8.3 The Borel-Weil-Bott theorem and cohomology

The Borel-Weil-Bott theorem connects representation theory to geometry. It shows how irreducible representations of compact Lie groups relate to cohomology of vector bundles on flag manifolds. This deep result bridges algebra and topology.

Understanding this theorem is key for grasping representation theory of Lie groups. It provides a powerful tool for constructing and analyzing representations, with applications in math physics and algebraic geometry.

Borel-Weil-Bott Theorem Statement

Geometric Realization of Irreducible Representations

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• The Borel-Weil-Bott theorem provides a geometric realization of the irreducible representations of a compact, connected Lie group $G$ in terms of the cohomology of certain homogeneous vector bundles over the flag manifold $G/B$, where $B$ is a Borel subgroup of $G$
• For a dominant integral weight $\lambda$, the theorem states that the irreducible representation of $G$ with highest weight $\lambda$ can be realized as the space of global sections of a certain line bundle $L_\lambda$ over $G/B$
• The theorem establishes a deep connection between the representation theory of Lie groups and the geometry and topology of flag manifolds
• The flag manifold $G/B$ is an example of a homogeneous space, i.e., a manifold on which a Lie group acts transitively

Cohomology and Vanishing Results

• The theorem provides a vanishing result for the higher cohomology groups of $L_\lambda$ when $\lambda$ is dominant
  • In this case, the representation is completely determined by the space of global sections $H^0(G/B, L_\lambda)$
• For non-dominant weights, the theorem gives a precise description of the cohomology in terms of certain "dot actions" of the Weyl group on the weight lattice
  • The dot action is defined as $w \cdot \lambda = w(\lambda + \rho) - \rho$, where $\rho$ is half the sum of the positive roots
  • The cohomology group $H^i(G/B, L_\lambda)$ is isomorphic to the irreducible representation with highest weight $w \cdot \lambda$, where $w$ is an element of the Weyl group with length $i$
• The study of the cohomology of homogeneous spaces is a rich and active area of research in algebraic geometry and representation theory, with connections to other fields such as number theory and mathematical physics

Representations of Compact Lie Groups

Constructing Irreducible Representations

• To apply the Borel-Weil-Bott theorem, one must first identify the dominant integral weights of the Lie group $G$, which correspond to the irreducible representations
• For each dominant integral weight $\lambda$, construct the associated line bundle $L_\lambda$ over the flag manifold $G/B$ using the Borel-Weil construction
• The space of global sections of $L_\lambda$, denoted $H^0(G/B, L_\lambda)$, is an irreducible representation of $G$ with highest weight $\lambda$
• When $\lambda$ is dominant, the higher cohomology groups $H^i(G/B, L_\lambda)$ vanish for $i > 0$, and the representation is completely determined by $H^0(G/B, L_\lambda)$

Understanding Dominant Integral Weights

• To apply the Borel-Weil-Bott theorem effectively, one must be able to identify the dominant integral weights of a given compact Lie group and understand their relationship to the root system and Weyl group
• Dominant integral weights are weights that lie in the fundamental Weyl chamber, which is determined by the choice of a set of positive roots
• The dominant integral weights are in one-to-one correspondence with the irreducible representations of the Lie group $G$
• Examples of dominant integral weights for classical Lie groups:
  • For $SU(n)$, the dominant integral weights are $n$-tuples of integers $(\lambda_1, \ldots, \lambda_n)$ satisfying $\lambda_1 \geq \lambda_2 \geq \cdots \geq \lambda_n$
  • For $SO(2n+1)$, the dominant integral weights are $n$-tuples of integers or half-integers $(\lambda_1, \ldots, \lambda_n)$ satisfying $\lambda_1 \geq \lambda_2 \geq \cdots \geq \lambda_n \geq 0$

Borel-Weil-Bott Theorem vs Cohomology

Cohomology of Homogeneous Vector Bundles

• The Borel-Weil-Bott theorem relates the representation theory of $G$ to the cohomology of certain vector bundles over the homogeneous space $G/B$
• The theorem can be generalized to other homogeneous spaces, such as generalized flag manifolds $G/P$, where $P$ is a parabolic subgroup of $G$
• The cohomology of homogeneous vector bundles over $G/P$ can be used to construct and study representations of $G$, analogous to the case of $G/B$
• Examples of generalized flag manifolds:
  • Grassmannians $Gr(k, n)$, which parametrize $k$-dimensional subspaces of an $n$-dimensional vector space
  • Orthogonal Grassmannians $OG(n, 2n)$, which parametrize $n$-dimensional isotropic subspaces of a $2n$-dimensional vector space with a non-degenerate symmetric bilinear form

Geometric Context and Applications

• Familiarity with the structure and properties of flag manifolds and their generalization to other homogeneous spaces is crucial for understanding the geometric context of the Borel-Weil-Bott theorem
• The theorem has important applications in various areas of mathematics, such as algebraic geometry, complex analysis, and mathematical physics
• In algebraic geometry, the Borel-Weil-Bott theorem is used to study the cohomology of vector bundles over flag varieties and their generalizations, which play a central role in the classification of algebraic varieties
• In complex analysis, the theorem provides a way to construct and study certain spaces of holomorphic functions on complex manifolds, such as the Bergman and Hardy spaces
• In mathematical physics, the Borel-Weil-Bott theorem is used to analyze the representation theory of Lie groups that arise in the study of symmetries in physical systems, such as gauge theories and conformal field theories

Representation Theory Applications

Classical Lie Groups

• The ability to apply the Borel-Weil-Bott theorem to concrete examples, such as the classical Lie groups (e.g., $SU(n)$, $SO(n)$, $Sp(n)$), is important for developing a deep understanding of the representation theory of compact Lie groups
• For the special unitary group $SU(n)$, the theorem can be used to construct and study the irreducible representations corresponding to the dominant integral weights, which are given by Young diagrams or partitions
• For the special orthogonal group $SO(n)$, the theorem provides a way to realize the irreducible representations in terms of the cohomology of certain vector bundles over the orthogonal flag manifold $SO(n)/T$, where $T$ is a maximal torus
• For the symplectic group $Sp(n)$, the Borel-Weil-Bott theorem can be applied to study the irreducible representations in terms of the cohomology of vector bundles over the symplectic flag manifold $Sp(n)/T$

Interdisciplinary Connections

• Recognizing the connections between the Borel-Weil-Bott theorem and other areas of mathematics can lead to new insights and applications of the theorem
• In number theory, the Borel-Weil-Bott theorem has been used to study the cohomology of certain arithmetic groups and their representations, which play a role in the Langlands program
• In mathematical physics, the theorem has been applied to the study of the representation theory of infinite-dimensional Lie groups, such as loop groups and affine Kac-Moody groups, which arise in the context of conformal field theory and integrable systems
• The Borel-Weil-Bott theorem has also found applications in the study of the geometry and topology of certain moduli spaces, such as the moduli space of stable vector bundles over a Riemann surface, which is related to the geometric Langlands correspondence