Dirac cohomology, unitary representations and a proof of a conjecture of Vogan

Abstract

Let G G be a connected semisimple Lie group with finite center. Let K K be the maximal compact subgroup of G G corresponding to a fixed Cartan involution θ \theta . We prove a conjecture of Vogan which says that if the Dirac cohomology of an irreducible unitary ( g , K ) (\mathfrak {g},K) -module X X contains a K K -type with highest weight γ \gamma , then X X has infinitesimal character γ + ρ c \gamma +\rho _{c} . Here ρ c \rho _{c} is the half sum of the compact positive roots. As an application of the main result we classify irreducible unitary ( g , K ) (\mathfrak {g},K) -modules X X with non-zero Dirac cohomology, provided X X has a strongly regular infinitesimal character. We also mention a generalization to the setting of Kostant’s cubic Dirac operator.

Dirac cohomology of Wallach representations

“…This is one of the main results of [Huang and Pandžić 2002], conjectured by Vogan. The implication (2) ⇒ (3) is obvious.…”

“…We denote by E γ the irreduciblẽ K -module with highest weight γ ∈ t * . Theorem 3.1 [Huang and Pandžić 2002]. Let X be an irreducible unitary (g, K )-module with infinitesimal character ∈ t * .…”

“…A careful analysis of this fact led Vogan [1997] to the notion of Dirac cohomology (see also [Huang and Pandžić 2002] = 0 has nonzero solutions in X ⊗ S are precisely the ones with nonzero Dirac cohomology. These representations are extremal in the sense that Dirac inequality becomes equality on some of theK -types of X ⊗ S. Once such aK -type E is fixed, there are only a few possible irreducible modules X with H D (X ) ⊇ E. Namely, the main result of [Huang and Pandžić 2002], conjectured by Vogan, says that such E determines the infinitesimal character of X .…”

Algebraic and analytic Dirac induction for graded affine Hecke algebras

“…These constructions are analogs of the Dirac operator and Dirac cohomology for representations of real reductive groups. One of the main results concerning Dirac cohomology for (g, K)-modules, conjectured by Vogan and proved by Huang and Pandžić [18], says that, if it is nonzero, the Dirac cohomology of a (g, K)-module uniquely determines the infinitesimal character of the module. A graded Hecke algebra analog is proved in [3,Theorems 4.2 and 4.4].…”

Cubic Dirac Operators and the Strange Freudenthal–de Vries Formula for Colour Lie Algebras

“…In [13], Kostant studies Dirac operators in a more general setting and shows that the introduction of a cubic term in the usual Dirac operator is necessary for non-symmetric pairs. Huang and Pandžić calculated the cohomology associated to Dirac operators as conjectured by Vogan ( [9]) and then Dirac cohomologies have been studied in various situations (e.g., [15], [11], [16]). The theory of Dirac operators has been extended to other algebraic structures: for example this has been studied for Lie superalgebras in [21], [10], [12], [28] and for Hecke algebras and rational Cherednik algebras in [2], [5], [6].…”