Dirac series for some real exceptional Lie groups

A non-vanishing criterion for Dirac cohomology

“…Let us consider the group F4 s in atlas [1,24], whose Dirac series are classified in [3]. We adopt the notation as in Section 6 of [3]. In particular, {ξ 1 , .…”

“…Then the corresponding irreducible unitary representation, realized as p below in atlas, has zero Dirac cohomology since a + c = 0. This representation has infinitesimal character Λ = ξ 1 + ξ 3 , which satisfies the HP condition due to (19) of [3]. Here recall that we should reverse the λ and ν parts in an atlas parameter of a representation to be compatible with the tables in Section 6 of [3].…”

“…This representation has infinitesimal character Λ = ξ 1 + ξ 3 , which satisfies the HP condition due to (19) of [3]. Here recall that we should reverse the λ and ν parts in an atlas parameter of a representation to be compatible with the tables in Section 6 of [3]. This is because Section 6 of [3] labels the simple roots of FI = F 4(4) in a way opposite to atlas.…”

“…Here recall that we should reverse the λ and ν parts in an atlas parameter of a representation to be compatible with the tables in Section 6 of [3]. This is because Section 6 of [3] labels the simple roots of FI = F 4(4) in a way opposite to atlas. Now let us realize p as a cohomologically induced module.…”

“…This notion was introduced by Vogan [23]. Motivated by his conjecture on Dirac cohomology proven by Huang and Pandžić [9], we say that a weight Λ ∈ h * f satisfies the Huang-Pandžić condition (HP condition for short henceforth) if (3) {δ − ρ (j) n } + ρ K = wΛ, where δ is any highest weight of some K-type, 0 ≤ j ≤ s − 1, and w ∈ W (g, t f ). Note that if Λ satisfies the HP condition, then it must be real in the sense of Definition 5.4.1 of [21].…”