Bx consists of n − 1 points and has a canonical free Sn-action.

−1 sr Proof. Look at the eigenspaces. Note that ν|˜gsr = µ (g ) commutes with the Sn-action. Actually do the proof. 2.38. Theorem (Universal resolution for general g). Let g be an arbitrary Lie algebra. Then, we have incidence variety g˜ = {(x, b) ∈ g × B | x ∈

7 b} such that the following diagram commutes g˜ µ ν

g h

ρ π h/W where ν : g˜ → h is given by (x, b) 7→ x mod [b, b] (recall h = b/[b, b]) and ρ: g → h/W induces embedding ∼ W G C[h] → C[g] ,→ C[g] for Weyl group W . (9/4/2017) Lecture 2 (presented by Chris Leonard). Recall that, given a torus T ≤ G, we get a Cartan subalgebra that acts on itself by conjugation? This gives rise to a Weyl group WT = NG(T )/T . 2.39. Lemma. If b, b0 ∈ B, then there is a canonical isomorphism ∼ b/[b, b] → b0/[b0, b0] which is independent of choice of conjugation. 2.40. Definition. We deﬁne H = b/[b, b] to be the abstract Cartan subalgebra. 2.41. Proposition. Normally, we take a root system on the pair (h, b) but this gives rise to a root system on H and an abstract Weyl group W. Thus, the W action on h gives rise to a W action on H. 2.42. Theorem (Universal resolution for general g revisited). Let g be an arbitrary Lie algebra. Then, we have incidence variety g˜ = {(x, b) ∈ g × B | x ∈ b} such that the following diagram commutes g˜ µ ν

g H

ρ π H/W where ν : g˜ → H is given by (x, b) 7→ x mod [b, b] 2.43. Theorem (Chevalley’s Restriction Theorem). ∼ G W C[h] → C[h] T = C[H]W for abstract Weyl group W.

8 2.44. Corollary. From the above, using the Nullstenllensatz (Proposi- tion 2.2.2 in [CG00]), we get g = Specm C[g] → Specm C[H]W = H/W where this composition is ρ in the Universal Resolution. Make more sense of how this and 2.45. Definition. We say h ∈ H is regular if |W · h| = |W|. We define Hreg to be the collection of regular elements of H. Finally, we define g˜sr = the above two µ−1(gsr) = ν−1(Hreg), where µ, ν are from the Universal Resolution above. theorems fit together. Cur- 2.46. Proposition. ([CG00] Prop 3.1.36) µ: g˜sr → gsr is a principal rently, this is a −1 sr sr W-bundle, that is W acts freely on µ (x) and so g˜ looks like g × W. mess.

Proof. There exists a unique Cartan subalgebra h = Zg(x). We then recall that W acts freely and transitively on Borel subalgebras containing −1 h, so W acts freely on µ (x), the set of all Borel subalgebras containing x. Why? Must all Borel subalge- 3. The Nilpotent Cone bras containing x also contain h? 3.1. Definition. Let N = {nilpotent elements in g}. We define the nilpotent cone to be N˜ = µ−1(N ) = {(x, b) ∈ N × B | x ∈ b} where µ:g ˜ → g is the projection from 2.42. 3.2. Proposition. The projection π : N˜ → B makes N˜ into a vector bundle over B with fiber π−1(b) = n. 3.3. Proposition. Fix B, b. We can define a group action of B on G × b given by b · (g, x) = (gb−1, bxb−1)

3.4. Definition. Let G ×B b be the set of orbits of G × b under the B-action deﬁned above.

3.5. Proposition. G×B b is a G-equivariant vector bundle whose ﬁbres are b G ×B b (g, x)

G/B g.B/B 3.6. Proposition. The following diagrams commute ∼ ∼ G ×B b g˜ G ×B n N˜

G/B ∼ B G/B ∼ B

From this, we can conclude that N˜ is a smooth variety. Why?!

9 We now go into some necessary background for understanding our nilpotent cone.

3.1. Symplectic Manifolds. 3.7. Definition. A symplectic manifold M is a holomorphic manifold with a non-degenerate closed 2-form ω such that, for all x ∈ M,

ωx : TxM × TxM → C is symplectic. 2n 3.8. Example. Consider C with coordinates p1, . . . , pn, q1, . . . , qn and P 2n ω = dpi ∧ dqi. Then, with correct choice of basis for TxC , we get 0 I ω = x −I 0 3.9. Theorem. Locally, all symplectic manifolds look like the example above. 3.10. Definition. We define O(M) to be the set of all regular functions on M, that is, all 0-forms. 3.11. Definition. Let V (M) be the set of all vector fields on M. That is, η ∈ V (M) is given by η : M → TM with smooth choice x 7→ ηx ∈ TxM. 3.12. Definition. We define ζ : O(M) → V (M) by

ωx(−, (ζf )x) = dfx What is this? ζ = ζ(f)? Also, 3.13. Deﬁne {, } on O(M) by f Definition. where does the {f, g} = ω(ζf , ζg) − go? 3.14. Theorem. O(M) is a Poisson algebra. That is, O(M) is an asso- ciative algebra, a Lie algebra, and {a, −}: O(M) → O(M) satisﬁes a Liebniz rule: {a, b · c} = {a, b}· c + b ·{a, c}. 2 3.15. Example. Take M = C with coordinates p, q and ω = dp ∧ dq. 0 1 Then, we can represent ω = . Write η ∈ V (M) as x −1 0 φ(p, q) η = : M → T M = 2 ψ(p, q) (p,q) C

2 ∂f ∂f Then, ω(−, η) = (ψ, −φ): M × C → C and gives df = ( ∂p , ∂q ). We then Why is the do- 2 note that main M × C ? ∂f ! − ∂q −q 0 −p ζ = =⇒ ζ 2 = , ζ 2 = , ζ = f ∂f q /2 0 −p /2 −p pq q ∂p

10 Thus, we get q2 p2 −q 0 { , − } = ω , = pq 2 2 0 −p q2 q2 {pq, } = q2 = 2 2 2 p2 p2 {pq, − } = p2 = −2 2 2

So, we have a Lie algebra embedding sl2 ,→ O(M) given by q2 e 7→ 2 p2 f 7→ − 2 h 7→ pq Say something about 3.16. Definition. For η ∈ V (M), deﬁne Lη : O(M) → O(M) by ζ mapping

(Lηf)(m) = dfm(ym) O(M) → V (M) and getting a Lie algebra. 3.17. Proposition. The map V (M) → Der(O(M)) deﬁned by η 7→ 0 Lη is an isomorphism of Lie algebras where we deﬁne [η, η ] ∈ V (M) and Where did this L[η,η0] = [Lη,Lη0 ]. ym come from? What is it? 3.2. The Moment Map.

3.18. Proposition. A Lie group G acts on symplectic manifold M via a symplectic action φm : G → M given by g 7→ gm. This induces TeG = g action on TmM for all m given by Lie algebra homomorphism g → V (M). Write this out more explicitly. 3.19. Definition. A symplectic G-action is Hamiltonian if g

∃H ζ O(M) V (M)

We denote H(x) = Hx. 3.20. Definition. We deﬁne the moment map µ: M → g∗ by

µ(m): g → C x 7→ Hx(m)

2 3.21. Example. Take M = C and let SP2 = SL2 act on M. Then, we get Hamiltonian H : sl2 → O(M) by example 3.15. We also have a canonical

11 ∼ ∗ ∗ ∗ ∗ isomorphism between sl2 → sl2 given by e 7→ e , f 7→ f , h 7→ h . Thus, we have moment map µ: M → g given by 1 pq −p2 µ(p, q) = ∈ N because det = 0 2 q2 −pq q2 ∗ p2 ∗ ∗ 2 This is precisely equal to 2 e − 2 f + pqh , so µ: C → N is a 2-fold cover ramiﬁed at the origin. ramiﬁed?! 3.22. Proposition. If M is a manifold, then the cotangent bundle T ∗M is symplectic. What does it mean for a 3.23. Proposition. The action C on M induces an action of G on T cotangent bun- T ∗M, which gives us our Hamiltonian. dle to be sym- 3.24. Example. Take P ≤ G. Then, G acts on G/P and so G acts on plectic? T ∗(G/P ) . Let p⊥ ⊆ g∗ where p⊥ is the annihilator of p. What is CT ? 3.25. Proposition. ([CG00] Prop 1.4.9/10) In the case of the above What are these example, there is a natural G-equivariant isomorphism actions? ∗ ∼ ⊥ T (G/P ) G ×p p how?

G/P and the moment map is given by ∗ µ: G×p → g (g, α) 7→ gαg−1 This is great, because it is ex- 3.3. Return to the Nilcone. plicit, but where ∼ 3.26. Lemma. Recall that we can identify g → g∗ via the Killing form. does this come from? 3.27. Proposition. ([CG00] Prop 3.22/3) There is a natural G-equivariant isomorphism Recall this. N˜ =∼ T ∗B and µ: R∗B → N is the moment map from G acting on B and it is surjective. 3.28. Definition. The moment map above is called Springer’s resolution. Proof of proposition. Understand why T ∗ B =∼ T ∗(G/B) these are actually isomor- ∼ ⊥ = G ×B b phisms. ∼ = G ×B n =∼ N˜

12 3.29. Proposition. ([CG00] Prop 3.2.5) x ∈ g is nilpotent if and only if G G P (x) = 0 for all P ∈ C[g]+ (which is polynomials in C[g] with no constant term).

3.30. Definition. x ∈ g∗ is nilpotent if and only if P (x) = 0 for all ∗ G P ∈ C[g ]+.

Proof of proposition. Consider the universal resolution 2.42,

g˜

g H ρ

H/W

−1 with ρ (0) = N . Does this follow immediately 3.31. Corollary. ([CG00] Cor 3.2.8) N is an irreducible variety of or am I missing dimension 2 dim n. something obvi- 3.32. Proposition. ([CG00] Prop 3.2.10) Regular nilpotent elements ous here? for a single open Zariski-dense orbit in cN, denoted Oprin, the principal orbit.

3.33. Example. Let g = sln. Then, the orbit is the orbit of   0 1          1        0

3.34. Proposition. Any regular nllpotent element is contained in a unique Borel subalgebra.

3.35. Remark. Given µ: N˜ → N , µ is a resolution of singularities of N , that is it restricts to an isommorphism on an open dense subset.

(9/11/2017) Lecture 3: The Steinberg Variety (presented by Chris Chung).

3.36. Corollary. ([CG00] Cor 3.2.25) N = F O is an algebraic strat- iﬁcation of N , each Ois a smooth locally closed subvariety of N .

13 4. The Steinberg Variety

4.1. Definition. We deﬁne the Steinberg variety to be 0 0 0 Z := N˜ ×N N˜ = {(x, b, x , b ) ∈ N˜ × N˜ | x = x } which is isomorphic to {(x, b, b0) ∈ N × B × B | x ∈ b ∩ b0} 4.2. Proposition. From the standard resolution, we get the diagram Z π2 µZ N B × B

sign and so we have N˜ × N˜ =∼ T ∗B × T ∗B =∼ T ∗(B × B), that is, isomorphic up ∗ ∗ to a sign to account for the symplectic form ω = p1ω1 − p2ω2 where ω1 and ∗ ω2 account for the standard symplectic forms on T B. Thus, we can think of the following

(T ∗(B × B), ω) B × B p∗ 1 p2 ∗ p2 p1 ∗ ∗ (T B, ω1) (T B, ω2) B B ∗ or, alternatively, T (B × B) → B × B has ﬁbres n1 × n2 over (b1, b2), so T ∗(B × B) =∼ N˜ × N˜ where the map is given by

(n, b1, n, b2) 7→ (n, b1, −n, b2) 4.3. Proposition. ([CG00] Prop 3.3.4) Z is the union of conormal bundles to G-orbits in B × B. ∗ ∗ ∗ 4.4. Definition. For S ⊆ M, TS M = ker(T M|S → T S). For all ∗ ∗ x ∈ S, Tx S = {(x, ζ) ∈ Tx M | ζ(α) = 0, ∀α ∈ TxS}. 1−1 To prove this proposition, recall that W ↔ {G-diagonal orbits in B×B} given by w 7→ Yw.

Proof of Proposition. For Ω a G-orbit in B × B through (b1, b2), T ∗ (B × B) ⊆ Z. Recall N˜ ∼ T ∗B ∼ G × n = G × b⊥ where n = [b, b]. Yw = = B B Then, for a ζ ∈ T ∗ (B × B), ζ = (x , b , x , b ) ∈ g∗ × B × g∗ × B with Yw 1 1 2 2 ⊥ ⊥ x1 ∈ b1 , x2 ∈ b2 . Then, for any α ∈ T(b1,b2)Ω, α = (u.b1, u.b2) for u ∈ g and thus hx1, ui + hx2, ui = 0 for all u ∈ g. Thus, we get x1 = −x2. So, ζ = (x1, b1, −x1, b2) ∈ Z. 4.5. Corollary. ([CG00] Cor 3.3.5) G ∗ Z = TYw (B × B) w∈W

14 Moreover, the irreducible components of Z are TYw (B × B). 4.6. Theorem. Fix a Borel b ⊆ g with n = [b, b] and Bsuch that Lie(B) = b and b⊥ = n ⊆ g. Then, for any G-orbit O ⊆ g and x ∈ O ∩ b, we get O ∩ (x + n) is a lagrangian subvariety of O. 4.7. Definition. For a symplectic manifold M, Z ⊆ M is lagrangian subvariety of M if, for all x ∈ Z, TxZ ⊆ TxM is a lagrangian subspace of TxM. 4.8. Definition. A lagrangian subspace W of symplectic space V is a 1 subspace with dim W = 2 dim V and on which the symplectic form is 0.

Proof Idea of Theorem. Focus on x nilpotent. 1 4.9. Lemma. ([CG00] 3.3.8) dim(O ∩ n) ≤ 2 dim O, that is, O ∩ n is isotropic.

Proof. Let n = dim n = dim B so dim T ∗B = dim Z = 2n by the identification of Z with a disjoint union of conormal spaces and defintion −1 of a conormal bundle. Furthermore, let ZO = µZ (O) for O ⊆ N , that is 0 ZO = {(x, b, b ) ∈ Z | x ∈ O}. Then, for any x ∈ O, we have fibration

Bx × Bx ZO ⊆ Z

−1 Bx = µ (x) O and so, by a subadditivity property of ﬁbrations,

dim O + 2 dim Bx ≤ dim ZO.

Furthermore, because ZO ⊆ Z,

dim ZO ≤ dim Z = 2n Given x ∈ O and x ∈ n ⊆ b, we deﬁne −1 S := {g ∈ G | gbg ∈ Bx} It turns out that S is B-stable and so we get isomorphism ∼ B/S → Bx Bg 7→ gbg−1 and thus 1 dim O + 2 dim B ≤ 2n =⇒ dim S − dim B + dim O ≤ n. x 2 Note that O ∩ n = {gxg−1 | g ∈ S}. This leads us to the fact that ∼ S/ZG(x) → O ∩ n

15 and so dim S − dim ZG(x) = dim(O ∩ n), and from there 1 dim S + dim O ≤ n + dim B = dim G 2 1 =⇒ dim(O ∩ n) + dim O ≤ dim G − dim Z (x) = dim O by orbit-stabilizer. 2 G Thus, rearranging our inequality, we are done. Proof of Theorem. Given the lemma above, it suﬃces to show that 1 O ∩ n is coisotropic, that is dim(O ∩ n) ≥ 2 dim O. Consider W ⊆ V → W ⊥ω ⊆ W We can view O as a symplectic manifold with Hamiltonian B-action. This ∗ gets us a function µB : O → b and thus gives us the identiﬁcation O ∩ n = −1 µB (0), which is coisotropic ([CG00] Thm 1.5.7).

4.10. Example. Take G = SLn. Then, let

O = {x | x nilpotent, rank x = 1} ⊆ sln n Note that, in this case x + n = x since x is nilpotent. If we take V = C , then x is of the form v ⊗ w∗ : V → V given by u 7→ w∗(u)v. The nilpotence of x = v ⊗ w∗ tells us that w∗(v) = tr(x) = 0. Thus, we get a surjection from the set ∗ ∗ {v ⊗ w | w (v) = 0} O the orbit of trace free rank 1 linear maps. and thus dim O = (2n − 1) − 1 = 2n − 2. In coordinates, this gives us X O = {x = (aij) | x 6= 0, aij = αiβj, αiβj = 0} So, let n be the set of upper triangular nilpotent matrices. Then, X O ∩ n = {x = (aij) | x 6= 0, aij = 0 for i > j, aij = αiβj, αiβj = 0} Thus, irreducible components of O ∩ n are

 0 a1,k+1 . . . a1,n    . .   . .    0 . .       0 ak,k+1 . . . ak,n    | rows are all proportional  0    .    ..      0  Therefore, since all the rows are proportional, there are only n − 1 choices for entries, so it has dimension n − 1 and is thus a Lagrangian subspace. So, we have found O ∩ (x + n) = O ∩ n as a lagrangian subvariety. sr ∼ 4.11. Example. Consider Ox for x ∈ g . So, O = G/T . Take b such that x ∈ b, n = [b, b] and B such that Lie(B) = b. Then, x ∈ b ∩ gsr so

16 x+n = Bx ⊆ O is a single B-orbit and, according to our theorem, x+n∩O is a Lagrangian subvariety of O. To see this, we note that 1 1 dim(O ∩ (x + n)) = dim(x + ) = dim n = dim G/T = dim O N 2 2 Now, taking O˜ = µ−1(O), we get that µ: O˜ → O is an unramified cover with #W leaves. Note π1(O) = 0. Thus, we get −1 ∼ ˜ ˜ ZO = µZ (O) = O ×O O Using a resolution, we get m := 2 dim n = 2 dim B = dim N What was the last part of this 4.12. Corollary. The irreducible components of ZO have the same dimension example sup- posed to show? dim ZO = dim Z = m ∼ ∼ Proof. Recall π : N˜ = G ×B n → B = G/B which restricts to the Understand this ∼ isomorphism O˜ = G×B (O ∩n). This is a fibration and the dimension of the proof better. 1 ˜ fibres is 2 dim O. Irreducible components of O have dimension dim(G/B) + 1 2 dim O. This gives us that irreducible components of ZO have dimension 1 2 dim O˜ − dim O = 2(dim G/B + dim O) − dim O = 2 dim G/B = m 2 ∼ 4.13. Remark. Later, we will use this to show Hm(Z; Q) = Q[W]. F 4.14. Corollary. Z = O ZO is a partition of Z into locally closed subsets of dimension dim Z. 4.15. Corollary (Robin Schensted Correspondance). The number of nilpotent orbits of g is finite.

Proof. Let us examine the irreducible components of ZO. Let x ∈ O. Understand this Then, O = G/Gx by orbit stabilizer. Furthermore, Gx acts on Bx, giving us proof better. ˜ ∼ a G-equivariant isomorphism O = G ×Gx Bx. ˜ O G ×Gx Bx

∼ µ

O G/Gx ˜ ˜ ∼ So, we get ZO = O ×O O = G ×Gx (Bx × Bx), which allows us to see that the irreducible components of ZO are G ×Gx (B1 × B2) where B1, B2 are irreducible components of Bx. Thus,

dim O + dim B1 + dim B2 = 2 dim B

17 and if we take B1 = B2, we get 1 dim B = dim B − dim O x 2 which is the dimension of irreducible points of Bx. Thus, there is a one-to- one correspondance

1-1 α β {Irreducible components of ZO} ←→ {Gx-orbits on pairs (Bx , Bx )} ◦ α where Cx = Gx/(Gx) and Cx acts on {Bx }.

(9/18/2017) Lecture 4: Lagrangian Construction of the Weyl Group (presented by Thomas Sale).

5. The Lagrangian Construction of the Weyl Group

5.1. Definition. Throughout this talk, let 1 m := 2 dim n = dim Z = dim (N˜ × N˜ ) R R 2 R Now, we must review some material 5.0.1. Borel-Moore Homology ([CG00] 2.6).

5.2. Definition. We deﬁne the Borel-Moore Homology to be BM ˆ H∗ (X) = H∗(X, {∞}) where Xˆ = X ∪ {∞} is the 1-point compactiﬁcation of X.

BM 5.3. Remark. Note that H∗ (X) = H∗(X, X \X) for X any compacti- ciation of X.

5.4. Proposition. Let M be a smooth not-necessarily compact oriented manifold and X ⊆ M with dimR M = N. Then, BM ∼ N−i Hi (X) = H (M,M \ X) 5.5. Definition. Let us deﬁne the intersection pairing to be a billinear pairing

BM BM ˜ ∩ BM ˜ Hi (Z) × Hj (Z) Hi+j−N (Z ∩ Z)

∼ ∼

HN−i(M,M \ Z) × HN−j(M,M \ Z˜) ∪ H2N−j−i(M, (M \ Z) ∪ (M \ Z˜)) where ∪ is the standard cup product on cohomology and Z, Z˜ are closed subspaces of M.

18 5.6. Definition. We deﬁne the Kunneth formula for the Borel-Moore homology via the Kunneth formula for standard homology.

BM BM BM H∗ (M1) × H∗ (M2) H∗ (M1 × M2) ∼ ∼

H∗(M1, M1 \ M1) × H∗(M2, M2 \ M2) H∗(M1 × M2, M1 × M2 \ (M1 × M2 ∪ M1 × M2))

5.1. Set-Theoretic Composition ([CG00] 2.7).

5.7. Definition. Let Z12 ⊆ M1 × M2 and Z23 ⊆ M2 × M3. Then, we deﬁne

Z12◦Z23 := {(m1, m3) ∈ M1×M3 | ∃m2 ∈ M2 such that (m1, m2) ∈ Z12 and (m2, m3) ∈ Z23} Now, we can deﬁne convolution products on Borel-Moore homology with d = dimR M2. Note that, from this point onward, we are dropping the BM and all homology is Borel-Moore homology. 5.8. Definition. We deﬁne the convolution product to be a map

∗: H1(Z12) × Hj(Z23) → Hi+j−d(Z12 ◦ Z23)

(c12, c23) 7→ c12 ∗ c23 where c12 ∗ c23 = (p13)∗((c12 [M3]) ∩ ([M1] c23)) and p13 : M1 × M2 × M3 → M1 × M3 the standard projection map that is also proper.

5.9. Example. Let f, g be smooth functions f : M1 → M2 and g : M2 → M3. Then, Graph(f) ◦ Graph(g) = Graph(g ◦ f) Thus, [Graph(f)] ∗ [Graph(g)] = [Graph(g ◦ f)] 5.2. Lagrangian Construction of the Weyl Group. 5.10. Definition. For the Steinberg variety Z ⊆ N˜ × N˜, we deﬁne

H(Z) := Hm(Z, Q), the top Borel-Moore homology ˜ where m = dimR(Z) = dimR(N), as above. 5.11. Remark. Note that H(Z) is an algebra with respect to the convolution ∗ since Z ◦ Z = Z. Take M1 = M2 = M3 = N˜ to get

∗: Hm(Z) × Hm(X) → H2m−m(Z ◦ Z) 5.12. Proposition. dim H(Z) = |W|

Proof. Recall that Z = tZO (4.14). The number of these subsets is precisely |W|. This gives the desired result.

19 5.13. Theorem. There is a canonical algebra isomorphism ∼ H(Z) = Q[W] To prove this theorem, we must give a few definitions and lemmas. Let us fix h ⊆ b ⊆ g, w ∈ W, and regular semisimple h ∈ b. 5.14. Definition. Consider subset S ⊆ h. We define S −1 g˜ := ν (S) = G ×B (S + n) 5.15. Example. Note that h g˜ = g˜ = G ×B (h + n) = G ×B b and 0 g˜ = G ×B n = N˜ h w(h) h 5.16. Definition. We define Λw ⊆ g˜ × g˜ to be the graph of the sr W-action ong ˜ . Thus, h 0 0 0 0 Λw = {(x, b, x, b ) | x ∈ b ∩ b , ν(x, b ) = h, (b, b ) ∈ Yw} ⊆ g˜ × g˜ 5.17. Proposition. The projection map π : g˜ → B given by (x, b) 7→ b 2 2 h induces a map π : g˜ × g˜ → B × cB given by π (Λw) = Yw Proof. Note that h π×π Λw ,→ g˜ × g˜ → B × B This image is G-stable with respect to the diagonal action and, by the set h 2 h characterization of Λw, π (Λw) = Yw. 2 h 5.18. Lemma. The fibrations π :Λw → Yw and G ×B∩w(B) (h + n ∩ w(n)) → G/(B ∩ w(B)) are equivalent. why? Proof. Identify G/B × G/w(B) with B × B via the assignment −1 −1 (g1B, g2wB) 7→ (g1bg1 , g2w(b)g2 ) Finish this? 5.19. Lemma. For w, y ∈ W, w(h) h h [Λy ] ∗ [Λw] = [Λyw]

h Proof. Since Λw is a graph, then, for w, y ∈ W, w(h) h h Λy ◦ Λw = Λyw Thus, we get the desired result. 5.20. Definition. We deﬁne w Hw := Graph(H → H) ⊆ H × H

g˜ ×Hw g˜ := {(y, x) ∈ g˜ × g˜ | ν(y) = w(ν(y))} ν := ν × ν| w ˜g×Hw ˜g

20 and

Λw := (g˜ ×Hw g˜) ∩ (g˜ ×g g˜) −1 −1 −1 ˜ ˜ 5.21. Remark. Note that νw (0) = ν (0)×ν (0) = N ×N , the graph over special point 0. Also note that −1 ˜ ˜ Λw ∩ νw (0) ⊆ (g˜ ×g g˜) ∩ (N × N ) ⊆ Z 5.22. Definition. We deﬁne reg reg w reg Hw := Graph(H → H ) and thus reg −1 reg Λw := Λ2 ∩ νw (Hw ) sr sr = (g˜ ×Hw g˜) ∩ (˜g ×g g˜ ) = Graph(˜gsr →w g˜sr)

reg Now, we seek to specialize [Λw ] at 0 ∈ H.

5.23. Remark. Note that H\Hreg is the root hyperplanes and codimR(H\ Hreg) = 2. 5.24. Definition. Let ` ⊆ Hreg be a real vector space with dimension less than or equal to 2. Then, we deﬁne `∗ := ` \{0} = ` ∩ Hreg 5.25. Proposition. The following diagrams commute ˜ ˜ ˜ ˜ w(`) ` w(`) ` g ×Hw g g × g g˜ ×`w g˜ g˜ × g˜

νw ν×ν νw ν×ν w Graph(H → H) H × H Graph(` →w w(`)) ` × w(`)

Proof. The first diagram follows from the definition of νw. The second diagram follows from the natural projection g˜ → ` from the definition of νw in 5.20. 5.3. Specialization. In our case, specialization will be the process This entire spe- ` cialization stuff H∗(Λ ) → H∗(Z) w makes no sense `∗ 0·h [Λw ] 7→ [Λw ] ∈ Hm(Z) to me. 0·h 5.26. Lemma. ([CG00] 3.4.11) [Λw ] does not depend on h. Proof. The proof follows from the transitivity of specialization. See [CG00] 2.6.38. The specialization at ` equals the specialization at Rh for any h ∈ `∗. For h, h0 ∈ Href , draw a polygonal path in Hreg from h to h0 with 0 vertices h = h1, h2, . . . , hm = h . Then, the specialization at spanR(h1, hi+1) equals the specialization at Rh1 also equals the specialization at Rhi+1.

21 5.27. Lemma. Specialization commutes with convolution. (See [CG00] 2.7.23) 5.28. Lemma. h w(h) h 0 0 0 lim([Λyw = [Λy ] ∗ [Λw]]) = ([Λyw = [Λy] ∗ [Λw]]) h→0 0 5.29. Proposition. {[Λw] | w ∈ W} is a basis of H(Z). Proof. H(Z) has basis {T ∗ = T ∗ (B × B) | w ∈ }. w Yw W So, let X ∗ ΛY = nywTw w∈W Now, we have 2 h π (Λy ) ⊆ Yy lim h→0 2 0 π (Λy) ⊆ Yy

0 ∗ thus implying that [Λy] involves only Tw such that Yw ⊆ Yy, which leads to the Bruhat order. This is a mess! Understand it 5.30. n = 1 for any w ∈ . Lemma. ww W better

Proof. Over an open subset U ⊆ B × B containing Yw, lim h h→0 Λw = G ×B∩w(B) (h + n ∩ w(n)) G ×B∩w(B) (n ∩ w(n))

Yw Yw

Thus, nww = 1.

5.31. Remark. Kashiwara and Saito showed that the nyw are not the Kazhdan-Lusztig numbers.

(9/25/2017) Lecture 5 (presented by Ramanujan Santharoubane).

6. Geometric Analysis of H(Z)

Previously, for the Springer variety Z, we have shown that H(Z) =∼ Q(W) and C ⊗Q H(Z) = C[W] is a semisimple Lie algebra. By the Artin- Wedderburn theorem, we have M C ⊗Q H(Z) = EndC(Eα) α

22 where {Eα} is a complete set of g irreducible representations. Our goal for this section is to understand this decomposition.

Recall that, for the Springer resolution

N˜ ˜ ˜ µ Z = N ×N N N and, for Y ⊆ N , −1 −1 Zy = µ (Y ) ×Y µ (Y ) ⊆ N˜ × N˜

Now, ZY ◦ Z = ZY = Z ◦ ZY .

6.1. Proposition. H(ZY ) has a H(Z)-bimodule structure. (10/9/2017) Lecture 6 (presented by Liron Speyer).

(10/9/2017) Lecture 7 (presented by George H. Seelinger).

7. Applications of the Jacobson-Morozov Theorem 7.1. Statement and Immediate Applications. 7.1. Definition. Let (e, f, h) be a triple of elements in some semisimple Lie algebra g over a ﬁeld of characteristic 0. We say (e, f, h) is a sl2-triple if [h, e] = 2e, [h, f] = −2f, [e, f] = h.

We recall from a ﬁrst course in semisimple Lie algebras the following proposition. 7.2. Proposition. A Lie algebra g decomposes into a direct sum of ﬁnite-dimensional subspaces, each of which is isomorphic to Vj, the j + 1- dimensional simple sl2-module with highest weight j. Elaborate

To each of these simple sl2-modules, we can associate an sl2-triple in g. However, the Jacobson-Morozov Theorem can provide us with a partial converse to this fact. 7.3. Theorem (Jacboson-Morozov Theorem). [CG00, 3.7.1] Let g be a complex semisimple Lie algebra. For any nilpotent element e ∈ g, there exists h, f ∈ g such that {e, f, h} is an sl2-triple.

Thus, there exists a Lie algebra homomorphism γ : sl2 → g such that 0 1 0 0 1 0 7→ e, 7→ f, 7→ h 0 0 1 0 0 −1 Moreover, h is a semisimple element and f is a nilpotent element of g.

23 7.4. Remark. This triple (e, f, h) associated with e is not necessarily unique. The following proposition, due to Kostant, tells us to what extent the associated triple is unique.

7.5. Proposition (Kostant’s Theorem). [CG00, 3.7.3] Let ZG(e) denote the centralizer in G of e. Then, the above homomorphism γ : sl2(C) → g is determined uniquely up to conjugation by an element in the unipotent radical of the group ZG(e).

Proof of Jacobson-Morozov Theorem when g = sln(C). [CG00, p 184] By the Jordan canonical form, any nilpotent element is conjugate to a direct sum of Jordan blocks. Thus, we need only show the theorem is true when e is a single m by m Jordan block. Since e is nilpotent, all of its eigenvalues are 0, so   0 1         e =  1        0 Then, we take  m − 1 0 ··· 0   0 m − 3 ··· 0    h =  . ..   . .  0 · · · −m + 1  0 0   m − 1 0 0     0 2(m − 2) 0    f =  ..   .     −2(m − 2) 0 0  −(m − 1) 0 From this, it is straightforward to check that the set (e, f, h) actually forms an sl2-triple. 7.6. Lemma. [CG00, 3.2.16] e ∈ g is nilpotent if and only if

(e, x) = tr(ade · adx) = 0 for all x ∈ Zg(e)

Proof. (=⇒). Assume e is nilpotent and x ∈ Zg(e). Since ade and adx commute , for large enough k, why? k k k (ade · adx) = ade · adx = 0 by the nilpotency of x. Thus, ade · adx is also nilpotent and thus has trace 0.

24 (⇐=). Now assume (e, x) = 0 for all x ∈ Zg(e). We note that ade is skew- ⊥ symmetric with respect to the Killing form and so im(ade) = ker(ade) , where ⊥ stands for the annihilator with respect to (, ). Thus, we have

7.7. Lemma. e ∈ im(ade) is equivalent to (e, Zg(e)) = 0 is equivalent to there exists an element h ∈ g such that [h, e] = e.

However, we can also show there exists a semisimple element s ∈ g such that [s, e] = e. To do this, take h as above and write its Jordan decomposition h = s + n where s semisimple and n nilpotent. Since e is an eigenvector for adh, then e is an eigenvector for ads and adn. However, adn is nilpotent and thus only has eigenvalues 0, so adn e = 0. Therefore, ads e = adh e = e and so we have found semisimple s ∈ g such that [s, e] = e.

So, take h ∈ g to be semisimple such that [h, e] = e. Then, we can decompose g into adh-eigenspaces: M g = gα, gα := {x ∈ g | adh(x) = α · x} α∈C where h ∈ g0 and e ∈ g1. We also have relation adh ◦ ade = ade ◦(1 + k adh) which tells us ade takes gα to gα+1. Thus, ade takes gα to gα+k and k since there are only ﬁnitely many non-zero spaces gα, we get ad e = 0 for suﬃciently large k. Thus, e is nilpotent. 7.8. Lemma. If e ∈ g is nilpotent, then there exists a semisimple h ∈ g such that [h, e] = e.

Proof. If e ∈ g is nilpotent, then by the lemma above, (e, Zg(e)) = 0. Thus, following the “only if” part of the proof above, we can ﬁnd such an h.

Proof of Full Theorem. [CG00, pp191–2] Fix a nilpotent e ∈ g. Note that if dim g = 3, then we are done since g = sl2.

(1) Decomposing a non-nilpotent x ∈ Zg(e) (assuming such an x exists) into its Jordan decomposition x = s + n and using [x, e] = 0 =⇒ [s, e] = 0 (see [Hum72, p ] ), we argue by induction on dim g to Fill in page reduce to the case where the subalgebra Zg(e) consists of nilpotent number elements only. Since [s, e] = 0, we have s ∈ Zg(e), so Zg(e) contains a non-zero semisimple element. Furthermore, the centralizer of such an element is a proper reductive Lie subalgebra of t ⊆ g. Since e commutes with the semisimple component of t, we have e ∈ t. Thus, this semisimple component of t is a Lie algebra of dimension less than dim g containing e, so we are done by the inductive hypothesis. So, we must deal with the case of Zg(e) consisting only of nilpotent elements.

25 (2) Show there is a semisimple h ∈ g such that [h, e] = 2e. By the lemma above, for any nilpotent element e ∈ g, we have a semisimple h ∈ g such that [h, e] = e =⇒ [2h, e] = 2e. (3) Fix an h ∈ g such that [h, e] = 2e as above and introduce weight space decomposition M g = gα, gα := {x ∈ g | adh(x) = α · x}. α∈C where h ∈ g0, e ∈ g2 and adh ◦ ade = ade ◦(2 + adh) =⇒ ade takes gα to gα+2. Thus, if we ﬁnd f ∈ g−2 such that ade(f) = h, we are done. Since ade shifts the gradation by 2, this amounts to showing that h ∈ im(ade). However, from the lemma in the proof above, we see this holds if and only if (h, Zg(e)) = 0. To prove this, we use the Jacobi identity and the fact that [h, e] = 2e to get that, for x ∈ Zg(e), [e, [h, x]] = −[h, [x, e]] − [x, [e, h]] = −[h, 0] + [x, 2e] = 0

and so [h, Zg(e)] ⊆ Zg(e). So, we have that C · h + Zg(e) is a Lie subalgebra of g. Now, by the ﬁrst part of the proof, we can assume x is nilpotent and using Engel’s theorem, this tells us that Zg(e) is nilpotent and thus C · h + Zg(e) is a solvable Lie algebra. Thus, by Lie’s theorem, for all x ∈ C · h + Zg(e), we can put adx in the upper triangular form so that, for x ∈ Zg(e), adx is strictly upper triangular. Therefore, for any x ∈ Zg(e), adh · adx is strictly upper triangular and so tr(adh · adx) = 0 =⇒ (h, Zg(e)) = 0 =⇒ h ∈ im(ade) gives us the existence of an f ∈ g−2 such that ade(f) = h. 7.9. Remark. In [CM93], parts (1) and (2) of the proof are morally equivalent. However, for part (3) in [CM93, pp 38–39], they proceed by contradiction, but follow a somewhat similar proof. In [CG00, p 190], they claim that the standard proof is quite elementary but involves several tricky lemmas. However, this proof made no use of anything terribly sophisticated in geometry, so I do not quite understand this statement. 7.10. Corollary. Given a nilpotent e ∈ g, there exists a rational homomorphism γ : SL2(C) → G such that its diﬀerential dγ : sl2(C) → g sends 0 1 to e. 0 0

Proof. SL2(C) is simply connected. Thus, any Lie algebra homomor- 1 phism can be extended to a (unique) Lie group homomorphism .

1This theorem is a highly non-trivial diﬀerential geometry proof, but proved in [Hal03, section 3.6]

26 Recall that any simple ﬁnite dimensional sl2(C)-module V looks like f f f highest weight · · ··· · lowest weight e e e where the dots correspond to h-eigenspaces, each of dimension 1, and V = ker f ⊕ im e where ker f is the rightmost vertex. This decomposition holds for any, not necessarily simple, sl2-module.

7.11. Corollary. Let the sl2-triple (e, f, h) act on a ﬁnite dimensional vector space V . Assume that v ∈ V is such that f · v = 0 and h · v = −m · v. Then, m is a non-negative integer and we have em+1 · v = 0.

Proof. This follows since every sl2(C)-module is symmetric about the 0-eigenspace. 7.12. Corollary. Any nilpotent element of a semisimple Lie algebra g is acting as a nilpotent operator on any ﬁnite dimensional g-module.

Proof. By the Jacobson-Morosov Theorem, any nilpotent element e is part of an sl2-triple and thus, by the corollary above, for each v ∈ V , there m+1 is an m ∈ N such that e · v = 0. Thus, since V is ﬁnite-dimensional, e is a nilpotent operator on V .

7.13. Corollary. Fix a nilpotent e ∈ g and corresponding sl2-triple (e, f, h). Then,

(a) All the eigenvalues of the operator adh : Zg(e) → Zg(e) are nonnegative integers. (b) If all the eigenvalues of the operator adh : g → g are even, then dim Zg(e) = dim Zg(h).

Proof. From the sl2-triple, we have an embedding sl2(C) ,→ g. The adjoint action on g of the image of the embedding makes g an sl2(C)-module. Then, by the decomposition of sl2(C)-modules, we see that all the eigenvalues of adh : g → g are integers and the weight space decomposition into adh-eigenspaces yields a Z-grading on the Lie algebra g: M g = gi, [gi, gj] ⊆ gi+j, ∀i, j ∈ Z i∈Z Furthermore, by the Jacobi identity, Zg(e) is adh-stable ([e, [h, x]] = −[h, [x, e]]− [x, [e, h]] = −[h, 0] − [x, 2e] = 0), so we have M Zg(e) = Zgi (e),Zgi (e) = Zg(e) ∩ gi i∈Z Finally, since ker(e) = Zg(e), then Zg(e) ∩ im(f) = {0}, so Zg contains only highest weight vectors. From the general theory, we know that if v0 is 2 a highest weight vector, then h.v0 = [dim(span{v0, f.v0, f .v0,...}) − 1]v0, but such an eigenvalue must be a nonnegative integer. If all the eigenvalues

27 are even, then repeated application of f to each element of Zg(e) will land it in g0 = Zg(h) since f : gm → gm−2 for all m ∈ Z.

7.14. Theorem. ([CG00] cor 3.2.13 and thm 3.7.13). Let e1, . . . , en be a set of root vectors in n for g = h ⊕ n ⊕ n−, one for each simple root P determined by b. Then, x = ei is a regular nilpotent element in g. 7.15. Remark. This theorem can be proved using the fact that nreg is a single B-orbit consisting of regular nilpotent elements in g since n ∈ nreg. However, we will use the corollary above to prove it instead.

Proof of Theorem. Let α1, . . . , αn be the set of simple roots determined by b and let nα1 ,..., nαn for the corresponding root spaces in n (so that, in particular, ei ∈ nαi ). Let n−α1 ,..., n−αn denote the corresponding negative root spaces and e−1, . . . , e−n the corresponding negative root vectors. From the general structure theory of semisimple Lie algebras, there is, for each 1 ≤ i ≤ n, a unique multiple hi of [ei, e−i] such that αi(hi) = 2. Furthermore, the hi’s form a basis of h. Now, ﬁx h ∈ h so that αi(h) = 2 for P all i and deﬁne complex numbers µ1, . . . , µn by the equation h = µihi. P Let y := µie−i. Now, for any simple roots α, β where α 6= β, α − β is not a root. Thus, [ei, e−j] = 0 whenever i 6= j because . Now, we calculate why? X X X [h, x] = [h, ei] = [h, ei] = αi(h)ei = 2x Where did this i come from? X [h, y] = µi(−αi)(h)e−i = −2y i X X X [x, y] = µj[ei, e−j] = µi[ei, e−i] = µihi = h i,j i i

Thus, (x, y, h) is an sl2-triple and all the eigenvalues of h on g are even since αi(h) = 2. By the corollary above, we obtain dim Zg(x) = dim Zg(h) = dim h and therefore x is regular. 7.2. Consequences using Transversal Slices. This whole section is com- 7.16. ([CG00] 3.2.19) A locally closed (in the ordinary Definition. pletely confus- Hausdorﬀ topology) complex analytic subset S ⊆ X containing the point y ing and I do will be called a transverse slice to Y at y if there is an open neighborhood of not understand y (in the ordinary Hausdorﬀ topology), U ⊆ X, and an analytic isomorphism ∼ anything except f :(Y ∩ U) × S → U such that f restricts to the tautological maps of the Kostant’s theo- factors rem. ∼ ∼ f : {y} × S → S and (Y ∩ U) × {y} → Y ∩ U. 7.17. Lemma. ([CG00] 3.2.20) Let G be an algebraic group, V a smooth algebraic G-variety, X a G-stable algebraic subvariety in V , and SV a locally-closed complex-analytic submanifold which is transverse to O at y ∈

28 O. Then the intersection with X of a small enough open neighborhood of y in SV is a transverse slice to O in X.

Proof. See [CG00]. 7.18. Proposition. ([CG00] 3.7.15)Fix a nilpotent e ∈ g and corresponding sl2-triple (e, f, h). Let O be the G-conjugacy class of e. Then, the aﬃce space e + Zg(f) is transverse to the orbit O in g. Moreover, we have O ∩ (e + Zg(f)) = e.

7.19. Remark. The aﬃne space e + Zg(f), or its intersection with N , is often referred to as the standard slice to the orbit O at the point e. This is due to the fact that there exists a small enough neighborhood U containing e such that N ∩ (e + Zg(f)) ∩ U is a transverse slice to O in N .

7.20. Corollary. ([CG00] 3.7.19) The ﬁber µ−1(e) ⊆ N˜ is a homo- −1 topy retract of the variety S˜ = µ (e + Zg(f)) ⊆ N˜.

7.21. Lemma. ([CG00] 3.7.21) Let U ⊆ ZG(e) be a unipotent normal L subgroup corresponding to the Lie algebra u = i>0 Zgi (e). (a) We have u = ker(e) ∩ im(e). (b) The aﬃne space h + u is stable under the adjoint U-action; moreover, h + u = U · h is a single U-orbit.

0 0 Proof of Kostant’s Theorem. Let (e, f, h) and (e, f , h ) be two sl2- triples. If h = h0, then

0 0 0 0 [f , e] = h = h = [f, e] =⇒ [f − f, e] = 0 and f − f ∈ Zg(e) However, f, f 0 ∈ g in the grading g = L g into ad -eigenspaces. Thus, −2 i∈Z i h 0 0 f − f ∈ Zg−2 (e) = 0 and so f = f.

0 0 Now, for any two sl2-triples (e, f, h) and (e, f , h ), we get

0 0 0 [h , e] = [h, e] = 2e =⇒ [h − h, e] = 0 =⇒ h − h ∈ Zg(e) = ker(e) Furthermore, we have

h0 − h = [e, f 0 − f] ∈ im(e) =⇒ h0 − h ∈ ker(e) ∩ im(e)

So, by Lemma 7.21(a), we get h0 ∈ h + u and thus, by Lemma 7.21(b), there is a u ∈ U such that h0 = uhu−1. Therefore,

e = u−1eu, h = u−1h0u, f 00 := u−1f 0u

00 00 and so (e, h, f ) is an sl2-triple for e. So, by above, f = f and we are done.

29 7.3. Correspondance between nilpotent orbits and G-conjugacy classes of sl2-triples. This exposition is borrowed from chapter 3 of [CM93] and modiﬁed given what we already have proven.

7.22. Proposition. For semisimple Lie algebra g, there is a bijection

Hom(sl2, g) \{0} ↔ {standard triples in g}

Proof. Given a nonzero homomorphism φ: sl2 → g, we get standard triple {φ(e), φ(f), φ(h)}. Conversely, an sl2-triple gives an injection of sl2 ,→ g. 7.23. Proposition. There is a map deﬁned by

Ω: {Gad-conjugacy classes of standard triples in g} → {nonzero nilpotent orbits in g}

{e, f, h} 7→ Oe

Proof. This map is induced by the fact that Hom(sl2, g) \{0} and standard triples in g are invariant under the action of Gad. 7.24. Proposition. The map Ω is surjective.

Proof. By the Jacobson-Morozov theorem, every nilpotent e ∈ g be- longs to some sl2-triple. Thus, there is a conjugacy class of triples {e, f, h} mapping to Oe. 7.25. Proposition (Kostant’s Theorem, rephrased). Let g be a complex semisimple Lie algebra. Any two standard triples (e, f, h) amd (e, f 0, h0) with the same nilpositive element are conjugate by an element of Gad, the adjoint group. Thus, the map

Ω: {Gad-conjugacy classes of standard triples in g} → {nonzero nilpotent orbits in g} is injective.

7.26. Remark.

7.27. Theorem. The map Ω is a one-to-one correspondance between the set of Gad-conjugacy classes of standard triples in g and the set of nonzero nilpotent orbits in g.

7.28. Example. Consider g = sln(C). Then, Gad = PSLn and we can pick orbit representatives   Jd1 (0)  Jd (0)  e =  2  [d1,...,dk]  ..   . 

Jdk (0)

30 where [d1, . . . , dk] is a partition of n. As discussed in the sln proof of the Jacobson-Morozov theorem, if we let  r   0 0   r − 2   µ1 0       ..   .. ..  ρr(h) =  .  ρr(f) =  . .       −r + 2   0 0  −r µr 0 where µi = i(r + 1 − i) for 1 ≤ i ≤ r, we can associate sl2-triple to e[d1,...,dk] given by

h[d1,...,dk] = ρd1 (h) ⊕ · · · ⊕ ρdk (h), f[d1,...,dk] = ρd1 (f) ⊕ · · · ⊕ ρdk (f)

Conversely, given an sl2-triple in sln, {e, f, h}, the corresponding orbit is simply Oe.

31 Bibliography

[CG00] N. Chriss and V. Ginzburg, Representation Theory and Complex Geometry, 2000. [CM93] D. H. Collingwood and W. M. McGovern, Nilpotent Orbits in Semisimple Lie Algebras, 1993. [Hal03] B. C. Hall, Lie Groups, Lie Algebras, and Representations: An Elementary In- troduction, 2003. First edition, corrected second printing, 2004. [Hum72] J. E. Humphreys, Introduction to Lie Algebras and Representaton Theory, 1972. Third printing, revised.