Bott, R., Homogeneous vector bundles, Ann. of Math. 66 (1957), 203--248.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....that the Kostant Auslander theorem ensures actually a canonical bijection between the unitary dual and the space O rigg (G) of rigged orbits. The method of orbits also gives all the irreducible representations of a connected and simply connected compact Lie group G by the Borel Weil Bott theorem [9]. In this case G is discrete and the canonical bijection established by the method of orbits between the unitary dual and the space O(G) picks out a countable set of coadjoint orbits that satisfy the integrality condition (i.e. the integral of the Kirillov 2 form over an arbitrary ....

Bott, R.: Homogeneous vector bundles. Ann. Math. 66, 203-248 (1957)

.... 1 (x) v e , and the regular functions on the latter space are given by the symmetric algebra of its dual. In other words, SG Pe ve ) is the sheaf of sections of G Pe S(v e ) It follows that W is the sheaf of sections of G Pe (W S(v e ) Now, a theorem of Bott [5] almost computes the right hand side of (8) for us. Speci cally, if U is a completely reducible representation of P e and U = G Pe U is the corresponding vector bundle over G=P e , then (9) H q (G=P e ; U) M ( M )2 b G H q (p e ; l e ; Hom(M ; U) M : In our case, W S(v ....

R. Bott, Homogeneous vector bundles, Ann. of Math. 66 (1957), 203-248.

....roots) is l(w) j Delta (w)j ( Hump] Lemma 10.3A) The sign of w (the determinant of the action of w on h) is sgn(w) Gamma1) l(w) Finally, recall that ae Gamma w Delta ae = X ff2 Delta (w) ff 2 X (H) 12 JEFFREY ADAMS, JING SONG HUANG, AND DAVID A. VOGAN, JR. Theorem 3. 10 ([Bott], page 228, or [Kost] Theorem 6.4) Suppose we are in the setting of Definition 3.8; use also the notation of Definition 3.9. Suppose 2 X (H) is G dominant, and V is the irreducible algebraic representation of G of highest weight . Similarly, suppose 2 X (H) is L dominant, and E is the ....

R. Bott, Homogeneous vector bundles, Ann. of Math. 66 (1957), 203--248.

....Our result applies to most of the familiar examples of direct limits of classical groups. We also introduce new examples involving iterated embeddings of the classical groups and see exactly how our results hold in those cases. Section 0. Introduction. The classical Bott Borel Weil Theorem [6] realizes representations of compact Lie groups as cohomology spaces of holomorphic vector bundles. Many of the geometric realizations of various kinds of representations are based on developments of this concept. In this paper we carry the geometric idea of the Bott Borel Weil Theorem over to ....

Bott, R., Homogeneous vector bundles, Annals of Math. 66 (1957), 203--248.

....H p (Z; O Z ) vanishes for p 1 [3] the spectral sequence is easy to handle and we get H q (X; OX ) E 0;q 1 = E 0;q 2 = H q (E; OE ) q C n : In much the same way we can compute the cohomology H j (W; OW ) H j (W; Phi Theta Z ) etc. via vanishing results of Bott [4]. Then using the sequences 0 OW p Theta CP 1 OX1[X2 0 0 D Theta W dp Gamma p Theta CP 1 0; we are able to find H j (W; D) It turns out that the obstruction space H 2 (W; D) is non trivial. Therefore we study the possible obstructions using Kuranishi theory [13] ....

R. Bott. Homogeneous vector bundles, Ann. Math. 66 (1958) 203--248.

....) H q (X, U(g)# C # p n # ) U(g)# C H q (X, # p n # ) Let # : W # Z be the length function on the Weyl group W of # with respect to the set of reflections corresponding to simple roots # in # . Let W (p) w # W #(w) p and n(p) Card W (p) By a lemma of Bott [5] (which follows easily from the Borel Weil Bott theorem) H q (X, # p n # ) 0 if p #= q; and H p (X, # p n # ) is a linear space of dimension n(p) with trivial action of G. Now, a standard spectral sequence argument implies that (ii) holds, and that #(X,D h ) has a finite ....

R. Bott, Homogeneous vector bundles, Annals of Math. 66 (1957), 203--248.

....manifold (M; Suppose that is integral, and let L be a G equivariant holomorphic line bundle with first Chern class [ The space of global holomorphic sections of L is a representation of G; this is called the Kahler quantization. The first natural extension of this idea, suggested by Bott [B1], is to replace the space of holomorphic sections by the alternating sum of sheaf cohomology groups, P i ( Gamma1) i H i (M; O L ) where O L is the sheaf of germs of holomorphic sections of L. Date: May 4, 1997. A. Canas da Silva was partially supported by a NATO fellowship. Her research ....

R. Bott, Homogeneous vector bundles, Ann. of Math. (2) 66 (1957), 203-248.

....deformation theory is a computation of the relevant cohomology groups on the twistor spaces. It is done via Leray s spectral sequence established by the link between the twistor spaces and flag manifolds. The computation is possible due to the vanishing theorems of Borel Hirzebruch and Bott [2] [3]. From this computation, we find that the obstruction space to deformations is non trivial. Therefore, in Section 4 we complete our calculation of the deformations through Kuranishi theory. The result is: Theorem Suppose G is a compact semi simple Lie group of rank r. Then the local moduli at a ....

....this formula. The summand C n is generated by the holomorphic action of the group T 2n Gammar Theta B. It is naturally isomorphic to a C = H 0 (E; Theta E ) On the product of elliptic curves E, the space H 1 (E; OE ) is isomorphic to the dual a C of H 0 (E; Theta E ) By Bott [3], H 0 (Z; Theta Z ) is isomorphic to g C . Therefore, Proposition 3 yields H 1 (W; Theta W ) a C Omega (a Phi g Phi sp(1) C ; and H 1 (W; D) a C Omega (a Phi g) C ; 20) In particular, the vector space H 1 (W; Theta W ) is complex linearly spanned by ....

R. Bott. Homogeneous vector bundles, Ann. Math. 66 (1958) 203-- 248.

....with complete precision. 1.3 The 4 dimensional case In any dimension, starting from the Kirillov Kostant Poisson structure on the dual of so(n) one can use geometric quantization to construct a Hilbert space describing the states of a quantum bivector. By the Bott Borel Weil theorem [10], this Hilbert space always turns out to be the direct sum of all the irreducible unitary representations of so(n) The 4 dimensional case is particularly simple, since we can reduce it to the previously treated 3 dimensional case using the isomorphism so(4) so(3) Phi so(3) This ....

R. Bott, Homogeneous vector bundles, Ann. Math. 66 (1957), 203-248.

....and H are connected complex Lie groups with G semisimple and H parabolic. U and K are connected compact Lie groups; U is semisimple and K is the centralizer of a torus. Further, any K invariant Hermitian metric on F is Kahler. The equivariant holomorphic vector bundles on G=H are now homogeneous [4] [47] that is, they are given by representations (ae; V ae ) of H : ae 7 V ae = G Theta H V ae ; inducing an isomorphism R(H) KG (G=H) The canonical extension e V ae is then of the form e V ae = P Theta H V ae M = P=H : For (ae 0 ; V ae 0 ) 2 R(G) the associated bundle ....

....the total differential d on p;q C (m C ) K vanishes since [m; m] ae k . Hence we have (cf. 22] IV) H p;q (F ) A p:q (F ) U = p;q C (m C ) K : 3.14) These formulas are very useful for explicit computations. According to Bott s generalization of the Borel Weil theorem [4] , H 0; F; V ae ) is an irreducible U module, if the induced highest weight of an irreducible representation (ae ; V ae ) 2 R(K) is non singular. The degree of the non vanishing cohomology group is given by the index of the induced highest weight of ae . If the induced highest weight of ae ....

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Bott, R.: Homogeneous vector bundles. Ann. Math. 66, (1957), 203--248.

....space of a semisimple complex Lie group G, P being a parabolic subgroup, and let E be a homogeneous vector bundle on X. Assume that E satisfies the following positivity condition: the highest weights of the associated P module are dominant. It is then a standard consequence of Bott s theorem [3, 5, 19], that E is spanned by global sections and has no cohomology in positive degrees, H q (X; E) 0 for q 0. The aim of this paper is to prove extensions of this vanishing property to Dolbeault cohomology, H p;q (X; E) H q (X; Omega p X Omega E) One of the motivations for understanding ....

....Theorem for Forms 4 p TX of holomorphic p forms on X is naturally homogeneous and the weights of the associated P module are P( Omega p X ) f Gamma P fi2S fi j S ae Phi X ; #S = pg. A fundamental tool for calculating the cohomology of a homogeneous vector bundle is Bott s Theorem [3]. We shall use the following version of this theorem, see for example [5, 19] Lemma 1.1 Let E be a homogeneous vector bundle on X = G=P . If is a highest weight of the G module H q (X; E) 2 (H q (X; E) then = oe( ffi) Gamma ffi where 2 P (E) oe( ffi) is the ....

Bott, R., Homogeneous vector bundles, Ann. Math. 66 (1957), 203--248.

....sets C = fx 2 g j C i (x) i ; i 2C ; i = 1; rank(g)g: 3.2.75) Their dimension is therefore dim C (g) Gamma rank(g) Coadjoint orbits are very important to the representation theory of (semi)simple Lie groups. In order to see why let us recall the Borel Weil Bott (BWB) theorem [30]. Let G be a compact semisimple Lie group with maximal torus T and let R be a finite dimensional irreducible representation of G. The space of highest weight vectors of R is 1 dimensional (call it V ) and furnishes a representation of T . The product space G Theta V is a trivial line bundle over ....

R. Bott, Homogeneous vector bundles, Ann. Math. 66 (1957) 203.

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Bott, R., Homogeneous vector bundles, Ann. of Math. 66 (1957), 203--248.

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R. Bott, Homogeneous vector bundles, Ann. of Math. 66 (1957), 203-248.

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Bott, R., Homogeneous vector bundles, Ann. of Math. 66 (1957), 203--248.

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