By Jing-Song Huang
This monograph provides a entire remedy of significant new principles on Dirac operators and Dirac cohomology. Dirac operators are commonly used in physics, differential geometry, and group-theoretic settings (particularly, the geometric development of discrete sequence representations). The comparable notion of Dirac cohomology, that is outlined utilizing Dirac operators, is a far-reaching generalization that connects index idea in differential geometry to illustration idea. utilizing Dirac operators as a unifying topic, the authors reveal how one of the most vital leads to illustration thought healthy jointly whilst considered from this angle.
Key issues lined include:
* evidence of Vogan's conjecture on Dirac cohomology
* basic proofs of many classical theorems, corresponding to the Bott–Borel–Weil theorem and the Atiyah–Schmid theorem
* Dirac cohomology, outlined through Kostant's cubic Dirac operator, in addition to different heavily similar types of cohomology, akin to n-cohomology and (g,K)-cohomology
* Cohomological parabolic induction and $A_q(\lambda)$ modules
* Discrete sequence thought, characters, life and exhaustion
* sprucing of the Langlands formulation on multiplicity of automorphic varieties, with applications
* Dirac cohomology for Lie superalgebras
An very good contribution to the mathematical literature of illustration thought, this self-contained exposition bargains a scientific exam and panoramic view of the topic. the cloth may be of curiosity to researchers and graduate scholars in illustration concept, differential geometry, and physics.
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Extra info for Dirac Operators in Representation Theory
11. Assume λ is not an integer of parity ducible. + 1. Then Vλ, is irre- Proof. Let U ⊂ V be a nonzero g-invariant subspace. Since U is invariant under W , it must contain an eigenvector vn of W . Namely, from any linear combination x of two or more vn ’s which lies in U , we can obtain a shorter one by combining x and π(W )x. If we now act on vn by X and Y , we get all vk , as the scalars in the formulas deﬁning the action can never be zero. Hence U = V . 3 Inﬁnite-dimensional representations 25 If λ = k − 1 where k ≥ 1 is an integer of parity , then Vλ, contains two irreducible submodules, one with weights k, k + 2, k + 4, .
Since this holds for any i, we see that u must be a constant. We will be done if we prove that the only constants contained in Pin (V ) are 1 and −1. This can be proved by using another piece of structure that we describe in the following. 12. The principal antiautomorphism of C(V ). We denote by α the unique antiautomorphism of C(V ) equal to the identity on V . 4), α is given by α(1) = 1 and α(Z i1 . . Z ik ) = Z ik . . Z i1 . , on the image of k (V ) under the Chevalley map). , that α is an antiinvolution.
13, there are two kinds of irreducible highest weight (sl(2, C), S O(2))-modules: the ﬁnite-dimensional ones, and the highest weight discrete series (or limits of discrete series) representations. We already worked out the case when both factors are ﬁnite-dimensional. Suppose that V is an irreducible (inﬁnite-dimensional) module with highest weight −n and that W is an irreducible ﬁnite-dimensional module with highest weight k, where n and k are positive integers. It is clear that the weights of V ⊗ W are −n + k, −n + k − 2, .
Dirac Operators in Representation Theory by Jing-Song Huang