Abstract
In this chapter, we specialize to the case where \(\mathfrak{g}\) is a complex reductive Lie algebra. Let \(\mathfrak{t}\subset\mathfrak{g}\) be a Cartan subalgebra. We will discuss the Harish-Chandra projection \(\mathsf{hc}_{U}:\ U(\mathfrak{g})\to S(\mathfrak{t})\) for the enveloping algebra, its counterpart \(\mathsf{hc}_{\mathrm{Cl}}:\ \mathrm{Cl}(\mathfrak{g})\to\mathrm{Cl}(\mathfrak{t})\) for the Clifford algebra, and their interaction under the natural algebra morphism \(\gamma:\ U(\mathfrak{g})\to\mathrm{Cl}(\mathfrak{g})\). A number of classical results, such as the Freudenthal–de Vries “strange formula”, will be proved in this spirit. We discuss the ρ-representation and its interpretation in terms of the spin representation. Following Kostant’s work, we consider applications of the cubic Dirac operator 
for equal rank pairs. This includes the Gross–Kostant–Ramond–Sternberg results on multiplets of representations for equal rank Lie subalgebras, as well as aspects of Dirac induction.
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Notes
- 1.
Suppose \(\mathfrak{g}\) is simple, and let G ℝ be the compact, simply connected Lie group with Lie algebra \(\mathfrak{g}_{\mathbb{R}}\). Then the Lie algebras \(\mathfrak{k}_{\mathbb{R}}\) obtained by this procedure are the Lie algebras of centralizers K ℝ of elements g=exp(ξ), where ξ is a vertex of the Weyl alcove of G. Up to conjugacy, these are precisely the centralizers that are semisimple.
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Meinrenken, E. (2013). Applications to reductive Lie algebras. In: Clifford Algebras and Lie Theory. Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge / A Series of Modern Surveys in Mathematics, vol 58. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-36216-3_8
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