Dirac cohomology and unipotent representations of complex groups

Abstract

If π is a (g, K)−module, then D induces an operatorwhere Spin is a spin module for C(s). If π is unitary, then π ⊗ Spin admits a K−invariant inner product , such that D is self adjoint with respect to this inner product. It follows that D 2 ≥ 0 on π ⊗ Spin. Using the above formula for D 2 , we find that Cas g + ρ g 2 ≤ ∆(Cas k ) + ρ k 2 on any K−type τ occurring in π ⊗ Spin. Another way of putting this is dirineq dirineqfor any τ occurring in π ⊗ Spin, where Λ is the infinitesimal character of π. This is the Dirac inequality mentioned above. These ideas are generalized by Vogan [V] and Huang-Pandžić [HP1] as follows. For an arbitrary admissible (g, K) module π, we define Dirac cohomology of π asThe main result about H D is the following theorem conjectured by Vogan. t:basicTheorem 1.2.[HP1] Assume that H D (π) is nonzero, and contains an irreducibleConversely, if π is unitary and τ = xΛ − ρ k is the highest weight of a K−type occuring in π ⊗ Spin, then this K−type is contained in H D (π).Note that for unitary π, the multiplicity of τ in H D (π) is the same as the multiplicity of τ in π ⊗ Spin.These results might suggest that difficulties should arise in passing between K−types of π and K−types of π ⊗ Spin. For unitary π, the situation is however greatly simplified by the Dirac inequality. Namely, together with (1.1), Theorem 1.2 shows that the infinitesimal characters τ + ρ k of K−types in Dirac cohomology have minimal possible norm. This means that whenever such E τ appears in the tensor product of a K−type E µ of π and a K−type E γ of Spin, it necessarily appears as the PRV component [PRV], i.e.,where µ − denotes the lowest weight of E µ .Assume now that g and k have equal rank, e.g. g 0 = sp(2n, R) or g 0 = u(p, q), and assume (as we may) that Λ is g-dominant. Then the above x must belong toThe condition that xΛ is regular and integral for K puts further restrictions on both x and Λ which will be made precise later. We will make use of the following decomposition of the K−module Spin: spindec spindec (1.4) Spin = σ∈W 1 E σρg −ρ k .

On a Conjecture of Barbasch and Pandžić

“…Since it has infinitesimal character c = /2 (see e.g. (2.2) of [1]), in view of Proposition 2.3, it suffices to focus on the K-types with spin norm c . By (10), only the K-type E c has spin 3386 DONG norm c .…”

“…and Pandžić on page 5 of[1] from Theorem 2.1 below, to find all the irreducible unitary representations with nonzero Dirac cohomology, it suffices to consider those J L −s L such that 2 L is dominant integral regular for + 0 0 , where s ∈ W is an involution. Now let us state Conjecture 3.4 of[1].…”

Representations associated to small nilpotent orbits for complex Spin groups

“…For the case O(2n, C) (rather than Spin(2n, C)), the K-structure of the representations studied in this paper were considered earlier in [McG] and [BP1].…”

Unitary representations with non-zero Dirac cohomology for complex E 6

“…Our main results are classifications of G(R) d for G(R) on the following list: EI = E 6 (6) , EIV = E 6(−26) , FI = F 4 (4) , FII = F 4(−20) .…”