### the-orem

Throughout this work, we will mostly deal with homogeneous vector bundles and their cohomology. The problem of computing such coho-mology is completely solved by the Borel–Weil–Bott theorem. How-ever, applying such theorem often leads to cumbersome calculations, whose difficulty increases with the complexity of the automorphism group of the variety. The goal of the following section is to establish a comfortable notation in order to apply the Borel–Weil–Bott theorem to any homogeneous vector bundle on a homogeneous variety, in a simple algorithmic way. This algorithm is based on Weyl reflections, therefore we start from a shorthand notation which applies to every semisimple Lie algebra. Let us first introduce a uniformized notation for homogeneous varieties:

**Definition 2.4.1.** *Let*𝐺 *be a semisimple Lie group of rank* 𝑟*. We call*
𝐺-flag variety *any homogeneous variety* 𝑋 = 𝐺/𝑃 *where* 𝑃 ⊂ 𝐺 *is a*
*parabolic subgroup. We say that a*𝐺*-flag variety is a*𝐺-Grassmannian*if*
*it has Picard number one. We call*complete 𝐺-flag variety*the quotient*
𝐺/𝐵*.*

Let 𝐺 be a semisimple Lie group of rank 𝑟 and 𝐵 ⊂ 𝐺 its Borel sub-group. Then, as we discussed in Remark 2.2.11, there exist projections 𝐺/𝐵 −→ 𝐺/𝑃 from the complete 𝐺-flag variety. Let us summarize here some results on vector bundles on 𝐺/𝐵 and their pushforwards to

the other 𝐺-flags.

**Lemma 2.4.2.** *(Wey03, Proposition 4.1.3) Every line bundle* L *on a*
*complete*𝐺*-flag has the form*

L ' E𝜔 (2.4.1)

*for some weight*𝜔*.*

**Theorem 2.4.3**(Borel–Weil–Bott for line bundles). *Let*𝐺*be a *
*semisim-ple Lie group and* 𝐵 ⊂ 𝐺 *a Borel subgroup. Let*𝜆*be an integral weight*
*over*𝐺/𝐵 *and*E𝜆 *the associated line bundle. Call* 𝜌 *the sum of all *
*fun-damental weights. Then one and only one of the following situations*
*occur:*

◦ *There exists a nontrivial Weyl reflection*𝑆*such that*𝑆(𝜆+𝜌) −𝜌 =𝜆*.*
*Then*𝐻^{•}(𝐺/𝐵,E𝜆) =0.

◦ *There exists a unique Weyl reflection*𝑆*such that*𝑆(𝜆)*is a dominant*
*integral weight. Then*𝐻^{•}(𝐺/𝐵,E𝜆) =𝑉_{𝑆(𝜆+𝜌)−𝜌}[−𝑙(𝑆)].

The following lemma allows to use Theorem 2.4.3 to compute the co-homology of irreducible homogeneous vector bundles on any homoge-neous variety, and leads to the second formulation of Borel–Weil–Bott’s theorem (Theorem 2.4.5 in the following).

**Lemma 2.4.4.** *Every irreducible homogeneous vector bundle* F *on a*
𝐺*-flag variety*𝐺/𝑃*has the form*

F ' 𝜋_{∗}E𝜆 (2.4.2)

*where* E𝜆 *is a homogeneous line bundle for some weight* 𝜔*, and* 𝜋 :
𝐺/𝐵−→ 𝐺/𝑃*.*

*Proof.* For the sake of self-containedness of this exposition, we will
give a proof of this lemma, despite it is a well-known result. First,
let us observe that for every 𝑥 ∈ 𝐺/𝑃 one has 𝜋^{−}^{1}(𝑥) ' 𝑃/𝐵. Fix a
𝑃-dominant weight 𝜆. By Theorem 2.4.3, 𝑃-dominance of 𝜆 implies
that 𝐻^{0}(𝐺/𝐵,E𝜆) is the representation space of the representation of
𝑃 associated to the highest weight 𝜆, let us call such representation
space 𝑉_{𝜆}. In fact, one has E𝜆 ' 𝑃×^{𝐵} C where the 𝐵-action on C is
given by the character of weight 𝜆. Then, by Leray spectral sequence,
𝐻^{0}(𝜋^{−1}(𝑥),E𝜆|_{𝜋}−1)=𝑉_{𝜆}. Then, we can construct a homogeneous vector
bundle F𝜆 := 𝐺 ×^{𝑃}𝑉_{𝜆} over 𝐺/𝑃 and we see that 𝜋_{∗}E𝜆 = F𝜆. On the
other hand, every irreducible homogeneous vector bundle on 𝐺/𝑃 has
the form𝐺×^{𝑃}𝐸_{0}hence there exists a line bundleE𝜆such that𝜋_{∗}E𝜆 =F𝜆

where 𝜆 is a 𝑃-dominant weight.

**Theorem 2.4.5**(Borel–Weil–Bott for vector bundles). *Let*𝐺*be a *
*semisim-ple Lie group and* 𝑃 ⊂ 𝐺 *a parabolic subgroup. Let* 𝜆 *be an integral*
*weight over* 𝐺/𝐵 *and* E𝜆 *the associated vector bundle. Call* 𝜌 *the sum*
*of all fundamental weights. Then one and only one of the following*
*situations occur:*

◦ *There exists a nontrivial Weyl reflection*𝑆*such that*𝑆(𝜆+𝜌) −𝜌 =𝜆*.*
*Then*𝐻^{•}(𝐺/𝑃,E𝜆) =0.

◦ *There exists a unique Weyl reflection*𝑆*such that*𝑆(𝜆)*is a dominant*
*integral weight. Then*𝐻^{•}(𝐺/𝑃,E𝜆) =𝑉_{𝑆(𝜆+𝜌)−𝜌}[−𝑙(𝑆)]*.*

*Remark* 2.4.6. A very useful consequence of this result is that
irre-ducible homogeneous vector bundles on any homogeneous variety 𝐺/𝑃
have nonvanishing cohomology in at most one degree.

**Weyl reflections on the Dynkin diagram**

In this section we will describe a simple method to compute cohomol-ogy of any irreducible homogeneous vector bundle on a homogeneous variety, given the weight of the associated representation.

Let us consider a highest weight 𝜆 = 𝜆_{1}𝜔_{1}+ · · · +𝜆_{𝑛}𝜔_{𝑛} on a rank
𝑛 semisimple Lie algebra. Let us write the weight directly on the
Dynkin diagram in the following way:

𝜆_{1} 𝜆_{2} 𝜆_{3}
𝜆_{4}

𝜆_{5}

The action of Weyl reflection is described by Equation 2.2.7. Since
Dynkin diagrams are a graphical way to express the data contained in
the Cartan matrix, which is the data defining Weyl reflections as well,
we can write a set of simple rules which tell us how to perform a given
Weyl reflection, simply by reading the Dynkin diagram. Since we are
writing the highest weight on the Dynkin diagram, we can talk about a
Weyl reflection respect to a*node*referring to the reflection associated to
the fundamental weight which corresponds to that node. The following
rules can be deduced simply by using Equation 2.2.7 to perform the
computations explicitly.

◦ Weyl reflection with respect to a node connected by simple lines:

𝜆_{1} 𝜆_{2} 𝜆_{3} 𝑆_{𝜔}

========2 ⇒^{𝜆}^{1}^{+}^{𝜆}^{2} ^{-}^{𝜆}^{2} ^{𝜆}^{3}^{+}^{𝜆}^{2}

◦ Weyl reflection with respect to a node connected by an

outward-directed double line:

𝜆_{1} 𝜆_{2} 𝜆_{3} 𝑆_{𝜔}

========2 ⇒ ^{𝜆}^{1}^{+}^{𝜆}^{2} ^{-}^{𝜆}^{2} ^{𝜆}^{3}^{+}^{2}^{𝜆}^{2}

◦ Weyl reflection with respect to a node connected by an inward-directed double line:

𝜆_{1} 𝜆_{2} 𝜆_{3} 𝑆_{𝜔}

========2 ⇒ ^{𝜆}^{1}^{+}^{𝜆}^{2} ^{-}^{𝜆}^{2} ^{𝜆}^{3}^{+}^{𝜆}^{2}

◦ Weyl reflection with respect to a node connected by an outward-directed triple line:

𝜆_{1} 𝜆_{2} 𝑆_{𝜔}

========1 ⇒^{-}^{𝜆}^{1} ^{𝜆}^{2}^{+}^{3}^{𝜆}^{1}

◦ Weyl reflection with respect to a node connected by an inward-directed triple line:

𝜆_{1} 𝜆_{2} 𝑆_{𝜔}

========2 ⇒^{𝜆}^{1}^{+}^{𝜆}^{2} ^{-}^{𝜆}^{2}

*Example*2.4.7. Let us consider the flag variety 𝐹(2,3, 𝑛) and the
pro-jections 𝑝 and 𝑞 to its Grassmannians𝐺(2, 𝑛) and 𝐺(3, 𝑛). We get the
following diagram:

𝐹(2,3, 𝑛)

𝐺(2, 𝑛) 𝐺(3, 𝑛)

𝑝 𝑞

(2.4.3)

Let us call U the tautological bundle of 𝐺(2, 𝑛). It is a homogeneous
vector bundle of rank 2. With the notationO (𝑎, 𝑏) = 𝑝^{∗}O (𝑎) ⊗𝑞^{∗}O (𝑏),
we illustrate the method above computing𝐻^{•}(𝐹(2,3, 𝑛), 𝑝^{∗}U^{∨}(−2,1)).
First, the flag variety 𝐹(2,3, 𝑛) is a 𝐺 𝐿(𝑛)-homogeneous variety
de-scribed as 𝐺 𝐿(𝑛)/𝑃^{2}^{,}^{3}. The associated Dynkin diagram is:

The weight associated to 𝑝^{∗}U^{∨}(−2,1) is 𝜔 =𝜔_{1}−2𝜔_{2}+𝜔_{3}. We can
write it on the Dyinkin diagram in the following way:

1 −2 1 0 0

In order to apply the Borel–Weil–Bott algorithm, we first need to add to 𝜔 the sum of fundamental weights, obtaining the following:

2 −1 2 1 1

We can now start with Weyl reflections. Since the second coefficient
of our weight is negative, we apply 𝑆_{𝜔}

2 and we get:

1 1 1 1 1

This last weight is dominant, hence we can subtract back the sum of fundamental weights obtaining the trivial weight (0, . . . ,0) correspond-ing to the trivial representation of dimension 1. Since we used only one Weyl reflection, the cohomology is concentrated in degree one, therefore we conclude that:

𝐻^{𝑘}(𝐹(2,3, 𝑛), 𝑝^{∗}U^{∨}(−2,1)) =

C 𝑘 =1 0 𝑘 ≠1

(2.4.4)

*Remark* 2.4.8. A geometric interpretation of the result we got from
Example 2.4.7 is the following: since 𝐻^{•}(𝐹(2,3, 𝑛), 𝑝^{∗}U^{∨}(−2,1)) =
Ext^{•}( (O (1,−1), 𝑝^{∗}U^{∨}(−1,0)), the outcome of our computation tells
that there exists a unique extension between𝑝^{∗}U^{∨}(−1,0) andO (1,−1).
By the isomorphism U ' U^{∨}(−1) we associate such extension to the

pullback𝑞^{∗}Ueof the tautological bundle of𝐺(3, 𝑛), i.e. to the sequence:

0−→ 𝑝^{∗}U −→ 𝑞^{∗}U −→ O (e 1,−1) −→0 (2.4.5)