By Jing-Song Huang

This monograph provides a finished remedy of significant new principles on Dirac operators and Dirac cohomology. Dirac operators are primary in physics, differential geometry, and group-theoretic settings (particularly, the geometric development of discrete sequence representations). The similar thought of Dirac cohomology, that is outlined utilizing Dirac operators, is a far-reaching generalization that connects index conception in differential geometry to illustration idea. utilizing Dirac operators as a unifying subject matter, the authors show how the most very important leads to illustration conception healthy jointly while seen from this angle.

Key issues coated include:

* evidence of Vogan's conjecture on Dirac cohomology

* easy proofs of many classical theorems, similar to the Bott–Borel–Weil theorem and the Atiyah–Schmid theorem

* Dirac cohomology, outlined by way of Kostant's cubic Dirac operator, in addition to different heavily comparable different types of cohomology, comparable to n-cohomology and (g,K)-cohomology

* Cohomological parabolic induction and $A_q(\lambda)$ modules

* Discrete sequence thought, characters, lifestyles and exhaustion

* polishing of the Langlands formulation on multiplicity of automorphic kinds, with applications

* Dirac cohomology for Lie superalgebras

An first-class contribution to the mathematical literature of illustration conception, this self-contained exposition bargains a scientific exam and panoramic view of the topic. the fabric should be of curiosity to researchers and graduate scholars in illustration idea, differential geometry, and physics.

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**Additional resources for Dirac Operators in Representation Theory**

**Example text**

U i · · · ∧ u k , i where the hat on u i indicates that u i is omitted. This deﬁnes the action of V on S, and this action extends to all of C(V ). To see this, one can simply check that the relations are satisﬁed. Let us instead embed S into the algebra C(V ) as a left ideal, in such a way that the above action corresponds to left multiplication. Denote by u ∗top any nonzero element in top (U ∗ ), viewed as an element of C(V ) via the Chevalley map. Then since u ∗ u ∗top = 0 in C(V ) for any u ∗ ∈ U ∗ , and since C(U ) = (U ) as B is 0 on U , we see that the left ideal of C(V ) generated by u ∗top can be identiﬁed with (U )u ∗top , which is isomorphic to S in the obvious way.

Restricting our attention to x ∈ V , let x = λv + w where w ⊥ v. , the conjugation by v preserves V ⊂ C(V ), and the operation it induces on V is minus the reﬂection with respect to v ⊥ . To eliminate the minus sign, instead of conjugation by v one can consider the twisted conjugation: x → κ(v)xv −1 = vxv, x ∈ C(V ). This twisted conjugation again preserves V , and the induced transformation on V is now exactly the reﬂection with respect to v ⊥ . Moreover, if we denote by Pin (V ) the subgroup of the group of units in C(V ) generated by all v ∈ V of length 1, then we get a homomorphism ρ from Pin (V ) into the orthogonal group O(V ), deﬁned by ρ(u)x = κ(u)xu −1 , x ∈ V, for u ∈ Pin (V ).

Then S is the only irreducible C(V )-module up to isomorphism. Proof. Let S be any irreducible C(V )-module. 6). Since p12 = p1 , it can only have eigenvalues 0 and 1 on S . Moreover, since C(V ) is simple, S cannot have a nonzero annihilator in C(V ) and hence p1 is not identically 0 on S . Let us take some nonzero x ∈ S such that p1 x = x. Then all u i∗ annihilate x, since u i∗ p1 = ± 21n u ∗n . . (u i∗ )2 . . u ∗1 u 1 . . u n = 0. Now deﬁne φ : S → S by φ(u I ) = u I x, I ⊂ {1, . . , n}.