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Dirac cohomology, elliptic representations and endoscopy. (English) Zbl 1344.22005
Nevins, Monica (ed.) et al., Representations of reductive groups. In honor of the 60th birthday of David A. Vogan, Jr. Proceedings of the conference, MIT, Cambridge, MA, USA, May 19–23, 2014. Cham: Birkhäuser/Springer (ISBN 978-3-319-23442-7/hbk; 978-3-319-23443-4/ebook). Progress in Mathematics 312, 241-276 (2015).
Let \(G\) be a real reductive group and \(K\) a maximal compact subgroup of \(G\) with complexified Lie algebras \({\mathfrak g}\) and \({\mathfrak k}\) respectively. Let \(B\) be a nondegenerate invariant symmetric bilinear form on \({\mathfrak g}\) which restricts to the Killing form on the semisimple part of \({\mathfrak g}\). Write \({\mathfrak g}={\mathfrak k}\oplus{\mathfrak p}\) for the corresponding Cartan decomposition, \(U({\mathfrak g})\) for the enveloping algebra of \({\mathfrak g}\) and \(C({\mathfrak p})\) for the Clifford algebra of \({\mathfrak p}\) with respect to \(B\). Fix an orthonormal basis \(\{Z_j\}\) of \({\mathfrak p}\) with respect to \(B\). The element \(\sum_{i=1}^nZ_i\times Z_i\) of \(U({\mathfrak g})\otimes C({\mathfrak p})\) is invariant with respect to the diagonal action of \(K\) and is independent of the choice of the basis of \({\mathfrak p}\). Associated with a \(({\mathfrak g},K)\)-module \((\pi, X)\) is the Dirac operator \[ D:=\sum_{i=1}^n\pi(Z_i)\times \gamma(Z_i) \] where \(\gamma\) denotes the Clifford multiplication. \(D\) acts as a differential on \(\text{Ker}(D^2)\) and the induced cohomology is the Dirac cohomology \(H_D(X):=\text{Ker} D/\text{Im} D\cap\text{Ker} D\), which is a \({\tilde K}\)-module, where \({\tilde K}\) is the spin double cover of \(K\). In the late 1990’s, Vogan conjectured that if \(X\) has infinitesimal character \(\Lambda\) and \(H_D(X)\) contains a \(\tilde{K}\)-type with infinitesimal character \(\lambda\) then \(\Lambda\) and \(\lambda\) are conjugate under the Weyl group of \(G\). This statement is now a theorem proved in the early 2000’s by the author and P. Pandžić [J. Am. Math. Soc. 15, No. 1, 185–202 (2002; Zbl 0980.22013)]. This result was the beginning of a long and still ongoing series of results on Dirac cohomology and its deep impacts in representation theory.
In the paper under review, the author surveys various aspects of representation theory involving Dirac operators: computation of Dirac cohomology for various modules, connections between Dirac cohomology and other cohomologies, study of unitary elliptic representations and endoscopic transfer in the light of Dirac cohomology and Dirac index. The paper also contains several open problems and conjectures.
For the entire collection see [Zbl 1336.22001].
Reviewer: Salah Mehdi (Metz)

22E46 Semisimple Lie groups and their representations
22E47 Representations of Lie and real algebraic groups: algebraic methods (Verma modules, etc.)
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