Lie Algebra Cohomology and the Borel-Weil-Bott Theorem. 1 Lie algebra cohomology and cohomology of G/T with coefficients in a line bundle

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1 Le Algebra Cohomology and the Borel-Wel-Bott Theorem Math G4344, Sprng 2012 We have seen that rreducble fnte dmensonal representatons of a complex smple Le algebra g or correspondng compact Le group are classfed and can be constructed startng from an ntegral domnant weght. The domnance condton depends upon a choce of postve roots (or equvalently, a choce of nvarant complex structure on the flag manfold.) An obvous queston s that of what happens f we make a dfferent choce of postve roots, or start wth a non-domnant hghest weght. The Weyl group permutes possble choces of postve roots, at the same tme permutng hghest weghts. It turns out that there s a generalzaton of the Borel-Wel theorem whch descrbes the effect of these Weyl group permutatons. Ths s the Borel-Wel- Bott theorem, whch realzes representatons n other cohomology degrees, not just the degree-zero case of holomorphc sectons. Ths phenomenon s best understood n terms of the Le algebra cohomology of the nlpotent radcal subalgebra n + g. 1 Le algebra cohomology and cohomology of G/T wth coeffcents n a lne bundle Recall that one way to motvate Le algebra cohomology s by startng wth de Rham cohomology of a group. For G a compact, smple Le group we have the de Rham complex (Ω (G), d) of dfferental forms Ω (G) wth the de Rham dfferental d satsfyng d 2 = 0. By the de Rham theorem the cohomology of ths complex gves the topologcal cohomology of the group. One way to wrte these dfferental forms s as Ω (G) = Hom C (Λ (g), C (G)) To get Le algebra cohomology one smply replaces C (G) by an arbtrary representaton V of the Le algebra of G, so co-chans are C (g, V ) = Hom C (Λ (g), V ) Λ (g ) C V For more detals and the formula for d, see [1] or [4]. cocycles as Z (g, V ) = ker d C (g,v ) coboundares as B (g, V ) = Im d C 1 (g,v ) As usual, one defnes 1

2 and the cohomology as H (g, V ) More abstractly, one can get ths defnton as the derved functors of the nvarants functor n the category of U(g) modules. The nvarants functor s V Hom U(g) (C, V ) = V g Replacng the trval representaton C here by a certan free U(g) resoluton called the Koszul resoluton gves precsely the complex defned above. Note that for a compact Le group the nvarants functor s exact, and the complexfcaton doesn t change ths so for sem-smple complex Le algebras one has H (g, V ) = H (g, C) V g Here Le algebra cohomology carres no more nformaton about the representaton V than the dmenson of ts nvarant subspace. For non-sem-smple Le algebras the nvarants functor s no longer exact, and the Le algebra cohomology of a representaton s a more nterestng nvarant than just ts degree-zero pece, the nvarants. We wll be nterested here n such a case, takng the Le algebra cohomology wth respect to the nlpotent radcal n + of a sem-smple Le algebra. Recall that n our dscusson of the Borel-Wel theorem we were usng a complex lne bundle L λ over the flag manfold G/T (G s a compact smple Le group, T a maxmal torus). The ntegral weght λ labels a T representaton ρ λ on C. Sectons of ths lne bundle are explctly Γ(L λ ) = {f : G C, f(gt) = ρ λ (t 1 )f(g)} = (C (G) C λ ) T = (C (G)) λ and holomorphc sectons are the subspace of ths nvarant under the rght acton of n +. We are nterested now n usng the structure of G/T as a complex manfold (whch depends on the choce of postve roots) to defne a holomorphc verson of cohomology. The usual topologcal cohomology computes the derved functor of the functor of takng global sectons of the sheaf of locally constant functons. For a complex manfold, we nstead use the sheaf of local holomorphc functons, or more generally the sheaf of local holomorphc sectons of a holomorphc lne bundle such as L λ. Just as the de Rham theorem allows computaton of topologcal cohomology usng dfferental forms, the Dolbeault theorem says we can compute holomorphc cohomology usng the the b-graded complex (Ω 0, (G/T, L λ ), ) of dfferental forms wth coeffcents n lne bundle L λ, of degree n local varables dz (and degree 0 n the dz). In degree 0 we just get H 0 (G/T, L λ ) = Γ hol (L λ ) 2

3 the holomorphc sectons, but we can also get hgher cohomology, n degrees up to the complex dmenson of G/T. If one works out explctly what the Dolbeault complex s n ths case, generalzng the case of holomorphc sectons, one fnds (Ω 0, (G/T, L λ ), ) = ((Hom(Λ (n + ), C (G) C λ )) T, d) where T acts on n + by the adjont representaton, and the d s the d of Le algebra cohomology for n +, wth n + actng on C (G) by nfntesmal rght translaton. Note that one has a commutng acton of G on ths complex, comng from the left G acton on functons on G, so we wll get G representatons on the cohomology spaces H (G/T, L λ ) Recall that the way Borel-Wel works s that one uses Peter-Weyl to see that Γ(L λ ) = (C (G) C λ ) T = (C (G)) λ = (V µ ) V µ λ µ domnant and thus that so For hgher cohomology, one has Γ hol (L λ = (V λ ) H(Ω 0, (G/T, L λ ), ) = H((Hom(Λ (n + ), C (G) C λ )) T, d) = H( (V µ ) (Hom(Λ (n + ), V µ C λ )) T, d) = = µ domnant µ domnant µ domnant H(Ω 0, (G/T, L λ ), ) = (V µ ) (H (n +, V µ C λ )) T (V µ ) H (n +, V µ ) λ µ domnant (V µ ) H (n +, V µ ) λ Ths show that n ths case computng holomorphc cohomology comes down to computng n + Le algebra cohomology. For some more detals of ths argument, see for nstance [2]. 3

4 2 Kostant s Theorem The computaton of the Le algebra cohomology of the nlpotent radcal was done by Kostant n 1961, wth the result Theorem 1 (Kostant s Theorem). For a fnte dmensonal hghest-weght representaton V λ of a complex sem-smple Le algebra g H (n +, V λ ) = w W :l(w)= C w(λ+ρ) ρ There are at least four possble approaches to provng ths: One can use the BGG resoluton and the fact that for Verma modules H (g, V (µ)) s C µ for = 0, 0 for > 0. Ths requres knowng the BGG resoluton, whch s a stronger result snce t carres nformaton about homomorphsms between Verma modules. One can prove Borel-Wel-Bott by other (e.g. topologcal) methods, then use ths to prove Kostant s theorem. For an example of such a proof of Borel-Wel-Bott, see Jacob Lure s notes[3]. One can fnd explct elements n H (n +, V λ ) that represent the cohomology classes n Kostant s theorem. One way to do ths s to look for elements n C (n +, V λ ) = Λ (n + ) V λ that represent these cohomology classes. Note that the weghts of (n + ) are multples of α where α R +, the postve roots. A choce that gves the rght element n degree for each Weyl group element w such that l(w) = s: ω β1 ω β2 ω β V λ (wλ) where ω βj (n + ) β j for β j a postve root such that wβ j s a negatve root. V λ (wλ) s the transform by w of the hghest weght space. Th more dffcult part of ths sort of proof s showng that only these elements can occur. One way to do ths s to construct an analog of the Laplacan, and show that t acts lke the Casmr on cohomology (ths was Kostant s orgnal method). A generalzaton of ths uses the full center of the envelopng algebra, and the Casselman-Osborne lemma, whch says that the center much act on the hgher cohomology n just the way that the Harsh-Chandra smorphsm says t acts n degree zero cohomology (the hghest weght space). For more detals on ths argument see Goodman-Wallach[4]. One can replace the use of the exteror algebra and a Laplacan by closely related spnors, and a square-root of the Laplacan known as the Drac operator. We ll try and come back to ths argument after developng the technology of spnors and Clfford algebras n the next couple weeks. 4

5 3 Borel-Wel-Bott and the Weyl Character Formula Kostant s theorem gves the Borel-Wel-Bott theorem very drectly. Recall that H (G/T, O(L λ )) = µ (V µ ) H (n +, V µ ) λ where the sum s over domnant ntegral weghts µ. By Kostant s theorem we have H (n +, V µ ) λ = ( C w(µ+ρ) ρ ) λ w W :l(w)= and ths has a one-dmensonal contrbuton ff w(µ + ρ) ρ = λ or equvalently w(µ + ρ) = λ + ρ Note that the set of weghts of the form µ + ρ for µ domnant ntegral are n the nteror of the domnant Weyl chamber, and actng on these by Weyl group elements gves us sets of weghts n the nterors of the other Weyl chambers. Weghts λ such that λ + ρ s on the boundary of a Weyl chamber wll not occur. In summary, we have Theorem 2 (Borel-Wel-Bott). If λ + ρ s a sngular weght then for all we have H (G/T, O(L λ )) = 0 If λ+ρ s a non-sngular weght, there wll be an such that w(λ+ρ) = µ+ρ s n the nteror of the domnant Weyl chamber for a w : l(w) = and H (G/T, O(L λ )) = (V µ ) As usual, the smplest example s G = SU(2), G/T = CP 1, and the Borel- Wel-Bott theorem can be proved va Serre dualty, whch says that for lne bundles L on a curve C one has H 1 (C, L) = H 0 (C, L ω C ) where ω C s the canoncal bundle on C. In our case C = CP 1, and lne bundles L n are labeled by an nteger n wth ρ correspondng to n = 1. The canoncal bundle s L 2. For n 0 we have, as n the Borel-Wel theorem H 0 (CP 1, L n ) = (V n ) 5

6 where V n s the rreducble SU(2) representaton of dmenson n + 1. By Serre dualty H 1 (CP 1, L n ) = H 0 (CP 1, L n+2 ) whch s consstent wth Borel-Wel-Bott whch tells us that H 1 (CP 1, L n ) = (V n 2 ) when n < 1 and, n the sngular n = 1 case H 1 (CP 1, L 1 ) = H 0 (CP 1, L 1 ) = 0 So, for n > 0 one gets all rreducbles as holomorphc sectons, whereas for n < 1 one gets all rreducbles agan, but n hgher cohomology (H 1 ). Workng out what happens n the SU(3) case wll be on the current problem set. Another quck corollary of Kostant s theorem s the Weyl character formula. Recall that ths says that the character ch(v λ ) of a fnte-dm rreducble of hghest weght λ s gven by ch(v λ ) = w W ( 1)l(w) e w(λ+ρ) ρ w W ( 1)l(w) e w(ρ) ρ Ths follows from an applcaton of the Euler-Poncaré prncple, whch says that n the case of an Abelan nvarant lke the character, ts value on the alternatng sum of the cohomology groups (the Euler characterstc) s the same as ts value on the alternatng sum of whatever co-chans ones uses to defne cohomology,.e. here we have ( 1) ch(h (n +, V )) = ( 1) ch(c (n +, V )) Ths follows from two facts: the frst s that ch(c (n +, V )) = ch(z (n +, V )) + ch(b +1 (n +, V )) snce we have an exact sequence 0 Z (n +, V ) C (n +, V ) d B +1 (n +, V ) 0 (here Z (n +, V ) are the co-cycles on whch d = 0, B +1 (n +, V ) are the coboundares whch are n the mage of d. Snce we also have a second fact H (n +, V ) = Z (n +, V )/B (n +, V ) ch(h (n +, V )) = ch(z (n +, V )) ch(b (n +, V )) and ths together wth our frst fact gves the Euler-Poncaré prncple. 6

7 Recall that C (n +, V ) = Hom(Λ (n + ), V ) = Λ (n + ) V so we have ( 1) ch(c (n +, V λ )) = ( 1) ch(λ (n + ) )ch(v λ ) whereas Kostant s theorem tells us that the Euler characterstc s ( 1) ch(h (n +, V λ )) = ( 1) l(w) e w(λ+ρ) ρ w W Applyng the Euler-Poncaré prncple n the case λ = 0 gves w W( 1) l(w) e w(ρ) ρ = ( 1) ch(λ (n + ) ) and thus n the general case the Weyl character formula as w W( 1) l(w) e w(λ+ρ) ρ = ( ( 1) l(w) e w(ρ) ρ )ch(v λ ) w W References [1] Knapp, A., Le Groups, Le Algebras, and Cohomology, Prnceton, [2] Barchn, L., Untary Representatons Attached to Ellptc Orbts. A Geometrc Approach (Lecture 3) n1998 PCMI lectures on Representaton Theory of Le Groups. [3] Lure, J., A Proof of the Borel-Wel-Bott Theorem, [4] Goodman, R. and Wallach, N., Symmetry, Representatons and Invarants, GTM 255 Sprnger, Appendx E. 7