Flag Manifolds and Representation Theory. Winter School on Homogeneous Spaces and Geometric Representation Theory

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1 Flag Manifolds and Representation Theory Winter School on Homogeneous Spaces and Geometric Representation Theory Lecture I. Real Groups and Complex Flags 28 February 2012 Joseph A. Wolf University of California at Berkeley p. 1

2 Parabolic Subalgebras G: conn. simply conn. complex semisimple Lie group g: the Lie algebra of G H: Cartan subgroup (centralizer of a Cartan subalg. h) Σ = Σ(g,h): root system, so g = h+ α Σ g α Σ + = Σ(g,h) + is a choice of positive root system Ψ = Ψ(g,h): simple system of roots for Σ + Φ is an arbitrary subset of Ψ q = q Φ := q nilp +q red parabolic subalgebra where q nilp = Σ + \ Φ g α is its nilradical and q red = h+ Φ g α is the Levi (reductive) component Notation example: p. 2

3 Parabolic Subgroups Q = Q Φ = {g G Ad(g)q = q}: parabolic subgroup of G. A parabolic subgroup is its own normalizer Each G conjugacy class of parabolic subgroups contains just one of the 2 #Ψ groups Q Φ Extremes: Q is a Borel subgroup and Q Ψ = G Let π λ be an irreducible finite dimensional representation of G of highest weight λ. Let v λ be a highest weight vector and Φ = {ψ Ψ λ,ψ = 0}. Then Q Φ is the G stabilizer of the line v λ C. In the action of G on the assoc. complex projective space, Q Φ is the isotropy subgroup of G at [v λ ]. Let G 0 be a real form of G. Then parabolic subgroup of G 0 means a subgroup that is a real form of a parabolic subgroup of G. p. 3

4 Complex Flag Manifolds the Z = G/Q, Q parabolic in G, are the complex flag manifolds Q z and q z : isotropies of G and g at z Z G u : compact real form of G can assume that H u := G u H is a maximal torus in G u fact: G u Q is the centralizer of a subtorus of H u and is a compact real form L u of the reductive part Q red of Q. Complex flag manifolds Z = G/Q = G u /L u are characterized by any of Z = G/Q is a complex homogeneous projective variety Z = G u /L u is a compact simply connected homogeneous Kähler manifold Z = G u /L u where L u is the centralizer of a torus p. 4

5 Homogeneous Vector Bundles In general: let A/B be a homogeneous space and β : B End(E β ) a linear representation Define E β = A β E β where A β E β = (A E β )/{(ab,v) = (a,β(b)v)} for a A,b B. Then µ : E β A/B, by µ[a,v] = ab, is an A homogeneous vector bundle with typical fiber E β on which B acts by β Sections are given by f : A E β with f(ab) = β(b) 1 f(a). This works w/o change in the categories C 0, C k and C. For the holomorphic category, one must define a complex structure on E β for which there are sufficiently many local holomorphic sections. This means local sections annihilated by antiholomorphic vector fields on A/B. p. 5

6 Borel Weil Theorem Again, G is a connected complex semisimple Lie group, Q is a parabolic subgroup and Z = G/Q complex flag τ λ is an irreducible representation of Q with highest weight λ and representation space E λ E λ Z = G/Q is the associated homogeneous holomorphic vector bundle and O(E λ ) Z is the sheaf of germs of holomorphic sections. Borel Weil Theorem. Let H 0 (Z;O(E λ )) denote the space of global holomorphic sections of E λ Z. If λ is Σ + dominant, i.e. λ,ψ 0 for all ψ Ψ, then the action of G on the H 0 (Z;O(E λ )) is the irreducible representation of G with highest weight λ. p. 6

7 Proof of Borel Weil Recall q = q red +q nil with g u q = l u where l = q red. { Λ := ν h 2 ν,ψ ψ,ψ }, integer 0 highest weights ν of irreducible representations π ν of G. H 0 (Z;O(E λ )) = (L 2 (G u /L u ) E λ ) qnil = ν Λ ((V ν Vν E λ ) l ) qnil = ν Λ ((V ν Vν E λ ) qnil ) l using the Peter Weyl Theorem and because l = q red normalizes q nil = ν Λ (V ν (V ν E λ ) qnil ) l = ν Λ (V ν E ν E λ ) l because q nil ignores E λ and pushes V ν to E ν = ν Λ V ν (E ν E λ ) l = V λ as G module. p. 7

8 Bott Borel Weil Theorem W = W g,h : Weyl group, generated by reflections in the roots, w α : ν ν 2 ν,α α,α α ρ: half the sum of the positive roots if w W then length l(w) = #{α Σ + w(α) Σ + } if λ h nonsingular (λ α for α Σ) then q(λ) = l(w) where w = w λ W s.t. w(λ),α > 0 for every α Σ + Bott Borel Weil Theorem Let λ h, highest weight of a representation τ λ of Q, defining E λ Z. If λ+ρ is singular then every H q (Z;O(E λ )) = 0. If λ+ρ is nonsingular then H q (Z;O(E λ )) = 0 for q q(λ+ρ), and H q(λ+ρ) (Z;O(E λ )) is the irreducible G module of highest weight w λ+ρ (λ+ρ) ρ. p. 8

9 Proofs Bott s proof of the Bott Borel Weil Theorem combines the Borel Weil Theorem, the Leray spectral sequence, and the Kodaira Vanishing Theorem. It is an analytic proof based on the theory of compact Kähler manifolds. Kostant s proof is based more on the structure of parabolic subalgebras q g (new at the time) and his idea of looking at Lie algebra cohomology. More recent proofs are based on Lie algebra cohomology and/or the Bernstein Gelfand Gelfand resolution of certain modules Because of time constraints that s all I ll say about it. p. 9

10 Real Group Orbits G 0 : real form of G Q z Q z and q z q z : complexifications of real isotropies G 0 Q z and g 0 q z fact: g 0 q z contains a Cartan subalgebra of g 0 consequence: only finitely many G 0 orbits on Z consequence: G 0 has open orbits on Z if G 0 (z) is open then Cartan H 0 G 0 Q z is fundamental (maximally compact for CSG of G 0 ) If H 0 G 0 Q z is a fundamental Cartan denote W(g,h) h 0 = {w W(g,h) w(h 0 ) = h 0 } open orbits are given by a double coset space of the Weyl group, W(g 0,h 0 )\W(g,h)/W(g,h) h 0 p. 10

11 Examples G 0 = SU(p,q) defined by h(z,w) = p 1 z iw i q 1 z p+iw p+i Z = {q dim subspaces of C p+q } G 0 orbits given by signature (+, ) of h Open: D i = {z Z signh z = (i,q i)} D 0 = {W C p q I WW >> 0} bounded symmetric domain Closed: S = {z Z nullh z = min(p,q)} Bergman-Shilov boundary of D 0 Z = CP n and G 0 = SL(n+1;R). Orbits: G 0 (u+iv) with u,v R n+1 If u,v linearly independent: open orbit G 0 ([e 1 + 1e 2 ]) If u,v linearly dependent: closed orbit G 0 ([e 1 ]) p. 11

12 Compact Subvarieties D = G 0 (z): open G 0 orbit on Z = G/Q K 0 : maximal compact subgroup of G 0 C 0 : unique K 0 orbit on D that is a complex submanifold Properties: Can assume C 0 = K 0 (z). Then the base cycle C 0 = K(z) = K/(K Q) is a flag manifold The cycle space M D of D is the component of C 0 in {gc 0 g G and gc 0 D} M D is a contractible Stein manifold of known structure Suppose that G 0 /K 0 is a bounded symmetric domain B. Then M D is B or B (hermitian holomorphic case) or B B (hermitian non holomorphic case) More precisely: p. 12

13 Hermitian Types Let G 0 be simple and of hermitian type: the symmetric space G 0 /K 0 is an irreducible bounded symmetric domain B. If D is of holomorphic type in the sense that µ and ν in the double fibration µ G 0 /(Q r 0 K 0) ν D = G 0 /Q r 0 B = G 0 /K 0 can be made simultaneously holomorphic, then M D is holomorphically equivalent to B or to B. If D is of nonholomorphic type in the sense that µ and ν cannot be made simultaneously holomorphic, then M D = B B. p. 13