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 2 G, x 2 g g:x = Add(φg)I (x) −1 where φg : G ! G maps h 7! ghg for h 2 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, con- nected, 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 0 1 0 1 ∗ ∗ ∗ 0 1 ∗ B C B C B C B = B C ;T = B C ;U = B C @ A @ A @ A 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 2 g, we define Zg(x) = fy 2 g j [x; y] = 0g;ZG(x) = fg 2 G j g:x = xg 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 2 g is regular if dim Zg(x) = rank g. 2.14. Definition. We say x 2 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 2 g has a Jordan decomposition x = s + n where s 2 g is semisimple and n 2 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 0 1 B C B C B C B C x = B 1 C B C B C @ A 0 Such an element is regular in sln because Zg(x) is spanned by the basis 2 n−1 n−1 fx; x ; : : : ; x g, which is the same size as the basis feii − ei+1;i+1gi=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 2 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 0 1 λ 0 B 1 C B C x = B C @ A 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 2 gsr () P (x) 6= 0 Proof. Take x 2 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 sub- spaces of g. 2.24. Proposition. B ⊆ Gr as a closed subvariety of solvable subalge- bras. 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 2 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 = fF = (0 ⊆ F1 ⊆ F2 ⊆ · · · ⊆ Fn = C )g 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 fB-orbits on Bg fG-diagonal orbits on B × Bg 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 w2W w w these proposi- cell. tions! So W is a fixed point set of the ∗ action on B. w 2 W is w · B=B, C Understand this w 2 W . So, B = B · (wB=B) gives us T W better. Why G G G=B = B = BW = BwB=B does this follow w2W and why do we care? 2.30. Corollary. Each B-orbit in B contains exactly one point of the form wB=B, w 2 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 = fx 2 sln j x(Fi) ⊆ Fi; 8ig 1 2 For example, if F is the coordinate flag, that is F = 0 ⊆ C ⊆ C ⊆ · · · ⊆ n C , then 0 1 ∗ b b B C F = = @ A 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 : 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 fxig, but if x 2 sln, we know that xi = 0 since tr x = 0. So, take n−1 ∼ X C = f(x1; : : : ; xn) j xi = 0g 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~ := f(x; F ) 2 sln × B j x(Fi) ⊆ Fig 2.35. Proposition. The following diagram commutes g~ µ ν n−1 n g C ⊆ C φ 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.