Dirac operators and Lie algebra cohomology

Dirac cohomology is a new tool to study unitary and admissible representations of semisimple Lie groups. It was introduced by Vogan and further studied by Kostant and ourselves \cite{V2}, \cite{HP1}, \cite{Kdircoh}. The aim of this paper is to study the Dirac cohomology for the Kostant cubic Dirac operator and its relation to Lie algebra cohomology. We show that the Dirac cohomology coincides with the corresponding nilpotent Lie algebra cohomology in many cases, while in general it has better algebraic behavior and it is more accessible for calculation.


Introduction
In 1928 Dirac discovered a matrix valued first-order differential operator as a square root of the Laplacian operator in order to understand elementary particles. Since then this operator and its various analogues are called Dirac operators in the scientific community. They have played an important role in physics as well as in mathematics; as an example, let us mention the famous Atiyah-Singer Index Theorem. Applications to representation theory of reductive Lie groups were started by Parthasarathy [P], who defined a Dirac operator on the symmetric space G/K and used it for geometric realization of most discrete series representations. This approach was further developed by Atiyah and Schmid [AS], who constructed all discrete series representations as kernels of the Dirac operator acting on the associated spinor bundles.
On the other hand, Lie algebra cohomology (with respect to nilpotent subalgebras) has been a fundamental invariant of representations of Lie algebras and groups starting from Kostant's 1962 paper [K1], which gives an algebraic proof of the Bott-Borel-Weil theorem. Lie algebra cohomology is related to characters, asymptotics of matrix coefficients, embeddings into various "standard" modules and various geometric realizations of representations.
It has been clear for decades that the Dirac operators are formally similar to the differentials of the de Rham or Dolbeault cohomology. There have been two problems with using that analogy in representation theory. First, the Dirac operators on non-symmetric homogeneous spaces are not as nicely behaved. Second, the index of the Dirac operator behaves well only for some unitary representations, like the discrete series; so the algebraic tools of representation theory do not work well with it. The first problem was resolved by Kostant [K2], who introduced the modified cubic Dirac operator that works well also on non-symmetric homogeneous spaces. The second problem was resolved by Vogan [V2], who introduced the concept of Dirac cohomology which applies both to unitary and nonunitary representations: it is the kernel of the Dirac operator D modulo the intersection of the image and the kernel. He made a conjecture on the infinitesimal character of irreducible representations with nonzero Dirac cohomology, which was proved in [HP1]. Kostant generalized this result to the case of his cubic Dirac operator and applied it to the topology of homogeneous spaces [K4]. Recently, Kumar [Ku] and Alekseev and Meinreken [AM] found further generalizations of the results in [HP1] and [K4]. They put these results into the broader setting of non-commutative equivariant cohomology.
The aim of this paper is to explore the relation between Dirac cohomology and (nilpotent) Lie algebra cohomology, in those situations when both can be defined. Our results show that Dirac cohomology coincides with the corresponding Lie algebra cohomology for a large family of unitary representations, including the discrete series representations, and for all unitary representations in certain special cases. On the other hand, Dirac cohomology seems to have a better algebraic behavior.
We have also been able to use Dirac cohomology and the proved Vogan's conjecture to obtain improvements of some classical results. In [HP2] we describe how to simplify certain parts of Atiyah-Schmid's construction of discrete series representations [AS] and sharpen the Langlands' formula on automorphic forms [L], [HoP]. The details will appear in a forthcoming book by the first and second named authors. This suggests that Dirac cohomology may be useful also in other situations, as a new tool for tackling problems not accessible to classical cohomology theories.
Let us now describe our main results on Dirac cohomology and its relation to (nilpotent) Lie algebra cohomology more precisely. Let G be a connected semisimple Lie group with finite center and complexified Lie algebra g. Let θ be a Cartan involution of G (and g), let K = G θ be the corresponding maximal compact subgroup, and let g = k ⊕ p be the complexified Cartan decomposition. As usual, the corresponding real forms will be denoted by g 0 , k 0 , etc. (One could also work in a more general setting of a reductive group G in the Harish-Chandra class.) If G is of hermitian type, then p decomposes as a sum of two abelian subalgebras, p + and p − . We prove a Hodge decomposition for p − -cohomology and p + -homology for all unitary (g, K)-modules, and show that they are both isomorphic to the Dirac cohomology up to a twist by a modular character. We note that in this setting Enright [E] found an explicit formula for the p − -cohomology of unitary highest weight modules.
For general g, and any two reductive subalgebras r 1 ⊂ r 2 ⊂ g to which the Killing form restricts nondegenerately, we show that Kostant's cubic Dirac operator D(g, r 1 ) can be decomposed as a sum of two anti-commuting Dirac operators D(g, r 2 ) + D ∆ (r 2 , r 1 ). Here ∆ denotes a certain "diagonal embedding". Decompositions of this kind also appear in [AM].
In particular, we consider the case when r ⊂ k is a reductive subalgebra, which is real, i.e., r is the complexification of r 0 = r ∩ g 0 . Then we can show that for any admissible (g, K)-module, the Dirac cohomology with respect to D(g, r) is the same as the kernel of D ∆ (k, r) on the Dirac cohomology with respect to D(g, k). Furthermore, the Dirac cohomology with respect to D(g, r) is also the same as the Dirac cohomology with respect to D(g, k) of the kernel of D ∆ (k, r).
In particular, if g and k have equal rank, then r can be a Levi subalgebra l of a θ-stable parabolic subalgebra q = l ⊕ u of g, with l ⊂ k. It follows that the Dirac cohomology with respect to D(g, l), which is closely related to theū-cohomology, has the advantage of being much easier to calculate. For instance, let t be a compact Cartan subalgebra, and t ⊕ n be a Borel subalgebra. Then we show how to (easily) explicitly calculate the Dirac cohomology of the discrete series representations with respect to the Dirac operator D(g, t). Comparing the obtained result with Schmid's formula forn-cohomology in [S], we see that they are the same up to an expected modular twist. In case G is of hermitian type, the Dirac cohomology with respect to D(g, l) coincides with theū-cohomology or u-homology (up to a modular twist) for all unitary representations.
We now describe the organization of our paper. In Sections 2 and 3 we decompose the Kostant cubic Dirac operator as sum of two "half Dirac operators", which correspond to the differentials ofū-cohomology and u-homology. Section 3 also contains a new proof of the Casselman-Osborne theorem on Lie algebra cohomology using an approach similar to that of [HP1]. This is aimed at explaining the formal similarity of the two results. In Section 4 we prove a Hodge decomposition for p − -cohomology or p + -homology for unitary representations in the Hermitian case, with the Dirac cohomology providing the "harmonic representatives" of both. The same proof applies to any finite-dimensional representation of an arbitrary semisimple group. In Section 5 we define the relative Dirac operators and show that in some cases Dirac cohomology can be calculated in stages. In Section 6 we develop this further and show how to use it to calculate Dirac cohomology explicitly. In particular, we compare the Dirac cohomology for a Levi subalgebra with theū-cohomology in some cases. In Section 7 we obtain a Hodge decomposition and equality of Dirac andū-cohomology for arbitrary unitary modules in the Hermitian case. We conclude the paper by showing that the homological properties of Dirac cohomology are quite different from those of Lie algebra cohomology. In fact, under certain conditions, we show that there is a six-term exact sequence of Dirac cohomology corresponding to a short exact sequence of (g, K)-modules. So Dirac cohomology resembles a K-theory, rather than a cohomology theory. This work was initiated by David Vogan [V3]. We would like to thank him for many interesting and stimulating conversations. We believe that the results in this paper are not the end of the theory of Dirac cohomology in representation theory, but rather the beginning of further investigations and applications. For example, the results of this paper should be related to the results of Connes and Moscovici [CM] in a similar way as the results of [HP1] are related to [AS].

Construction of certain Dirac operators
Let g be a complex semisimple Lie algebra, q = l ⊕ u a parabolic subalgebra,q = l ⊕ū the opposite parabolic subalgebra, and s = u ⊕ū. Then g = l ⊕ s.
Furthermore, the restrictions of the Killing form B to l and s are non-degenerate, and the above decomposition is orthogonal. Since u andū are isotropic subspaces in perfect duality under B, we can identifyū with u * ; this identification is l-invariant. Let u 1 , . . . , u n be a basis of u, and let u * 1 , . . . , u * n be the dual basis ofū. Let C(s) be the Clifford algebra of s. Unlike in [HP1], we will use the same defining relations as Kostant, namely vw + wv = 2B(v, w); in particular, if B(v, v) = 1, then v 2 = 1, and not −1 like in [HP1]. Of course, over C there is no substantial difference between the two conventions.
We are going to make use of the well known principle of constructing invariants by contracting dual indices. The aim is to construct a family of interesting l-invariants in U (g) ⊗ C(s). These will include Kostant's cubic Dirac operator D, but we will get D as a sum of four members of the family, and we will also be able to combine them in different ways, to get other operators with properties similar to the properties of D. For example, we will have nice expressions for their squares. The form of this principle we need is the statement of the following lemma; the proof is quite easy and essentially reduces to the fact that under the identification Hom(u, u) ∼ = u * ⊗ u, the identity map corresponds to the sum i u * i ⊗ u i . Lemma 2.1. Let ψ : s ⊗2k → U (g) ⊗ C(s) be a linear map which is l-equivariant with respect to the adjoint actions. Then is independent of the chosen basis u i and l-invariant. Here I = (i 1 , . . . , i k ) ranges over all k-tuples of integers in {1, . . . , n}, u I = u i 1 ⊗ · · · ⊗ u i k , and u * For example, ψ can be composed of the obvious inclusions s ֒→ g ֒→ U (g) and s ֒→ C(s), products, commutators in g and the Killing form B(., .). Here are several examples of this kind which we will study in the following: Note the symmetry obtained by exchanging the roles of u andū. To see how the Dirac operator fits in here, note that one can build an orthonormal basis (Z i ) of s from u i and u * i , by putting for j = 1, . . . , n. Then it is easy to check that Also, we can rewrite a, a − and e as follows: Also, the sum in the definition of a is clearly twice the same sum over only those i, j for which i < j. The only issue is thus to pass from the Clifford product to the wedge product. For this, we use (1.6) in [K2]. First, since u is isotropic, The second two terms here are clearly zero if k is different from i and j. For k = i, the second term is u j while the third is zero. For k = j, the second term is zero while the third is −u i . It follows that we will be done if we can show However, using Lemma 2.1, we see that Since there are no nonzero l-invariants in u, this sum must be 0. Analogously, So we proved the first equality for a. Now in the form with wedge product, we can clearly commute u * k in front of u i and u j , and then we obtain the second equality by reversing the above argument.
The formulas for e are obvious from the defining relations of C(s).
Consider now the basis (b j ) j=1,...,2n of s, given by b 1 = u 1 , . . . , b n = u n , b n+1 = u * 1 , . . . , b 2n = u * n ; the dual basis is then In particular, we obtain Kostant's cubic Dirac operator as We are also particularly interested in the elements Note that D = C +C − . We will use the fact that commuting with E operates on C, C − , D and D − in the following way: Proposition 2.4. Proof. The second two relations follow from the first two, and the first two are immediate from the following lemma.
Lemma 2.5. Commuting with e in the Clifford algebra C(s) acts as I on u and as −I onū.
Proof. Clearly, for j = i, [u * j u j , u i ] = 0. For j = i, we calculate Namely, since u is isotropic, u 2 = 0 for any u ∈ u. The first claim now follows, and the second is analogous. Kostant [K2], Theorem 2.16, has calculated Here Ω g denotes the Casimir element of Z(g) ⊂ U (g). Further, Ω l ∆ is the Casimir element for the diagonal copy l ∆ of l, embedded into U (g) ⊗ C(s) via where α : l → so(s) → C(s) is the action map followed by the standard inclusion of so(s) into C(s) using the identification so(s) ∼ = 2 s. Finally, C is the constant ||ρ|| 2 − ||ρ l || 2 .
Using this result and the above remarks, we can now quickly calculate the squares of C, C − and D − : Proof. From Kostant's expression for D 2 , it is clear that D 2 commutes with all l-invariant elements of U (g) ⊗ C(s). In particular, D 2 commutes with E, and using Proposition 2.4 we see Since D + D − = 2C and D − D − = 2C − , it follows that On the other hand, Proposition 2.4 implies that So we see that C 2 = (C − ) 2 = 0, and also D 2 + (D − ) 2 = 4C 2 = 0. Now we can completely describe the Lie superalgebra D spanned by E, C, C − and Ω = D 2 inside the superalgebra U (g) ⊗ C(s). Here U (g) ⊗ C(s) is an associative superalgebra with Z 2 -grading of the Clifford factor; so it is also a Lie superalgebra in the usual way, with the supercommutator [a, b] D is a subalgebra of the Lie superalgebra U (g) ⊗ C(s), with the commutation relations from Propositions 2.4 and 2.6. Namely, Ω is central, and [E, C] The last relation is obtained as follows: Note that we can regard D as a Z-graded Lie superalgebra with, C − , E, Ω and C of degrees −1, 0, 0 and 1 respectively. Note also that the subalgebra of D spanned by C, C − and Ω is a Heisenberg superalgebra.
The Lie superalgebra D can be identified with gl(1, 1), i.e., the endomorphisms of the superspace C ⊕ C, with the first C even and the second C odd. In Kac's classification [Kac] it is denoted by l(1, 1). It was extensively used by physicists under the name supersymmetric algebra. It is a completely solvable Lie superalgebra, and its irreducible finite-dimensional representations are described in [Kac].
To finish this section, let us note that D is independent not only of the choice of basis (u i ) but also of the choice of u ⊂ s. On the other hand, E, C and C − do depend on the choice of u.

3.ū-cohomology and u-homology
We retain the notation from previous sections. Let V be an admissible (g, K)-module.
be the set of p-cochains of the complex defining theū-cohomology of V . The differential d : We have the following identifications : Like in the previous section, we fix a basis (u i ) i=1,...,n of u and denote the dual basis of u by (u * i ) i=1,...,n .
Lemma 3.1. Through the above identifications, the differential d : Proof. This is a straigtforward calculation, starting from the fact that the identification The space V ⊗ p u is also the space of p-chains for the u-homology of V , with differential ∂ : V ⊗ p u → V ⊗ p−1 u given by : Note that we are tensoring with · u from the right and not from the left as usual; this is because we will have an action of U (g) ⊗ C(s) on V ⊗ · u which will be more natural in this order.
For reasons that will become apparent later, we will instead of ∂ consider the operator δ = −2∂. Of course, δ defines the same homology as ∂.
To get our Dirac operators act, we need to consider the U (g) ⊗ C(s) -module V ⊗ S, where S is the spin module for the Clifford algebra C(s). We will use the identification of · u with S, given explicitly in [K4] and [K3]. Namely, one can construct S as the left ideal in C(s) generated by the element u * top = u * 1 . . . u * n . One then has S = ( · u)u * top , which is isomorphic to · u as a vector space, and the action of C(s) is given by left Clifford multiplication. Explicitly, u ∈ u and u * ∈ū Namely, since u andū are isotropic, the Clifford and wedge products coincide on each of them; in particular, u * u * top = 0. The natural action of l on V ⊗ S is the tensor product of the restriction of the g-action on V and the spin action on S. On the other hand, the usual l action onū-cohomology and u-homology is given by the adjoint action on ·ū and · u. Thus, our identification of · u ⊗ V with V ⊗ S is not an l-isomorphism. However, as was proved in [K3], Prop.3.6, the two actions differ only by a twist with the one dimensional l-module Z ρ(ū) of weight ρ(ū).
This means that, if we consider d and δ as operators on V ⊗ S via the above identification, then as an l-module, the cohomology of d gets identified with H · (ū, V ) ⊗ Z ρ(ū) , while the homology of δ gets identified with H · (u, V ) ⊗ Z ρ(ū) .
Section 2 act as d and δ respectively. In particular, the cubic Proof. We are going to use the explicit formulas for the action of C(s) on S given above. From these formulas, it is immediate that the action of A coincides with the first (single) sum in the expression for d, while the action of we see that this is equal to minus twice the first (single) sum in the expression for the u-homology operator ∂.
It remains to identify the action of the cubic terms a and a − . We use the expression for a from Lemma 2.3, i.e., a = − 1 For a − we use its definition from 2.2. To write the calculation nicely, we introduce the following notation: If k > l then we change the sign and definê If k = l, we setŶ k,l = 0. This now allows us to write This is now clearly invariant for exchanging the roles of k and l, hence it is twice the same sum extending just over k < l, i.e., minus twice the second (double) sum in the expression for ∂.
It is now clear why we considered δ = −2∂ instead of just ∂; in this way we have the action of D being equal to d + δ.
Before we go on, let us note how the element E of Section 2 acts on V ⊗ S; it is in fact a degree operator up to a shift. This means E can be used to identify the degrees in which the cohomology (homology) is appearing.
Proposition 3.3. The element E of U (g) ⊗ C(s) from 2.2 acts on V ⊗ k u as multiplication by the scalar k − n. Consequently, it preserves the kernel and image of both d and δ, and hence acts on the k-th cohomology of d and the k-th homology of δ, by the same scalar k − n. In particular, E + n is the degree operator.
Proof. Using Lemma 2.5 we see that for any u ∈ u, eu = ue + u. On the other hand, by It now immediately follows that E preserves the kernel and image of d and δ, as these are homogeneous operators (of degree 1 and −1 respectively). (Note that this last assertion can also be obtained from the commutation relations of Proposition 2.4.) We will now state a result for the operators C and C − analogous to the one obtained for D in [HP1] and [K4]. A corollary will be the Casselman-Osborne Theorem. Our goal here is not to give a new proof of the Casselman-Osborne Theorem, the existing ones being completely satisfactory, but to shed some light on the formal similarity between the Casselman-Osborne Theorem and the main result of [HP1].
where ǫ x is 1 for even x and −1 for odd x. In the following we fix a compact group L with complexified Lie algebra l.
Proof. The result for d D is due to Kostant [K4]. The proof is the same as the proof of the main result of [HP1]. We give details for d C , the proof for d C − being entirely similar. As in [HP1], we use the standard filtration on U (g), which induces a filtration (F n A) n∈N on A := U (g) ⊗ C(s). This filtration being L-invariant, it induces in turn a filtration on A L . Clearly, The Z 2 -gradation on the Clifford algebra C(s) induces a Z 2 -gradation on A. We set We can assume i r are different from each other and likewise for j s .
If j = j s for all s, then the contribution to the sum is zero. If j = j s , then Since g = l ⊕ u ⊕ū and C(s) ≃ · s = · u ⊗ ·ū , one has It follows thatd C is (up to a sign depending on the · u-degree) equal to −2 Id ⊗dū, where dū is the Koszul differential for the vector spaceū. It is well known that dū is exact except at degree zero, where the cohomology is C, embedded as the constants. It follows thatd C is exact except at degree zero, where the cohomology is S(l) ⊗ S(u) ⊗ · u embedded in the obvious way. It remains to pass to the invariants: Lemma 3.5. The differentiald C on (S(g) ⊗ C(s)) L is exact except at degree zero. The zeroth cohomology is (S(l) ⊗ S(u) ⊗ · u) L = S(l) L ⊗ 1, embedded in the obvious way. More precisely To prove the lemma, we need to show that (S(l) ⊗ S(u) ⊗ · u) L = S(l) L ⊗ 1 ⊗ 1. To see this, we may choose some element h in the center of l, such that l is the centralizer of h in g, ad h has real eigenvalues, and u is the sum of the eigenspaces of ad h corresponding to the positive eigenvalues. Making h act on an element in (S(l) ⊗ S(u) ⊗ · u) L , we see that this element has to be in S(l) L ⊗ 1 ⊗ 1.
We can now finish the proof of the theorem. We proceed as in [HP1] by induction on the degree of the filtration.
Our goal is now to compute η l (z). Let h be a Cartan subalgebra of l, and let us denote respectively by W l and W g the Weyl groups of h in l and g.
We have Harish-Chandra isomophisms and an obvious inclusion i : With this notation we have: Lemma 3.6. For all z ∈ Z(g), η l (z) = µ g/l (z).
Proof. The proof is similar to the proof of Theorem 4.2 in [K4], but much simpler. We give only a sketch. Let b be a Borel subalgebra of g containingū, and suppose the Cartan algebra h has been chosen to lie in b.
Let V λ be the irreducible finite dimensional representation with highest weight λ (relative to b), and let v λ be a non-zero highest weight vector in V λ . Recall the element u * top in C(s) used to define the spin module S.
Since this is true for all dominant weights λ, and since these form a Zariski dense set in h * , we conclude that indeed µ g/h (z) = µ l/h (η l (z)).
Let V be a representation of g admitting an infinitesimal character χ V and let z ∈ Z(g). Then z acts on H · (ū, V ) by the scalar χ V (z).
Notice that z ⊗ 1 acts on Ker C/ Im C as η l (z). Namely, Ca + aC leaves Ker C and Im C stable, and induces the zero action on Ker C/ Im C.
Thus, the induced action of η l (z) on Ker is equal to the scalar multiplication by χ V (z). This is exactly the statement of the Casselman-Osborne Theorem (see [CO], or [V1], Theorem 3.1.5). Namely, our definition of µ g/l = η l differs from the map Z(g) → Z(l) from the Casselman-Osborne Theorem exactly by the above ρ-shift.

Hodge decomposition for p − -cohomology
In this section we study positive definite hermitian forms on V ⊗ S for unitary modules V , such that d and δ are minus adjoint to each other. Here unitarity is defined with respect to a fixed real form g 0 of g; we also fix a corresponding Cartan involution θ and Cartan decompositions g 0 = k 0 ⊕ p 0 and g = k ⊕ p.
Such a form however exists only in very special situations; if g is simple, then the pair (g, k) must be hermitian symmetric, l must be equal to k, and u andū must be the abelian subalgebras p ± of p. In that case, we get a Hodge type decomposition: p + -homology, p −cohomology and the Dirac cohomology are all represented by the space of "harmonics", Ker D = Ker D 2 .
A parallel result holds much more generally if V is a finite-dimensional module. Since this case is known, we include it only as a remark. However we start by studying a sufficiently general situation which applies to both cases.
Let r be a subalgebra of g to which B restricts non-degenerately and let s be the orthogonal complement to r with respect to B. We choose a maximal isotropic subspace s + of s. Since X, Y = −2B(X, θȲ ) (with¯denoting conjugation with respect to g 0 ) defines a positive definite Hermitian form on g and hence also on s + , the subspace s − = θs + intersects s + trivially. Let S = · s + be a spin module for the Clifford algebra C(s) corresponding to this polarization. We extend the form , to all of S in the usual way, using the determinant. Proof. We need to show that X · λ, µ = λ, −θX · µ for any X ∈ s and λ, µ ∈ S. Assume X ∈ s + . We can assume that λ = λ 1 ∧ · · · ∧ λ s and µ = µ 1 ∧ · · · ∧ µ t , with X, λ i and µ j all belonging to a fixed orthonormal basis of s + . Then both sides of the equality to be established are nonzero precisely when X is one of the µ j 's, while the other µ j 's are precisely (all) the λ i 's. In this case both sides are clearly the same. If X ∈ s − the claim follows by considering θX ∈ s + . In case s is odd-dimensional, the extra element is handled easily.
We now assume that r is real and θ-stable, i.e., r is the complexification of a subalgebra r 0 ⊂ g 0 with r 0 = r 0 ∩ k 0 ⊕ r 0 ∩ p 0 .
(the other terms are zero). Since the adjoint of , and the coefficients are real, we see that v is self-adjoint. (b) By the formulas from Lemma 2.3, it is enough to show that u i u j [u * i , u * j ] is adjoint to [u i , u j ]u * i u * j . This however follows immediately from Lemma 4.1: we can normalize the u i 's so that u Let now V be a unitary (g, K)-module and let q = l ⊕ u be a θ-stable parabolic subalgebra. Then the adjoint of the operator u i on V with respect to the unitary form is −ū i , while the adjoint of u i on S is either −ū i , orū i , depending on whether u i is in k or in p. If we consider the tensor product hermitian form on V ⊗ S, denoted again by , , and as before denote by A k and A p the k -respectively p -parts of A, then we see: We will use this corollary in Sections 5 and 7. Now we turn our attention to the case when A k and 1 ⊗ a do not appear in C. This is the already mentioned case, when l = k, and u is contained in p. Then u is forced to be abelian and we denote as usual u = p + , u = p − . In this case, the Dirac operator D = D(g, l) is the "ordinary" Dirac operator corresponding to k, there is no cubic part, and we conclude is anti-self-adjoint.
More generally, if l contains k (this can happen when g is not simple), then D(g, l) is anti-self-adjoint with respect to , .
For the rest of this section we assume that l = k. So D is anti-self-adjoint with respect to the positive definite form , on V ⊗ S. In particular, the operators D and D 2 have the same kernel on V ⊗ S.
By [P], Proposition 3.2 (or more generally, [K2], Theorem 2.16), we know that where Ω g and Ω k ∆ are the Casimir operators for g and diagonally embedded k, and C is the constant ||ρ g || 2 − ||ρ k || 2 . It follows that if Ω g acts on V by a constant, then Ω k ∆ is up to a constant equal to D 2 on V ⊗ S. Since Ω k ∆ acts by a scalar on eachK-type in V ⊗ S, the same is true for D 2 . So D 2 is a semisimple operator, i.e., V ⊗ S is a direct sum of eigenspaces for D 2 . In particular: Proof. We have seen that V ⊗ S is a direct sum of eigenspaces for D 2 . Clearly, the zero eigenspace is Ker D 2 , and the sum of the nonzero eigenspaces is Im D 2 .
It is now easy to obtain a variant of the usual Hodge decomposition. The following arguments are well known; see e.g. [W], Scholium 9.4.4. We first have In particular, the Dirac cohomology of V is equal to p − -cohomology and to p + -homology, up to modular twists: More precisely, (up to modular twists) the Dirac cohomology Ker D is the space of harmonic representatives for both p − -cohomology and p + -homology.
Remark 4.8. There is a variant of the above results for a finite dimensional module V . In this case one puts a positive definite form on V invariant under the compact form k 0 ⊕ ip 0 of g; this is sometimes called an admissible form. In this way the adjoint of u i on V will beū i if u i ∈ p and −ū i if u i ∈ k. Hence the adjoint of u i on V is u * i for all i. So this form combines with , on S better than a unitary form. In particular, d and δ are now adjoint to each other with respect to , , D is self-adjoint, and the above arguments, including Theorem 4.7, apply without change. There is no need here to assume that l = k; q = l ⊕ u can be any parabolic subalgebra of (any) g, with the assumption that l is θ-stable and real.
This case was however already known; it is implicit in [K3] and it is explicitly mentioned in [V3].
Moreover, we can generalize this further and prove self-adjointness of D for the more general setting when r ⊂ g is not necessarily a Levi subalgebra of a parabolic subalgebra, but any real and θ-stable reductive subalgebra (to which B then restricts nondegenerately). Namely, getting back to the setting of Lemma 4.2 (a) and its proof, we see that not only the cubic term v but also the linear term

Remark 4.9.
Here is an example which shows that a Hodge decomposition like in Theorem 4.7 does not hold generally if l is not compact, even for unitary modules in the Hermitian symmetric case. Let W 0 be the (g, K)-module of the even Weil representation of Sp(4, R), and let l be the gl(2) containing the compact Cartan subalgebra and corresponding to the short noncompact root. Then l is a Levi factor of a θ-stable parabolic subalgebra with abelian u. One can explicitly calculate everything and show that the Dirac cohomology is strictly larger than u-homology orū-cohomology (while the latter two are still equal).

Relative Dirac operators
In this section we compare various Dirac operators arising from two compatible decompositions g = r ⊕ s = r ′ ⊕ s ′ . Here both decompositions are like in the Kostant's setting for the cubic Dirac operator: r and r ′ are reductive subalgebras of g such that the Killing form B restricts to a nondegenerate form on each of them, while s respectively s ′ are the orthogonals of r respectively r ′ . Compatibility of the two decompositions means Clearly, all these decompositions are orthogonal.
Example 5.1. If l is a θ-stable Levi subalgebra of g, then r = l and r ′ = k satisfy the above conditions. Now r ∩ r ′ is also a reductive subalgebra of g to which B restricts nondegenerately. The corresponding decomposition is The corresponding Dirac operator will be denoted by D(g, r∩r ′ ). We note that this setting includes the case in which r ′ is a subalgebra of r, and therefore the setting r 1 ⊂ r 2 ⊂ g from the introduction.
To write down D(g, r ∩ r ′ ), take orthonormal bases Z i for s, and Z ′ j for r ∩ s ′ . Identify where⊗ denotes the Z 2 -graded tensor product. Now if W k is the union of the bases Z i and Z ′ j , then by Kostant's definition, D(g, r ∩ r ′ ) is the element Kostant's original definition uses exterior multiplication instead of Clifford multiplication in the second sum (with the Clifford and exterior algebras identified as vector spaces via Chevalley identification). For orthogonal vectors, there is however no difference between exterior and Clifford multiplication, so the above definition is the same as Kostant's. Taking into account the decomposition (1), we see Note that the terms with Z i , Z ′ j and Z ′ k do not appear, because B( ∈ r is orthogonal to s. We consider U (g) ⊗ C(s) as the subalgebra U (g) ⊗ C(s) ⊗ 1 of U (g) ⊗ C(s)⊗C(r ∩ s ′ ). In view of this, we see that the first and third sum in (2) combine to give D(g, r), the Kostant's cubic Dirac operator corresponding to r ⊂ g.
The remaining three sums come from the cubic Dirac operator corresponding to r ∩ r ′ ⊂ r. However, this is an element of the algebra U (r) ⊗ C(r ∩ s ′ ), and this algebra has to be embedded into U (g) ⊗ C(s)⊗C(r ∩ s ′ ) diagonally, not as U (r) ⊗ 1 ⊗ C(r ∩ s ′ ). Namely, we use the diagonal embedding U (r) ∼ = U (r ∆ ) ⊂ U (g) ⊗ C(s) from the setting g = r ⊕ s; the definition is the same as for r = l above Proposition 2.6. Thus we embed We will denote ∆(D(r, r ∩ r ′ )) by D ∆ (r, r ∩ r ′ ) and call it a relative Dirac operator. In case when there are several subalgebras and confusion might arise, we will use more precise notation ∆ g,r instead of ∆ and give up the notation D ∆ (.). This diagonal embedding has already been used by Kostant and Alekseev-Meinrenken; in particular, decompositions like the following one can be founded in [AM].
Proof. To prove (i), we need to describe the image under ∆ of and see that it matches the second, fourth and fifth sum in (2). In fact, it is obvious that the image under ∆ of the second sum in (3) equals the fifth sum in (2), and it remains to identify is the action map of r on s, followed by the standard inclusion of so(s) into C(s), given by where E ij are the matrix units relative to the basis Z i . Thus we are left with showing that the second sum in (4) equals the third sum in (2), i.e., that This will follow if we see that for any k. But this last equality is clear from the definition of α. Namely, the matrix of ad Z ′ k on s in the basis . To prove (ii), we use the fact that D(g, r) commutes with r ∆ , which is one of the most basic properties of D(g, r). It follows that the anticommutator It remains to see that also This follows from the definition of⊗, since all the C(s)-parts of the monomial terms of D(g, r), and also all Z ′ i Z ′ j Z ′ k ∈ C(r ∩ s ′ ), are odd. Example 5.3. In the setting of Example 5.1, we obtain D(g, l ∩ k) = D(g, l) + D ∆ (l, l ∩ k) = D(g, k) + D ∆ (k, l ∩ k), and both decompositions have anticommuting summands. If l is contained in k, which is possible if and only if g and k have equal ranks, then we get only one nontrivial decomposition, D(g, l) = D(g, k) + D ∆ (k, l). This case will be of special interest below.
We now want to use Theorem 5.2 to relate the Dirac cohomology of the various Dirac operators involved. In some cases one can apply the following easy fact from linear algebra. We define the cohomology of any linear operator T on a vector space V to be the vector space H(T ) = Ker T /(Im T ∩ Ker T ).
Lemma 5.4. Let A and B be anticommuting linear operators on a vector space V . Assume that A 2 diagonalizes on V , i.e. V = λ V λ , with A 2 = λ on V λ . Then the cohomology H(A + B) of A + B on V is the same as the cohomology of the restriction of A + B to V 0 = Ker A 2 .
Proof. Since A + B commutes with A 2 , its kernel, image and cohomology decompose accordingly to eigenspaces V λ . We thus have to prove that A + B has no cohomology on V λ for λ = 0. In other words, we are to prove that Ker ( Let v ∈ V λ be such that (A + B)v = 0, i.e., Av = −Bv. Then To apply this to Dirac cohomology, denote by H D (g, r; V ) the Dirac cohomology of a (g, K)-module V with respect to D(g, r); analogous notation will be used for other Dirac operators. The reader should bear in mind that H D (g, r; V ) is in fact the cohomology of the operator D(g, r) on the space V ⊗ S.
Corollary 5.6. Let r ⊂ k be a reductive subalgebra of g, with B| r×r nondegenerate. As usual, let s be the orthocomplement of r. Assume that either dim p is even, or dim s ∩ k is even 1 . Let V be an irreducible unitary (g, K)-module. Then H D (g, r; V ) = H D (k, r; H D (g, k; V )), i.e., the Dirac cohomology can be calculated "in stages", as the D(k, r)-cohomology of the D(g, k)-cohomology.
Proof. Since V is unitary, we can consider the form , on V ⊗ S p introduced in Section 4, where S p is the spin module for C(p). We can extend this form to all of V ⊗ S, by choosing any positive definite form on the spin module S s∩k for C(s ∩ k). Here we identify S = S p ⊗ S s∩k , which can be done by the assumption on dimensions. Let A = D(g, k) and B = D ∆ (k, r).
By Corollary 4.3, A is anti-self-adjoint, and consequently the conditions of Corollary 5.5 are satisfied. So the cohomology with respect to D(g, r) is the cohomology with respect to B of Ker with Lie algebra k acting through k ∆ . The Dirac cohomology of this module with respect to D(k, r) is thus identified with the cohomology with respect to B = D ∆ (k, r).

The case of compact Levi subalgebra
In this section we first consider a reductive subalgebra r 0 of g 0 contained in k 0 . Later on we will specialize to the case when r = l is a Levi subalgebra of a θ-stable parabolic subalgebra of g.
The first thing we will do in this situation is generalize Corollary 5.6 to nonunitary modules. Like there, we assume for simplicity that either dim p is even or dim s ∩ k is even, so that we can write the spin module as S = S p ⊗ S s∩k . The idea is to reverse the roles of D(g, k) and D ∆ (k, r). Namely, for any admissible (g, K)-module V , we can decompose V ⊗ S p into a direct sum of finite dimensional (unitary) modules for the spin double coverK of K. Hence, by Remark 4.8, there is a positive definite form , on It follows that B = D ∆ (k, r) is a semisimple operator, while for A = D(g, k) we still have that A 2 is semisimple. In this situation, we have the following lemma which complements Corollary 5.5: Lemma 6.1. Let A and B be anticommuting linear operators on a vector space V , such that A 2 and B are semisimple (i.e., can be diagonalized). Then H(A+B) is the cohomology (i.e., the kernel) of B acting on H(A).
Proof. Applying Lemma 5.4, we can replace V by Ker A 2 , i.e., assume A 2 = 0. On the other hand, by Corollary 5.5, H(A + B) is the cohomology of A acting on Ker B.
Since B is semisimple, we can decompose  Ker A, and assume that Bv ∈ Im A. This implies λv 1 − λv 2 is in Im A, so v 1 − v 2 ∈ Im A. This however can only happen if both v 1 and v 2 are in Im A, again because A exchanges V λ and V −λ . But then also v = v 1 + v 2 is in Im A, so v is zero in cohomology and we are done.
This now immediately implies the following theorem which says that Dirac cohomology with respect to a subalgebra r ⊂ k as above can be calculated "in stages".
Theorem 6.2. Let r 0 be a reductive subalgebra of g 0 contained in k 0 . Let V be an admissible (g, K)-module. Then the Dirac cohomology with respect to D(g, r) can be calculated as the Dirac cohomology with respect to D(k, r) of the Dirac cohomology with respect to D(g, k) of V . In other words: Also, we can reverse the order of taking Dirac cohomology, i.e., H D (g, r; V ) = H(D(g, k) H D (k,r;V ) ).
Proof. The first formula was explained above, and the second is a direct application of Corollary 5.5, with A = D ∆ (k, r) and B = D(g, k) (opposite from Corollary 5.6).
For the rest of this section we consider a θ-stable parabolic subalgebra q = l ⊕ u of g, with the Levi subalgebra l contained in k. In particular, there is a Cartan subalgebra t of g contained in l ⊂ k; so g and k have equal rank. The opposite parabolic subalgebra is q = l ⊕ū. As before, we denote s = u ⊕ū, so g = l ⊕ s.
We apply the above considerations to r = l. Since H D (g, k; V ) is a finite dimensional K-module, and k and l have equal rank, H D (k, l; H D (g, k; V )) is given by [K4], Theorem 5.1. (This can also be read off from Remark 4.8, by using Poincaré duality to pass to u-cohomology and then the better known Kostant's formula for u-cohomology; in fact, this is how Kostant proves it.) This gives H D (g, l; V ) very explicitly provided we know H D (g, k; V ) explicitly. For example, one can in this way calculate the Dirac cohomology of the discrete series representations with respect to the (compact) Cartan subalgebra t: Example 6.3. Let V = A b (λ) be a discrete series representation; here b = t ⊕ n is a Borel subalgebra of g containing a compact Cartan subalgebra t. The infinitesimal character of V is λ + ρ. Then the Dirac cohomology of V with respect to D(g, k) consists of a singlẽ K-type V (µ), whose highest weight is µ = λ + ρ n , where ρ n = ρ(u ∩ p). This is obtained from the highest weight of the lowest K-type of V , λ + 2ρ n , by shifting by −ρ n (the lowest weight of S).
This result is contained in the work of Parthasarathy and Schmid. One can also prove it as follows: it is shown in [HP1], Proposition 5.4, that thisK-type is contained in the Dirac cohomology. Since V has a unique lowest K-type, and since −ρ n is the lowest weight of the spin module, with multiplicity one, it follows that any otherK-type has strictly larger highest weight, and thus can not contribute to the Dirac cohomology.
We now apply the above mentioned Kostant's formula (Theorem 5.1 of [K4]) to calculate the Dirac cohomology with respect to D(k, t) (we again stress that k and t have equal rank): It follows from µ + ρ k = λ + ρ that Remark 6.4. Comparing with Schmid's formula in Theorem 4.1 of [S], we have (note that Schmid's n is ourn, and his λ is our λ + ρ.) In other words, n-cohomology of a discrete series representation coincides with the Dirac cohomology up to a ρ-shift. This fact is however not covered by our results in Sections 4 and 7. This indicates that it should be possible to generalize our results; however, the next example shows they do not hold in full generality.
Example 6.5. In the setting of Remark 4.9, one can show that H D (l, t; H D (g, l; W 0 )) is strictly larger than H D (g, t; W 0 ). Here t is the compact Cartan subalgebra, and the last cohomology can be calculated in stages, as H D (k, t; H D (g, k; W 0 )).
We now want to similarly analyze the "half-Dirac" operators C and C − . Let u 1 , . . . , u k be a basis for u ∩ k and let v 1 , . . . , v p be a basis for u ∩ p. These can be taken to be the root vectors corresponding to compact, respectively noncompact positive roots, with respect to some ∆ + (g, t) compatible with u. We normalize these bases so that the dual bases for u ∩ k respectivelyū ∩ p with respect to the Killing form are u * i = −ū i respectively v * i =v i . As before, D = D(g, l), C = C(g, l) = A + 1 ⊗ a and C − = C − (g, l) = A − + 1 ⊗ a − denote the Dirac operator for the pair (g, l) and its parts. We can now further decompose The third sum is 0, since we can assume [v j , v * j ] is in l and hence orthogonal to u * i . Namely, we can choose v j and v * j (and also u i and u * i ) to be root vectors with respect to t. Finally, the fourth sum is calculated by noting that since A completely analogous calculation applies to C − (k, l), so we proved: Proposition 6.6. Under the diagonal map ∆ : Note the unexpected feature of this result, the mixing of the positive and negative parts under the diagonal embedding. Namely, a p and a − p have opposite positions from the ones one would expect. So we do not have an analogue of Theorem 5.2 for C and C − , unless a p = a − p = 0. This last thing happens precisely when the pair (g, k) is hermitian symmetric. This is the reason why we are able to obtain results about u-cohomology only in hermitian case. Maybe this peculiar behavior has something to do with the fact that some of the most concrete results about n-cohomology, like [E], [C] or [A], are also obtained in hermitian situation only.
Another difficulty with the non-hermitian case is the fact that while we can write D(g, k) = A p + A − p , the two summands here are not differentials (they are also not Kinvariant). So there is no hope to get a Hodge decomposition like the one in Section 4. Of course, there is also no u ∩ p-homology or cohomology, since u ∩ p is not a Lie algebra. Yet there is perfectly well defined Dirac cohomology for D(g, k), and one can hope that it will somehow replace the nonexistent p − -cohomology.
What we do get without the hermitian assumption, is a copy of gl(1, 1) inside U (g) ⊗ C(s), spanned by C ∆ (k, l), C − ∆ (k, l), E k = ∆E(k, l) = − 1 2 ⊗ i u * i u i and ∆D(k, l) 2 . This follows immediately from the fact that ∆ is (obviously) a morphism of superalgebras.
Here we have used the more precise notation for the various diagonal embeddings to avoid confusion; note the equality ∆ g,k (∆ k,l (Ω l )) = ∆ g,l (Ω l ).
In other words, we see that the gl(1, 1) corresponding to the pair (g, l) sits diagonally in the direct product of the two copies of gl(1, 1) described above. Let us summarize the above discussion: Corollary 6.7. Assume (g, k) is a hermitian symmetric pair, let q = l ⊕ u be a θ-stable parabolic subalgebra of g, and assume that l ⊂ k and u ⊃ p + . Then there are two supercommuting copies of gl(1, 1) inside U (g) ⊗ C(s). One is spanned by C ∆ (k, l), C − ∆ (k, l), ∆E(k, l) and ∆D(k, l) 2 and the other is spanned by C(g, k), C − (g, k), E(g, k), and D(g, k) 2 . The diagonal of the product of these two super subalgebras is the copy of gl(1, 1) spanned by C, C − , E and D 2 from the end of Section 2.

Hodge decomposition forū-cohomology in hermitian case
In this section, (g, k) is a hermitian symmetric pair, q = l ⊕ u is a θ-stable parabolic subalgebra of g, and we assume l ⊂ k and u ⊃ p + .
Let V be a unitary (g, K)-module, and consider the form , on V ⊗ S introduced in Section 4. To apply the results of Section 4, we decompose and embed V ⊗ · p + as V ⊗ · p + ⊗ 1. The form , restricts to the analogous definite form on V ⊗ · p + .
Denote as before by D = D(g, l) the Dirac operator for the pair (g, l) and by C = C(g, l) and C − = C − (g, l) its parts coresponding to u andū. By Corollary 6.7, C = C ∆ (k, l) + C(g, k), and similarly for C − . Moreover, the copy of gl(1, 1) corresponding to the pair (g, k) supercommutes with the copy of gl(1, 1) corresponding to the pair (k, l).
By Corollary 4.3, the adjoints of C ∆ (k, l) and C(g, k) are respectively C − ∆ (k, l) and . We consider the positive semidefinite operator ∆ = CC adj + C adj C = [C, C adj ]. By the above remarks we have We know from Section 4 that V ⊗ · p + decomposes into eigenspaces of D(g, k) 2 for eigenvalues λ ≤ 0. Each eigenspace isK-invariant, and eachK-isotypic component of V ⊗ · p + is contained in an eigenspace. We assume V is admissible, so the eigenspaces are finite-dimensional.
Passing from V ⊗ · p + to V ⊗ S is tensoring with the finite-dimensional l-module · u ∩ k. On this last space, there is no action of U (g) or U (k ∆ ). So every eigenspace of D(g, k) 2 on V ⊗ · p + just gets tensored with · u ∩ k, and this gives the eigenspace on V ⊗ S for the same eigenvalue.
Since D ∆ (k, l) 2 commutes with D(g, k) 2 , it preserves these eigenspaces. Moreover, the Levi subgroup L ⊂ K corresponding to l is compact. So is then the double coverL, which acts on V ⊗ S. SinceL commutes with D(g, k) 2 , it also preserves its eigenspaces and hence these eigenspaces decompose intoL-irreducibles. Since D ∆ (k, l) 2 is up to a constant equal to the Casimir element of l ∆ , it follows that D ∆ (k, l) 2 diagonalizes on each eigenspace of D(g, k) 2 . To conclude: Lemma 7.1. V ⊗ S is a direct sum of eigenspaces for ∆. In particular, V ⊗ S = Ker ∆ ⊕ Im ∆.
We now turn our attention to the Dirac cohomology of D = D(g, l). In addition to the above considerations, we bring in Corollary 5.6, and note that for both D ∆ (k, l) and D(g, k) the cohomology is the same as the kernel or the kernel of the square. Thus we obtain: Theorem 7.8. The Dirac cohomology H D (g, l; V ) of a unitary (g, K)-module V is isomorphic to theū-cohomology of V and the u-homology of V up to appropriate modular twists. Moreover, all three cohomologies have the same set of harmonic representatives, Ker ∆.

Homological properties of Dirac cohomology
Let us start by showing that although we proved that in some cases Dirac cohomology of a unitary (g, K)-module with respect to D(g, l) can be identified withū-cohomology or u-homology, one should by no means expect that these notions agree for general (g, K)modules. Let us see that this is not the case even for (sl(2, C), SO(2))-modules.
Consider the module V which is a nontrivial extension of the discrete series representation W of highest weight −2 by the trivial module C: (In other words, V is a dual Verma module.) The k-weights of V (for the basis element They are 0 → C1 ⊗ u → Cv 0 ⊗ u 0 → Cv −2 ⊗ u → C1 ⊗ 1 → 0 → 0 → 0 for u-homology and 0 → C1 ⊗ 1 → Cv 0 ⊗ 1 → 0 → C1 ⊗ u → Cv −2 ⊗ u ⊕ Cv 0 ⊗ u → Cv −2 ⊗ u → 0 forū-cohomology. Here all arrows are the obvious ones except for the one labelled by 0. For Dirac cohomology, instead of a long exact sequence (which clearly does not makes sense in general, as Dirac cohomology is not Z-graded), there is a six-term exact sequence. In the above example, this sequence is We see that in this example the six-term sequence agrees with the u-homology long exact sequence only because of the presence of zeros; in general, all three sequences are different.
To define the six-term sequence in a more general situation, let us assume that the Dirac cohomology is Z 2 -graded. This happens whenever we are starting from g = r ⊕ s with s even-dimensional; this is automatic when r = l is a Levi subalgebra. Let be a short exact sequence of (g, K)-modules. Tensor this sequence by S, and denote the arrows still by i and p (they get tensored by the identity on S). Assuming that D 2 is a semisimple operator for each of the three modules, we can construct a six-term exact sequence The horizontal arrows are induced by i and p. The vertical arrows are the connecting homomorphisms, defined as follows. Let z ∈ Z ⊗ S represent a Dirac cohomology class, so Dz = 0. Choose y ∈ Y ⊗ S such that py = z. Since D 2 is semisimple, we can assume D 2 y = 0. Since pDy = Dpy = Dz = 0, we see that Dy = ix for some x ∈ X. Since D 2 y = 0, we see that Dx = 0, so x defines a cohomology class. This class is by definition the image of the class of z under the connecting homomorphism. Clearly, we changed parity when we applied D, and this defines both vertical arrows at once.
It is easy to see that this map is well defined, and that the resulting six-term sequence is exact. To conclude: Theorem 8.1. Let g = r⊕ s be an orthogonal decomposition, with r a reductive subalgebra and s even-dimensional. Let 0 → X → Y → Z → 0 be a short exact sequence of (g, K)modules and assume that the square of the Dirac operator D(g, r) is a semisimple operator for X, Y and Z. Then there is a six-term exact sequence corresponding to this short exact sequence, as described above.
Finally, let us comment on what can be done when s is odd dimensional, say dim s = 2n + 1. The usual spin modules S 1 and S 2 are not Z 2 -graded and thus it seems the above construction does not make sense. Recall that S 1 and S 2 are defined by writing s = C 2n ⊕ C, considering the spin module for C (C 2n ), and letting the last basis element of s act in two different ways (preserving the even and odd subspace instead of exchanging them).
We can instead consider the unique irreducible graded moduleS of C(s). It can be constructed as the restriction of the (unique) spin module for C (C 2n+2 ) to C(s) ⊂ C(C 2n+2 ). As a non-graded module,S decomposes as S 1 ⊕ S 2 . If we define Dirac cohomology using S in place of S 1 or S 2 , we double it, but we do get a Z 2 -grading. Then the above construction works also in the odd case. Thus, this is probably a more natural definition of Dirac cohomology in the odd case.
It remains to see what can be done if D 2 is not a semisimple operator. One possibility might be to consider a more general definition of Dirac cohomology in that setting.