- Email: [email protected]
Corrigendum to “Dirac cohomology and translation functors” [J. Algebra 375 (2013) 328–336]
Journal of Algebra 451 (2016) 577–582
Contents lists available at ScienceDirect
Journal of Algebra www.elsevier.com/locate/jalgebra
Corrigendum
Corrigendum to “Dirac cohomology and translation functors” [J. Algebra 375 (2013) 328–336] S. Mehdi a,∗ , P. Pandžić b a
Institut Elie Cartan de Lorraine, CNRS – UMR 7502, Université de Lorraine, France b Department of Mathematics, University of Zagreb, Croatia
a r t i c l e
i n f o
Article history: Received 17 July 2014 Available online 28 January 2016 Communicated by Shrawan Kumar MSC: primary 22E47 secondary 22E46
a b s t r a c t The statements of Theorem 1.3, Lemma 3.2 and Proposition 5.2 in [4] are incorrect. We give counterexamples to these statements and we offer a replacement for Theorem 1.3, under stronger assumptions. © 2012 Elsevier Inc. All rights reserved.
Keywords: (g, K)-module Dirac operator Dirac cohomology Translation functor
We first give a counterexample to Theorem 1.3 in [4]. We thank the referee for suggesting this counterexample. Let G be the Hermitian group SU (2, 1) with Cartan involution θ equal to the conjugate transpose inverse. The complexified Lie algebra of G is g = sl(3, C), the maximal compact subgroup corresponding to θ is K = S(U (2) × U (1)) and the corresponding Cartan decomposition of g is g = k ⊕ p. We fix a compact Cartan subalgebra h of g to be DOI of original article: http://dx.doi.org/10.1016/j.jalgebra.2012.11.022.
* Corresponding author. E-mail addresses: [email protected] (S. Mehdi), [email protected] (P. Pandžić). http://dx.doi.org/10.1016/j.jalgebra.2015.04.032 0021-8693/© 2012 Elsevier Inc. All rights reserved.
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the space of diagonal matrices in g. The set Δ of h-roots in g splits into subsets Δc of compact roots and Δn of non-compact roots. Let b be a θ-stable Borel subalgebra of g not containing either of the two abelian K-invariant subspaces p± of p. Let Δ+ ⊂ Δ be the positive root system corresponding to b and ρ the half sum of elements of Δ+ . The Harish-Chandra module of the nonholomorphic discrete series representation of G with Harish-Chandra parameter ρ is the cohomologically induced module Ab (0). The Dirac cohomology HD (Ab (0)) equals the irreducible finite-dimensional K-module with highest weight ρn , where ρn is the half + sum of elements of Δ ∩ Δn , see Proposition 5.4 in [2]. On the other hand, by Proposition 11.180 in [3], the limit of discrete series Ab (−ρ) with infinitesimal character equal to 0 is a non-zero irreducible unitary (g, K)-module. The module Ab (−ρ) can be obtained by translating Ab (0) to infinitesimal character 0. It has trivial Dirac cohomology since no W (g, h)-translate of the infinitesimal character 0 can be k-regular. Back to Theorem 1.3 in [4], we take λ = 0, ν = ρ, X0 = Ab (−ρ), Fρ the irreducible representation of G with highest weight ρ and Xρ = Ab (0). Since X0 is a translate of Xρ , it follows that Xρ embeds into X0 ⊗ Fρ (see Proposition 7.143 in [3]). Since HD (Xρ ) = 0 but HD (X0 ) = 0, we see that Theorem 1.3 of [4] does not hold. Next we provide an example showing that statements of Lemma 3.2 and Proposition 5.2 are not correct. Let Xλ be a lowest weight discrete series module for (g, K) = (sl(2, C), SO(2)). The K-types of Xλ are spanned by the weight vectors xλ+1 , xλ+3 , . . . , where the subscripts denote the weights. Let Fν be the finite-dimensional module with highest weight ν, spanned by the weight vectors f−ν , f−ν+2 , . . . , fν . Recall that the spin module is spanned by weight vectors s±1 . Then one checks that • Ker D12 = Ker D1 = xλ+1 ⊗ Fν ⊗ s−1 ; • Ker D2 = Xλ ⊗ f−ν ⊗ s−1 ⊕ Xλ ⊗ fν ⊗ s1 ; • Ker D1 ∩ Ker D2 = Cxλ+1 ⊗ f−ν ⊗ s−1 . (Ker D12 = Ker D1 follows from unitarity of Xλ , or can be obtained by a direct calculation.) Assuming that λ > ν, the translates of Xλ by Fν are the lowest weight discrete series modules Xλ−ν and Xλ+ν . One checks that ϕ(Ker DXλ−ν ) ⊆ Ker D1 ∩ Ker D2 , but ϕ(Ker DXλ+ν ) Ker D1 ∩ Ker D2 . This shows that Lemma 3.2 does not hold for Xλ+ν . Moreover, ϕ(Ker DXλ+ν ) β(Ker(DXλ ) ⊗ Ker(DFν ) ⊗ S ), so Proposition 5.2 also fails. The mistake in the proof of Lemma 3.2 was the claim that D1 + D2 = 0 on ϕ(Ker DXλ+ν ) implies D12 = D22 on ϕ(Ker DXλ+ν ). Namely, ϕ(Ker DXλ+ν ) need not be invariant under D1 or D2 . Note that in the above example Lemma 3.2 fails for one of the translates of Xλ , but it holds for the other translate. In the following, we show that a similar property holds in a much more general setting.
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Getting back to the general setting of [4], let Xλ be a (g, K)-module with infinitesimal character λ. We fix compatible choices of positive roots for g and k; in particular this fixes ρ = ρg and ρk . As a K-module, the spin module S decomposes as
S = 2m
Ewρ−ρk .
w∈W 1
Here W 1 denotes the subset of the Weyl group W = W (g, t) consisting of elements which take the fundamental chamber for g into the fundamental chamber for k, and m = [ 12 dim a]. Here and in the sequel, we denote by Ea the finite-dimensional K-module with highest weight a. Assume that PRV(Eγ ⊗ Ewρ−ρk ) ⊆ Ker DXλ
(1)
for some K-type Eγ in Xλ and for some element w ∈ W . Here PRV(Ea ⊗ Eb ) denotes the PRV component of Ea ⊗ Eb , i.e., the unique component of Ea ⊗ Eb with extremal weight a +w0 b, where w0 denotes the longest element of the Weyl group of k (see [6]). We note that in case Xλ is unitary, Ker DXλ = HD (Xλ ) can only contain PRV components as above. Let Fν be the irreducible finite-dimensional (g, K)-module with highest weight ν ∈ of t ⊆ h∗ ; here h is a fundamental Cartan subalgebra of g. Let E be the K-submodule Xλ ⊗ Fν ⊗ S defined as E = PRV(Eγ ⊗ Cartan(Ewν ⊗ Ewρ−ρk )),
(2)
where Cartan(Ea ⊗ Eb ) denotes the Cartan component of Ea ⊗ Eb , i.e., the unique component of Ea ⊗ Eb with highest weight a + b. The motivation for the above definition of E is the fact HD (Fν ) = 2m
Cartan(Eσν ⊗ Eσρ−ρk )
σ∈W 1
(see [1] or [5]). This fact implies that E ⊆ Ker D2 . We claim that, under a dominance assumption, E is also contained in Ker D1 ; this follows from (1) and the following lemma. Lemma 3. Let a, b, c be k-dominant weights such that a + w0 b + w0 c is k-dominant. Then PRV(PRV(Ea ⊗ Eb ) ⊗ Ec )
(4)
= PRV(PRV(Ea ⊗ Ec ) ⊗ Eb ) = PRV(Ea ⊗ Cartan(Eb ⊗ Ec )).
(5)
Proof. Since (5) is symmetric in b and c, it suffices to prove that (4) equals (5). The highest weights of (4) and (5) are both equal to a +w0 b +w0 c, so it is enough to prove that
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Ea+w0 b+w0 c has multiplicity one in Ea ⊗ Eb ⊗ Ec . This follows from the characterization of the PRV component as the unique component of the tensor product with infinitesimal character of the smallest possible norm. 2 Corollary 6. Assume that γ + w0 wν + w0 wρ + ρk is k-dominant.
(7)
Then the K-module E defined by (2) has highest weight γ +w0 wν +w0 wρ +ρk . Moreover, E appears with multiplicity one in Eγ ⊗ Ewν ⊗ Ewρ−ρk ⊂ Xλ ⊗ Fν ⊗ S, and is contained in Ker D1 ∩ Ker D2 . From now on, we assume that λ is regular and Xλ is irreducible.
(8)
Moreover, denoting by Δ(Fν ) the multiset of weights of Fν , and by Cλ the (open) integral Weyl group chamber of λ, we assume that ∀μ ∈ Δ(Fν ),
λ + μ ∈ Cλ .
(9)
(Recall that the integral Weyl group for λ consists of w ∈ W (g, h) such that λ − wλ is a sum of roots. See [7, Lemma 7.2.17].) The module Xλ defines a unique coherent family {Xλ+β }, where β ranges over the lattice of weights of finite-dimensional (g, K)-modules (see [7, Chapter 7]; note that since G is connected, we can base coherent families on the fundamental Cartan subalgebra instead of the split Cartan subgroup). By the assumption (9), for each weight μ of Fν , Xλ+μ is an irreducible (g, K)-module. Moreover, we have Xλ ⊗ Fν =
Xλ+μ .
(10)
μ∈Δ(Fν )
We thank David Vogan for communicating this fact to us [8]. We tensor (10) with the spin module S, and denote by E μ the corresponding projection of E onto Xλ+μ ⊗ S. Since by Corollary 6, E is contained in Ker D1 ∩ Ker D2 ⊆ Ker DXλ ⊗Fν , it follows that E μ ⊆ Ker DXλ+μ . We have proved Proposition 11. With the above notation, assume that the assumptions (1), (7), (8) and (9) hold. If μ ∈ Δ(Fν ) is such that E μ = 0, then the translate Xλ+μ of Xλ satisfies Ker DXλ+μ = 0. Note that since E = 0, there is at least one μ such that E μ = 0. If μ = 0 is the only such μ then the translate Xλ+μ is the same as Xλ . We could avoid this case by requiring
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that 0 is not a weight of Fν . Below we discuss other cases when one can say more about the μ satisfying E μ = 0. We now strengthen the assumption (1) by requiring PRV(Eγ ⊗ Ewρ−ρk ) ⊆ HD (Xλ ).
(12)
(Note that when Xλ is unitary, (1) is equivalent to (12).) In this case, we can use Vogan’s conjecture (see the end of Section 2 of [4]) to conclude that u(λ) = γ + w0 wρ + 2ρk for some u ∈ W 1 . Here we are assuming that γ + w0 wρ + ρk is k-dominant; this follows if we assume (7). We now replace λ by u−1 (λ); this does not affect the assumptions (8) and (9). So we have λ = γ + w0 wρ + 2ρk .
(13)
Suppose now that E μ is not only contained in Ker DXλ+μ but 0 = E μ ⊆ HD (Xλ+μ ).
(14)
Note that the K-type E μ is isomorphic to E, so its highest weight is given by Corollary 6. Therefore we can use Vogan’s conjecture again to conclude that, up to Weyl group conjugacy, λ + μ equals γ + w0 wν + w0 wρ + 2ρk , which equals λ + w0 wν by (13). By assumption (9), it follows that μ = w0 wν. In particular, there is at most one μ satisfying (14). Observe that if Xλ is unitary then E μ = 0 implies μ = 0. Indeed, if E 0 = 0 then (14) would be satisfied for μ = 0, so it would follow that λ + w0 wν is conjugate to λ, which is impossible. The above discussion implies the following result, which can be regarded as a substitute for Theorem 1.3 in [4]. Theorem 15. Let Xλ be an irreducible unitary (g, K)-module with regular infinitesimal character λ. Assume that Xλ contains a K-type with highest weight γ = λ − w0 wρ − 2ρk , such that (12) holds. Let Fν be the irreducible finite-dimensional (g, K)-module with highest weight ν ∈ t . Assume that (7) and (9) hold. Then (a) If (14) holds for some μ ∈ Δ(Fν ), then μ = w0 wν. In particular, Xλ+w0 wν has nonzero Dirac cohomology. (b) If all Xλ+μ , μ ∈ Δ(Fν ), are unitary, then (14) holds for some μ ∈ Δ(Fν ) and hence Xλ+w0 wν has nonzero Dirac cohomology.
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Note that the role of Lemma 3.2 in [4] is played by Corollary 6 together with projecting the K-type E onto the K-types E μ . The condition (3.3) of Lemma 3.2 is no longer needed, but our present argument requires the dominance conditions described above. Note also that the equation (5.3) in Proposition 5.2 of [4] can be replaced by E ⊆ β(Ker(DXλ ) ⊗ Ker(DFν ) ⊗ S ) which follows from Corollary 6 and the fact Ker D1 ∩ Ker D2 ⊆ β(Ker(DXλ ) ⊗ Ker(DFν ) ⊗ S ). (See the proof of Proposition 5.2 and Lemma 5.6 of [4].) References [1] J.-S. Huang, Y.-F. Kang, P. Pandžić, Dirac cohomology of some Harish-Chandra modules, Transform. Groups 14 (2009) 163–173. [2] J.-S. Huang, P. Pandžić, Dirac cohomology, unitary representations and a proof of a conjecture of Vogan, J. Amer. Math. Soc. 15 (2002) 185–202. [3] A.W. Knapp, D.A. Vogan, Cohomological Induction and Unitary Representations, Princeton University Press, Princeton, NJ, 1995. [4] S. Mehdi, P. Pandžić, Dirac cohomology and translation functors, J. Algebra 375 (2013) 328–336. [5] S. Mehdi, R. Zierau, The Dirac cohomology of a finite dimensional representation, Proc. Amer. Math. Soc. 142 (5) (2014) 1507–1512. [6] K.R. Parthasarathy, R. Ranga Rao, V.S. Varadarajan, Representations of complex semi-simple Lie groups and Lie algebras, Ann. of Math. (2) 85 (1967) 383–429. [7] D.A. Vogan, Representations of Real Reductive Groups, Birkhäuser, Boston, 1981. [8] D.A. Vogan, personal communication, 2013–2014.
Contents lists available at ScienceDirect
Journal of Algebra www.elsevier.com/locate/jalgebra
Corrigendum
Corrigendum to “Dirac cohomology and translation functors” [J. Algebra 375 (2013) 328–336] S. Mehdi a,∗ , P. Pandžić b a
Institut Elie Cartan de Lorraine, CNRS – UMR 7502, Université de Lorraine, France b Department of Mathematics, University of Zagreb, Croatia
a r t i c l e
i n f o
Article history: Received 17 July 2014 Available online 28 January 2016 Communicated by Shrawan Kumar MSC: primary 22E47 secondary 22E46
a b s t r a c t The statements of Theorem 1.3, Lemma 3.2 and Proposition 5.2 in [4] are incorrect. We give counterexamples to these statements and we offer a replacement for Theorem 1.3, under stronger assumptions. © 2012 Elsevier Inc. All rights reserved.
Keywords: (g, K)-module Dirac operator Dirac cohomology Translation functor
We first give a counterexample to Theorem 1.3 in [4]. We thank the referee for suggesting this counterexample. Let G be the Hermitian group SU (2, 1) with Cartan involution θ equal to the conjugate transpose inverse. The complexified Lie algebra of G is g = sl(3, C), the maximal compact subgroup corresponding to θ is K = S(U (2) × U (1)) and the corresponding Cartan decomposition of g is g = k ⊕ p. We fix a compact Cartan subalgebra h of g to be DOI of original article: http://dx.doi.org/10.1016/j.jalgebra.2012.11.022.
* Corresponding author. E-mail addresses: [email protected] (S. Mehdi), [email protected] (P. Pandžić). http://dx.doi.org/10.1016/j.jalgebra.2015.04.032 0021-8693/© 2012 Elsevier Inc. All rights reserved.
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the space of diagonal matrices in g. The set Δ of h-roots in g splits into subsets Δc of compact roots and Δn of non-compact roots. Let b be a θ-stable Borel subalgebra of g not containing either of the two abelian K-invariant subspaces p± of p. Let Δ+ ⊂ Δ be the positive root system corresponding to b and ρ the half sum of elements of Δ+ . The Harish-Chandra module of the nonholomorphic discrete series representation of G with Harish-Chandra parameter ρ is the cohomologically induced module Ab (0). The Dirac cohomology HD (Ab (0)) equals the irreducible finite-dimensional K-module with highest weight ρn , where ρn is the half + sum of elements of Δ ∩ Δn , see Proposition 5.4 in [2]. On the other hand, by Proposition 11.180 in [3], the limit of discrete series Ab (−ρ) with infinitesimal character equal to 0 is a non-zero irreducible unitary (g, K)-module. The module Ab (−ρ) can be obtained by translating Ab (0) to infinitesimal character 0. It has trivial Dirac cohomology since no W (g, h)-translate of the infinitesimal character 0 can be k-regular. Back to Theorem 1.3 in [4], we take λ = 0, ν = ρ, X0 = Ab (−ρ), Fρ the irreducible representation of G with highest weight ρ and Xρ = Ab (0). Since X0 is a translate of Xρ , it follows that Xρ embeds into X0 ⊗ Fρ (see Proposition 7.143 in [3]). Since HD (Xρ ) = 0 but HD (X0 ) = 0, we see that Theorem 1.3 of [4] does not hold. Next we provide an example showing that statements of Lemma 3.2 and Proposition 5.2 are not correct. Let Xλ be a lowest weight discrete series module for (g, K) = (sl(2, C), SO(2)). The K-types of Xλ are spanned by the weight vectors xλ+1 , xλ+3 , . . . , where the subscripts denote the weights. Let Fν be the finite-dimensional module with highest weight ν, spanned by the weight vectors f−ν , f−ν+2 , . . . , fν . Recall that the spin module is spanned by weight vectors s±1 . Then one checks that • Ker D12 = Ker D1 = xλ+1 ⊗ Fν ⊗ s−1 ; • Ker D2 = Xλ ⊗ f−ν ⊗ s−1 ⊕ Xλ ⊗ fν ⊗ s1 ; • Ker D1 ∩ Ker D2 = Cxλ+1 ⊗ f−ν ⊗ s−1 . (Ker D12 = Ker D1 follows from unitarity of Xλ , or can be obtained by a direct calculation.) Assuming that λ > ν, the translates of Xλ by Fν are the lowest weight discrete series modules Xλ−ν and Xλ+ν . One checks that ϕ(Ker DXλ−ν ) ⊆ Ker D1 ∩ Ker D2 , but ϕ(Ker DXλ+ν ) Ker D1 ∩ Ker D2 . This shows that Lemma 3.2 does not hold for Xλ+ν . Moreover, ϕ(Ker DXλ+ν ) β(Ker(DXλ ) ⊗ Ker(DFν ) ⊗ S ), so Proposition 5.2 also fails. The mistake in the proof of Lemma 3.2 was the claim that D1 + D2 = 0 on ϕ(Ker DXλ+ν ) implies D12 = D22 on ϕ(Ker DXλ+ν ). Namely, ϕ(Ker DXλ+ν ) need not be invariant under D1 or D2 . Note that in the above example Lemma 3.2 fails for one of the translates of Xλ , but it holds for the other translate. In the following, we show that a similar property holds in a much more general setting.
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579
Getting back to the general setting of [4], let Xλ be a (g, K)-module with infinitesimal character λ. We fix compatible choices of positive roots for g and k; in particular this fixes ρ = ρg and ρk . As a K-module, the spin module S decomposes as
S = 2m
Ewρ−ρk .
w∈W 1
Here W 1 denotes the subset of the Weyl group W = W (g, t) consisting of elements which take the fundamental chamber for g into the fundamental chamber for k, and m = [ 12 dim a]. Here and in the sequel, we denote by Ea the finite-dimensional K-module with highest weight a. Assume that PRV(Eγ ⊗ Ewρ−ρk ) ⊆ Ker DXλ
(1)
for some K-type Eγ in Xλ and for some element w ∈ W . Here PRV(Ea ⊗ Eb ) denotes the PRV component of Ea ⊗ Eb , i.e., the unique component of Ea ⊗ Eb with extremal weight a +w0 b, where w0 denotes the longest element of the Weyl group of k (see [6]). We note that in case Xλ is unitary, Ker DXλ = HD (Xλ ) can only contain PRV components as above. Let Fν be the irreducible finite-dimensional (g, K)-module with highest weight ν ∈ of t ⊆ h∗ ; here h is a fundamental Cartan subalgebra of g. Let E be the K-submodule Xλ ⊗ Fν ⊗ S defined as E = PRV(Eγ ⊗ Cartan(Ewν ⊗ Ewρ−ρk )),
(2)
where Cartan(Ea ⊗ Eb ) denotes the Cartan component of Ea ⊗ Eb , i.e., the unique component of Ea ⊗ Eb with highest weight a + b. The motivation for the above definition of E is the fact HD (Fν ) = 2m
Cartan(Eσν ⊗ Eσρ−ρk )
σ∈W 1
(see [1] or [5]). This fact implies that E ⊆ Ker D2 . We claim that, under a dominance assumption, E is also contained in Ker D1 ; this follows from (1) and the following lemma. Lemma 3. Let a, b, c be k-dominant weights such that a + w0 b + w0 c is k-dominant. Then PRV(PRV(Ea ⊗ Eb ) ⊗ Ec )
(4)
= PRV(PRV(Ea ⊗ Ec ) ⊗ Eb ) = PRV(Ea ⊗ Cartan(Eb ⊗ Ec )).
(5)
Proof. Since (5) is symmetric in b and c, it suffices to prove that (4) equals (5). The highest weights of (4) and (5) are both equal to a +w0 b +w0 c, so it is enough to prove that
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Ea+w0 b+w0 c has multiplicity one in Ea ⊗ Eb ⊗ Ec . This follows from the characterization of the PRV component as the unique component of the tensor product with infinitesimal character of the smallest possible norm. 2 Corollary 6. Assume that γ + w0 wν + w0 wρ + ρk is k-dominant.
(7)
Then the K-module E defined by (2) has highest weight γ +w0 wν +w0 wρ +ρk . Moreover, E appears with multiplicity one in Eγ ⊗ Ewν ⊗ Ewρ−ρk ⊂ Xλ ⊗ Fν ⊗ S, and is contained in Ker D1 ∩ Ker D2 . From now on, we assume that λ is regular and Xλ is irreducible.
(8)
Moreover, denoting by Δ(Fν ) the multiset of weights of Fν , and by Cλ the (open) integral Weyl group chamber of λ, we assume that ∀μ ∈ Δ(Fν ),
λ + μ ∈ Cλ .
(9)
(Recall that the integral Weyl group for λ consists of w ∈ W (g, h) such that λ − wλ is a sum of roots. See [7, Lemma 7.2.17].) The module Xλ defines a unique coherent family {Xλ+β }, where β ranges over the lattice of weights of finite-dimensional (g, K)-modules (see [7, Chapter 7]; note that since G is connected, we can base coherent families on the fundamental Cartan subalgebra instead of the split Cartan subgroup). By the assumption (9), for each weight μ of Fν , Xλ+μ is an irreducible (g, K)-module. Moreover, we have Xλ ⊗ Fν =
Xλ+μ .
(10)
μ∈Δ(Fν )
We thank David Vogan for communicating this fact to us [8]. We tensor (10) with the spin module S, and denote by E μ the corresponding projection of E onto Xλ+μ ⊗ S. Since by Corollary 6, E is contained in Ker D1 ∩ Ker D2 ⊆ Ker DXλ ⊗Fν , it follows that E μ ⊆ Ker DXλ+μ . We have proved Proposition 11. With the above notation, assume that the assumptions (1), (7), (8) and (9) hold. If μ ∈ Δ(Fν ) is such that E μ = 0, then the translate Xλ+μ of Xλ satisfies Ker DXλ+μ = 0. Note that since E = 0, there is at least one μ such that E μ = 0. If μ = 0 is the only such μ then the translate Xλ+μ is the same as Xλ . We could avoid this case by requiring
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that 0 is not a weight of Fν . Below we discuss other cases when one can say more about the μ satisfying E μ = 0. We now strengthen the assumption (1) by requiring PRV(Eγ ⊗ Ewρ−ρk ) ⊆ HD (Xλ ).
(12)
(Note that when Xλ is unitary, (1) is equivalent to (12).) In this case, we can use Vogan’s conjecture (see the end of Section 2 of [4]) to conclude that u(λ) = γ + w0 wρ + 2ρk for some u ∈ W 1 . Here we are assuming that γ + w0 wρ + ρk is k-dominant; this follows if we assume (7). We now replace λ by u−1 (λ); this does not affect the assumptions (8) and (9). So we have λ = γ + w0 wρ + 2ρk .
(13)
Suppose now that E μ is not only contained in Ker DXλ+μ but 0 = E μ ⊆ HD (Xλ+μ ).
(14)
Note that the K-type E μ is isomorphic to E, so its highest weight is given by Corollary 6. Therefore we can use Vogan’s conjecture again to conclude that, up to Weyl group conjugacy, λ + μ equals γ + w0 wν + w0 wρ + 2ρk , which equals λ + w0 wν by (13). By assumption (9), it follows that μ = w0 wν. In particular, there is at most one μ satisfying (14). Observe that if Xλ is unitary then E μ = 0 implies μ = 0. Indeed, if E 0 = 0 then (14) would be satisfied for μ = 0, so it would follow that λ + w0 wν is conjugate to λ, which is impossible. The above discussion implies the following result, which can be regarded as a substitute for Theorem 1.3 in [4]. Theorem 15. Let Xλ be an irreducible unitary (g, K)-module with regular infinitesimal character λ. Assume that Xλ contains a K-type with highest weight γ = λ − w0 wρ − 2ρk , such that (12) holds. Let Fν be the irreducible finite-dimensional (g, K)-module with highest weight ν ∈ t . Assume that (7) and (9) hold. Then (a) If (14) holds for some μ ∈ Δ(Fν ), then μ = w0 wν. In particular, Xλ+w0 wν has nonzero Dirac cohomology. (b) If all Xλ+μ , μ ∈ Δ(Fν ), are unitary, then (14) holds for some μ ∈ Δ(Fν ) and hence Xλ+w0 wν has nonzero Dirac cohomology.
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Note that the role of Lemma 3.2 in [4] is played by Corollary 6 together with projecting the K-type E onto the K-types E μ . The condition (3.3) of Lemma 3.2 is no longer needed, but our present argument requires the dominance conditions described above. Note also that the equation (5.3) in Proposition 5.2 of [4] can be replaced by E ⊆ β(Ker(DXλ ) ⊗ Ker(DFν ) ⊗ S ) which follows from Corollary 6 and the fact Ker D1 ∩ Ker D2 ⊆ β(Ker(DXλ ) ⊗ Ker(DFν ) ⊗ S ). (See the proof of Proposition 5.2 and Lemma 5.6 of [4].) References [1] J.-S. Huang, Y.-F. Kang, P. Pandžić, Dirac cohomology of some Harish-Chandra modules, Transform. Groups 14 (2009) 163–173. [2] J.-S. Huang, P. Pandžić, Dirac cohomology, unitary representations and a proof of a conjecture of Vogan, J. Amer. Math. Soc. 15 (2002) 185–202. [3] A.W. Knapp, D.A. Vogan, Cohomological Induction and Unitary Representations, Princeton University Press, Princeton, NJ, 1995. [4] S. Mehdi, P. Pandžić, Dirac cohomology and translation functors, J. Algebra 375 (2013) 328–336. [5] S. Mehdi, R. Zierau, The Dirac cohomology of a finite dimensional representation, Proc. Amer. Math. Soc. 142 (5) (2014) 1507–1512. [6] K.R. Parthasarathy, R. Ranga Rao, V.S. Varadarajan, Representations of complex semi-simple Lie groups and Lie algebras, Ann. of Math. (2) 85 (1967) 383–429. [7] D.A. Vogan, Representations of Real Reductive Groups, Birkhäuser, Boston, 1981. [8] D.A. Vogan, personal communication, 2013–2014.