Dirac index of some unitary representations of Sp(2n, \mathbb{R}) and SO^*(2n)
aa r X i v : . [ m a t h . R T ] F e b DIRAC INDEX OF SOME UNITARY REPRESENTATIONS OF Sp (2 n, R ) AND SO ∗ (2 n ) CHAO-PING DONG AND KAYUE DANIEL WONG
Abstract.
Let G be Sp (2 n, R ) or SO ∗ (2 n ). We compute the Dirac index of a large classof unitary representations considered by Vogan in Section 8 of [32], which include all weaklyfair A q ( λ ) modules and (weakly) unipotent representations of G as two extreme cases. Weconjecture that these representations exhaust all unitary representations of G with nonzeroDirac cohomology. In general, for certain irreducible unitary module of an equal rankgroup, we clarify the link between the possible cancellations in its Dirac index, and theparities of its spin-lowest K -types. Introduction
Let G be a connected linear Lie group with Cartan involution θ . Denote by b G the unitarydual of G , which stands for the set of all the equivalence classes of irreducible unitaryrepresentations of G . Classification of b G is a fundamental problem in representation theoryof Lie groups. An algorithm answer has been given by Adams, van Leeuwen, Trapa andVogan [1]. Including unitarity detecting, the corresponding software implementation atlas [34] now computes many aspects of problems in Lie theory.To get a better understanding of b G , Vogan introduced Dirac cohomology in 1997 [33]. Letus build up necessary notation to recall this notion. Assume that K := G θ is a maximalcompact subgroup of G . Choose a maximal torus T of K . Write the Lie algebra of G , K and T as g , k and t , respectively. Let g = k ⊕ p be the Cartan decomposition on the Lie algebra level. Let a be the centralizer of t in p ,and put A = exp( a ). Then H = T A is the unique maximally compact θ -stable Cartansubgroup of G , and h := t ⊕ a is the fundamental Cartan subalgebra of g . We will write g = g ⊗ R C and so on.Fix a non-degenerate invariant symmetric bilinear form B on g . We may also write B as h· , ·i . Then k and p are orthogonal to each other under B . Fix an orthonormal basis { Z , . . . , Z m } of p with respect to the inner product on p induced by B . Let U ( g ) bethe universal enveloping algebra, and let C ( p ) be the Clifford algebra. As introduced by Mathematics Subject Classification.
Primary 22E46.
Key words and phrases.
Dirac cohomology, Dirac index, parity of spin-lowest K -type, weakly unipotentrepresentation. Parthasarathy [24], the
Dirac operator is defined as(1) D := m X i =1 Z i ⊗ Z i ∈ U ( g ) ⊗ C ( p ) , which is independent of the choice of the orthonormal basis { Z i } mi =1 . Let Ad : K → SO ( p )be the adjoint map, and p : Spin( p ) → SO ( p ) be the universal covering map. Then e K := { ( k, s ) ∈ K × Spin( p ) | Ad( k ) = p ( s ) } is the spin double cover of K . Let π be any ( g , K ) module. The Dirac operator D acts on π ⊗ Spin G , where Spin G is a spin module for the Clifford algebra C ( p ). The Dirac cohomology is defined [33] as the following e K -module:(2) H D ( π ) := Ker D/ (Ker D ∩ Im D ) . Fix a positive root system ∆ + ( k , t ) once for all, and denote the half sum of roots in ∆ + ( k , t )by ρ c . We will use E µ to denote the k -type (that is, an irreducible representation of k ) withhighest weight µ . Abuse the notation a bit, E µ will also stand for the K -type as well as the e K -type with highest weight µ .The following Vogan conjecture, proved by Huang and Pandˇzi´c [17], is foundational forcomputing H D ( π ). Theorem 1.1. ( Huang-Pandˇzi´c [17])
Let π be any irreducible ( g , K ) module with infin-itesimal character Λ ∈ h ∗ . Assume that H D ( π ) is non-zero, and that E γ is contained in H D ( π ) . Then Λ is conjugate to γ + ρ c by some element in the Weyl group W ( g , h ) . Let b G d be the Dirac series of G . That is, the members of b G with non-zero Dirac co-homology. Classification of Dirac series is a smaller project than the classification of theunitary dual. Yet it is still worthy of pursuing since Dirac series contains many interest-ing unitary representations such as the discrete series [17], certain A q ( λ ) modules [16] andbeyond. Moreover, due to the research announcement of Barbasch and Pandˇzi´c [4], Diracseries should have applications in the theory of automorphic forms.Recently, Dirac series has been classified for complex classical Lie groups [5, 9, 10] and GL ( n, R ) [11]. For other classical groups such as U ( p, q ), Sp (2 n, R ) and SO ∗ (2 n ) whoseunitary dual is unknown, it is still very hard to achieve the complete classification of theirDirac series. In this case, the Dirac index can offer some help: it is much easier to computeand whenever it is non-zero, the Dirac cohomology must be non-zero.Let us build up a bit more notation for introducing Dirac index. Unless stated otherwise,we further assume that G is equal rank henceforth. Then h = t and a = 0. Put t R = i t and t ∗ R = i t ∗ . Let C ⊆ t ∗ R be the dominant Weyl chamber for ∆ + ( k , t ). Choose a positive rootsystem ∆ + ( g , t ) = ∆ + ( k , t ) ∪ ∆ + ( p , t ) . Let C g ⊆ t ∗ R be the dominant Weyl chamber for ∆ + ( g , t ). Let(3) W ( g , t ) = { w ∈ W ( g , t ) | w C g ⊆ C} . IRAC INDEX OF SOME UNITARY REPRESENTATIONS OF Sp (2 n, R ) AND SO ∗ (2 n ) 3 The set W ( g , t ) has cardinality s := | W ( g , t ) | / | W ( k , t ) | . Any positive root system (∆ + ) ′ ( g , t )of ∆( g , t ) containing ∆ + ( k , t ) has the form(∆ + ) ′ ( g , t ) = ∆ + ( k , t ) ∪ (∆ + ) ′ ( p , t )with (∆ + ) ′ ( p , t ) = w ∆ + ( p , t ) for some w ∈ W ( g , t ) . Denote the half sum of roots in ∆ + ( g , t )(resp., ∆ + ( p , t )) by ρ (resp., ρ n ). The notation ρ ′ and ρ ′ n will be interpreted similarly. Then ρ = ρ c + ρ n , ρ ′ = ρ c + ρ ′ n . Let p + = X α ∈ ∆ + ( p , t ) g α , p − = X α ∈ ∆ + ( p , t ) g − α . Then p = p + ⊕ p − and Spin G ∼ = ^ p + ⊗ C − ρ n . Any weight in Spin G has the form − ρ n + h Φ i , where Φ is a subset of ∆ + ( p , t ) and h Φ i standsfor the sum of the roots in Φ. Now put(4) Spin + G = even ^ p + ⊗ C − ρ n , Spin − G = odd ^ p + ⊗ C − ρ n . The Dirac operator D interchanges π ⊗ Spin + G and π ⊗ Spin − G . Thus the Dirac cohomology H D ( π ) breaks up into the even part H + D ( π ) and the odd part H − D ( π ). The Dirac index of π is defined as the virtual e K -module(5) DI( π ) := H + D ( π ) − H − D ( π ) . By Remark 3.8 of [23], if (∆ + ) ′ ( g , t ) is chosen instead of ∆ + ( g , t ), one hasDI ′ ( π ) = ( − + ) ′ ( p , t ) \ ∆ + ( p , t )) DI( π ) . Therefore, the Dirac index is well-defined up to a sign. Moreover, by Proposition 3.12 of[22], DI( π ) = π ⊗ Spin + G − π ⊗ Spin − G . It turns out that the Dirac index preserves short exact sequences and has nice behavior withrespect to coherent continuation [22, 23]. This idea is pursued in [12] to compute the Diracindex of all weakly fair A q ( λ )-modules for G = U ( p, q ).In this paper, we study a larger class of unitary representations constructed in [32, Sec-tion 8] (see Theorem 2.5 below). This includes all weakly fair A q ( λ )-modules and unipotentrepresentations as two extreme cases. We will compute the Dirac index for all such repre-sentations for G = Sp (2 n, R ) and SO ∗ (2 n ), and conjecture that composition factors of theserepresentations should exhaust the Dirac series of G .The paper is organized as follows: In Section 2, we provide the preliminaries for Diracindex and cohomological induction. We also describe the class of representations that we areinterested in. In Sections 3 and 4, we study the Dirac index of all unipotent representationsof G = Sp (2 n, R ) and SO ∗ (2 n ). In Section 5, we compute the Dirac index of all unitaryrepresentations of G covered in Theorem 2.5. Finally, in Section 6, we reveal the relationbetween the possible cancellations in H D ( π ), and the parities of the spin-lowest K -types of π , where π is irreducible unitary. See Theorem 6.2. CHAO-PING DONG AND KAYUE DANIEL WONG Preliminaries
We continue with the notation in the introduction. In particular, G is equal rank.2.1. Dirac index.
In this section, we choose the
Vogan diagram for g as Appendix C ofKnapp [18]. Then we have actually chosen a∆ + ( g , t ) = ∆ + ( k , t ) ∪ ∆ + ( p , t ) . The Vogan diagram for g has a unique black dot, which stands for a simple root γ . Wedenote the fundamental weight corresponding to γ by e ζ . Put(6) ζ = 2 k γ k e ζ. The following result should be well-known, and it can be obtained by going through theclassification of real simple Lie algebras.
Lemma 2.1.
Let β be the highest root in ∆ + ( g , t ) . Then G/K is Hermitian symmetric ifand only if the unique black dot simple root has coefficient in β . Otherwise, the uniqueblack dot simple root must have coefficient in β . As a consequence, we always have thatfor any α ∈ ∆( g , t ) , (7) h α, ζ i is even ⇔ α ∈ ∆( k , t ) , h α, ζ i is odd ⇔ α ∈ ∆( p , t ) . Lemma 2.2.
For any w ∈ W ( g , t ) , the set Φ w := w ∆ − ( g , t ) ∩ ∆ + ( g , t ) must be containedin ∆ + ( p , t ) .Proof. Suppose there exists a w ∈ W ( g , t ) such that Φ w is not contained in ∆ + ( p , t ). Thenwe can find a root α ∈ ∆ + ( k , t ) such that α ∈ Φ w . By the definition of Φ w , we can furtherfind a root β ∈ ∆ + ( g , t ) such that α = − wβ . That is, − α = wβ . Now, h− α, wρ i = h wβ, wρ i = h β, ρ i > . Therefore, h wρ, α i <
0. This contradicts to the assumption that w ∈ W ( g , t ) since wρ should be dominant for ∆ + ( k , t ). (cid:3) The first named author learned the following result from Pandˇzi´c. It should be well-knownto the experts.
Lemma 2.3.
We have the following decompositions of the spin module into k -types: (8) Spin + G = M l ( w ) even E wρ − ρ c , Spin − G = M l ( w ) odd E wρ − ρ c , where w runs over the set W ( g , t ) .Proof. As we know, Spin G = ^ p + O C − ρ n ∼ = M w ∈ W ( g , t ) V wρ − ρ c . It remains to separate the even part and the odd part of the spin module. Take an arbitrary w ∈ W ( g , t ) . Then there exists a subset Φ of ∆ + ( p , t ) such that wρ − ρ c = ρ n − h Φ i , IRAC INDEX OF SOME UNITARY REPRESENTATIONS OF Sp (2 n, R ) AND SO ∗ (2 n ) 5 where h Φ i stands for the sum of roots in Φ. One deduces from the above equality that h Φ w i = h Φ i . Taking inner products of the two sides of the above equality with ζ , we have that(9) l ( w ) ≡ | Φ | (mod 2)by recalling (7) and Lemma 2.2. Note that any weight of E wρ − ρ c can be obtained bysubtracting some roots of ∆ + ( k , t ) from wρ − ρ c . Now the desired result follows from (7). (cid:3) Let π be an irreducible unitary ( g , K ) module. Assume that H D ( π ) is non-zero. Thenany e K -type E γ of H D ( π ) lives in either π ⊗ Spin + G or π ⊗ Spin − G . We assign a sign to E γ asfollows: the sign is +1 in the first case, it is − sign of E γ in H D ( π ).2.2. Cohomologically induced modules.
Firstly, let us fix an element H ∈ t R which isdominant for ∆ + ( k , t ), and define the θ -stable parabolic subalgebra(10) q = l ⊕ u as the nonnegative eigenspaces of ad( H ). The Levi subalgebra l of q is the zero eigenspace ofad( H ), while the nilradical u of q is the sum of positive eigenspaces of ad( H ). If we denoteby u the sum of negative eigenspaces of ad( H ), then g = u ⊕ l ⊕ u . Let L be the normalizer of q in G . Then L ∩ K is a maximal compact subgroup of L . Let z be the center of l .We choose a positive root system (∆ + ) ′ ( g , t ) containing ∆( u , t ) and ∆ + ( k , t ). Set∆ + ( l , t ) = ∆( l , t ) ∩ (∆ + ) ′ ( g , t ) . Denote by ρ L (resp., ρ Lc ) the half sum of positive roots in ∆ + ( l , t ) (resp., ∆ + ( l ∩ k , t )). Let ρ Ln = ρ L − ρ Lc . Denote by ρ ( u ) (resp., ρ ( u ∩ p ), ρ ( u ∩ k )) the half sum of roots in ∆( u , t )(resp., ∆( u ∩ p , t ), ∆( u ∩ k , t )). The following relations hold(11) ρ ( j ) = ρ L + ρ ( u ) , ρ c = ρ Lc + ρ ( u ∩ k ) , ρ ( j ) n = ρ Ln + ρ ( u ∩ k ) . The cohomological induction functors L j ( · ) and R j ( · ) lift an admissible ( l , L ∩ K ) module Z to ( g , K ) modules, and the most interesting case happens at the middle degree S :=dim( u ∩ k ). Assume that Z has real infinitesimal character Λ Z ∈ t ∗ R . After [19], we say that Z is in the good range (relative to q and g ) if(12) h Λ Z + ρ ( u ) , α i > , ∀ α ∈ ∆( u , t ) . We say that Z is in the weakly good range if(13) h Λ Z + ρ ( u ) , α i ≥ , ∀ α ∈ ∆( u , t ) . Moreover, Z is said to be in the fair range if(14) h Λ Z + ρ ( u ) , α | z i > , ∀ α ∈ ∆( u , t ) . We say that Z is in the weakly fair range if(15) h Λ Z + ρ ( u ) , α | z i ≥ , ∀ α ∈ ∆( u , t ) . CHAO-PING DONG AND KAYUE DANIEL WONG
When the inducing module Z is a one-dimensional unitary character C λ , we denote thecorresponding ( g , K )-module L S ( Z ) by A q ( λ ), which has infinitesimal character λ + ρ ′ . Goodrange A q ( λ ) modules must be non-zero, irreducible, and unitary [19]. They play an importantrole in the unitary dual. Indeed, as shown by Salamanca-Riba [28], any irreducible unitary( g , K )-module with a real, integral, and strongly regular infinitesimal character Λ must beisomorphic to an A q ( λ ) module in the good range. Here Λ being strongly regular means that h Λ − ρ ′ , α i ≥ , ∀ α ∈ (∆ + ) ′ ( g , t ) . Note that the module A q ( λ ) is weakly fair if(16) h λ + ρ ( u ) , α i ≥ , ∀ α ∈ ∆( u , t ) . A weakly fair A q ( λ ) module can be zero or reducible. However, whenever it is non-zero,it must be unitary [19]. More importantly, many singular unitary representations can berealized as (a composition factor of) a weakly fair A q ( λ ) module.Based on [22, 23], the following result computes the Dirac index of weakly fair A q ( λ ). Theorem 2.4. (Theorem 4.3 of [12])
The Dirac index of weakly fair A q ( λ ) is equal to DI( A q ( λ )) = X w ∈ W ( l , t ) det( w ) e E w ( λ + ρ ) , where (17) e E µ = ( if µ is ∆( k , t ) -singular det( w ) E wµ − ρ c if ∃ w ∈ W ( k , t ) s.t. wµ is dominant regular for ∆ + ( k , t ) . A larger class of unitary modules.
As stated in the previous section, all weaklyfair A q ( λ )-modules are unitarty. Indeed, [32] obtained a unitarity theorem for a larger classof representations. Theorem 2.5 ([32] Proposition 8.17) . Let G be a reductive Lie group, and q = l + u be a θ -stable parabolic subalgebra of g such that l ∼ = g A l + · · · + g A + g ′ m , where each g A t is of Type A , and g ′ m is not of Type A with rank m (here we allow l = 0 ,i.e., there are no Type A factors and l = g ; or m = 0 , i.e, all factors are of Type A ).Suppose Z is an ( l , L ∩ K ) module given by the tensor product of unitary characters ofType A and a weakly unipotent representation π u of G ′ m [32, Definition 8.16] , such that theinfinitesimal character Λ Z of Z is in the weakly fair range. Then L S ( Z ) is an unitary ( g , K ) module. If the weakly unipotent representation in Theorem 2.5 is taken to be the trivial repre-sentation, or g ′ m does not exist in l , then we are in the setting of A q ( λ ) modules in theprevious subsection. On the other extreme, if l = g = g ′ m , we obtain all (weakly) unipotentrepresentations.We are interested in representations in Theorem 2.5 with non-zero Dirac cohomology.A necessary condition for such a module to have non-zero Dirac cohomology is that theweakly unipotent representation π u of g ′ m has infinitesimal character satisfying Theorem IRAC INDEX OF SOME UNITARY REPRESENTATIONS OF Sp (2 n, R ) AND SO ∗ (2 n ) 7 G = U ( p, q ) and Sp (2 n, R ), the classification of all such π u along with their Diraccohomologies are given in [3].From now on, we focus on G = Sp (2 n, R ) and SO ∗ (2 n ). We begin by computing DI( π u )for all unipotent representations π u of G whose infinitesimal characters satisfy Theorem 1.1.From there, one can use the results in [12] to compute DI( L S ( Z )) in Theorem 2.5 in thespecial case when Z consists of trivial characters for each Type A factor, and π u for the G ′ m factor (Theorem 5.5).As for the general case of weakly fair L S ( Z ′ ) such that Z ′ consists of some Type A characters and the same π u for the G ′ m factor, then it can be placed inside a coherent family containing the representation L S ( Z ) given in the previous paragraph (c.f. [31, Theorem7.2.23]). Consequently, one can get DI( L S ( Z ′ )) from DI( L S ( Z )) by the translation principle(Proposition 5.8). The explicit formulas for DI( L S ( Z ′ )) are given in Corollary 5.16 andTheorem 5.19.3. Dirac index of unipotent representations of Sp (2 n, R )Let G be Sp (2 n, R ). We fix a Vogan diagram for sp (2 n, R ) as in Fig. 1, where the simpleroots are e − e , e − e , . . . , e n − − e n − , e n − − e n and 2 e n when enumerated from left toright, with 2 e n being dotted. Therefore, ζ = ( , . . . , ). Figure 1.
The Vogan diagram for sp (2 n, R )By fixing this Vogan diagram, we have actually fixed∆ + ( k , t ) = { e i − e j | ≤ i < j ≤ n } , and chosen ∆ + ( p , t ) = { e i + e j | ≤ i < j ≤ n } ∪ { e i | ≤ i ≤ n } . Then ρ c = ( n − , n − , . . . , − n − ), ρ n = ( n +12 , . . . , n +12 ), and ρ = ( n, n − , . . . , π u of Sp (2 n, R ) whose infinitesimal characters satisfy Theorem 1.1. Therefore, it includes allpossible unipotent representations of G with non-zero Dirac cohomology. In the next coupleof subsections, we will study the Dirac index of these representations.3.1. Dirac index of π u = X ( r, s ; ǫ, η ) . Let k be a positive integer such that r + s = 2 k ≤ n ,where r and s are non-negative integers. Let ǫ, η be 0 or 1. Set ǫ = 0 if r = 0, and set η = 0 if s = 0. The irreducible unipotent representation X ( r, s ; ǫ, η ) is constructed from thecharacter C ǫ,η of O ( r, s ) via theta lifting. Its K -types are as follows:(18) ( r − s , . . . , r − s a + ǫ, . . . , a r + ǫ, , . . . , , − b s − η, . . . , − b − η ) , where a ≥ · · · ≥ a r ≥ b ≥ · · · ≥ b s ≥ X ( r, s ; ǫ, η ) is Λ = Λ k = ( n − k, n − k − , . . . , k + 1 , k, k − , . . . , − k + 1) . CHAO-PING DONG AND KAYUE DANIEL WONG
As in [3], we call the last 2 k coordinates k, k − , . . . , − k + 1 the core of Λ, and call the first n − k coordinates n − k, n − k − , . . . , k + 1 the tail of Λ. For w ∈ W ( g , t ), w Λ − ρ c is thehighest weight of a e K -type if and only if the entries of w Λ are strictly decreasing. In sucha case, we must have that either(19) w Λ = ( i , . . . , i u , k, k − , . . . , − k + 1 , − j v , . . . , − j ) , or(20) w Λ = ( i , . . . , i u , k − , . . . , − k + 1 , − k, − j v , . . . , − j ) . Here u and v are two non-negative integers such that u + v = n − k . Moreover, i > · · · > i u , j > · · · > j v are such that { i , . . . , i u , j , . . . , j v } = { k + 1 , k + 2 , . . . , n − k } . Let τ := w Λ − ρ c be a e K -type. We call τ special if (19) holds, and call τ non-special if (20)holds. Let w K be the longest element of W ( k , t ), which is isomorphic to S n . Then one seesthat τ is special if and only if − w K τ is non-special.Let τ := w Λ − ρ c be a e K -type with w Λ given by (19). In particular, τ is special. Let i ′ t = i t − k and j ′ t = j t − k . Then { i ′ , . . . , i ′ u , j ′ , . . . , j ′ v } = { , , . . . , n − k } . Put(21) M ( τ ) := j ′ + · · · + j ′ v + r ( ǫ + v ) + r ( r − , and(22) N ( τ ) := j ′ + · · · + j ′ v + s ( η + v ) + s ( s − n ( n + 1)2 . When τ is special, it follows from the proof of [3, Theorem 3.7] that the sign of the e K -type E τ in H D ( X ( r, s ; ǫ, η )) (if it does occur) is ( − M ( τ ) . Indeed, in the setting of [3], we lookat all possibilities of σ ∈ W ( g , t ) such that(23) σρ = ( x , . . . , x s , i ′ , . . . , i ′ u , − j ′ v , . . . , − j ′ , − y r , . . . , − y ) , where x = η + u + 2 e b + s, . . . , x s = η + u + 2 e b s + 1; y = ǫ + v + 2 e a + r − , . . . , y r = ǫ + v + 2 e a r . for some integers e a ≥ · · · ≥ e a r ≥ e b ≥ · · · ≥ e b s ≥
0. For each possibility of σ satisfying(23), the PRV component [25] of E ( r − s ,..., r − s )+(2 f a + ǫ,..., f a r + ǫ, ,..., , − e b s − η,..., − e b − η ) ⊗ E σρ − ρ c appearing in X ( r, s ; ǫ, η ) (c.f. (18)) and Spin ± G (c.f. (8)) respectively contributes a copy of E τ in H D ( X ( r, s ; ǫ, η )). In order to determine the sign of E τ in DI( X ( r, s ; ǫ, η )), it sufficesto study the parity of l ( σ ) by Lemma 2.3. IRAC INDEX OF SOME UNITARY REPRESENTATIONS OF Sp (2 n, R ) AND SO ∗ (2 n ) 9 Table 1.
Signs of the e K -types in H D ( X ( r, s ; ǫ, η )) of Sp (12 , R ) τ τ τ τ − w K τ − w K τ − w K τ − w K τ r = 4 , s = 0 0 0 1 1 1 1 0 0 r = 3 , s = 1 ǫ + 1 ǫ ǫ + 1 ǫ η + 1 η η + 1 ηr = 2 , s = 2 1 1 0 0 0 0 1 1 r = 1 , s = 3 ǫ ǫ + 1 ǫ ǫ + 1 η η + 1 η η + 1 r = 0 , s = 4 0 0 1 1 1 1 0 0Note that for all σ satisfying (23), ρ − σρ = h Φ σ i is a sum of roots in ∆ + ( p , t ) by Lemma2.2. Therefore, by (7) and (9), we have l ( σ ) ≡ | Φ σ | ≡ h ρ − σρ, ζ i (mod 2) . On the other hand, a direct calculation of ρ − σρ using (23) gives h ρ − σρ, ζ i = j ′ + · · · + j ′ v + y + · · · + y r ≡ M ( τ ) (mod 2) . Therefore, Lemma 2.3 implies that each E τ in H D ( X ( r, s ; ǫ, η )) has the same sign ( − l ( σ ) =( − M ( τ ) .Still assume that τ is special. By Lemma 3.2 of [3], each E − w K τ in H D ( X ( r, s ; ǫ, η )) hasthe same sign, which equals to ( − n ( n +1)2 times the sign of E τ in H D ( X ( s, r ; η, ǫ )). Thissign turns out to be ( − N ( τ ) .To sum up, when passing from H D ( X ( r, s ; ǫ, η )) to DI( X ( r, s ; ǫ, η )), no cancellation hap-pens. For each special e K -type E τ , the sign of E τ (resp., E − w K τ ) in DI( X ( r, s ; ǫ, η )) is( − M ( τ ) (resp., ( − N ( τ ) ). The multiplicity of E τ , which can be zero, is known fromProposition 3.4 and Theorem 3.7 of [3]. Example 3.1.
Let us consider Sp (12 , R ) and take k = 2. Then Λ = (4 , , , , , − r + s = 4 and u + v = 2. We list all the four special e K -types as follows: • τ = (4 , , , , , − − ρ c = ( , , , , , ), u = 2, v = 0. • τ = (3 , , , , − , − − ρ c = ( , , , , , − ), u = 1, v = 1, j ′ = 2. • τ = (4 , , , , − , − − ρ c = ( , , , , , − ), u = 1, v = 1, j ′ = 1. • τ = (2 , , , − , − , − − ρ c = ( − , − , − , − , − , − ), u = 0, v = 2, j ′ = 2, j ′ = 1.Now the sign of each E τ i and E − w K τ i in H D ( X ( r, s ; ǫ, η )) is given in Table 1. For instance,it reads that the sign of E τ in H D ( X (3 , ǫ, η )) (if it does occur) is ( − ǫ +1 . Theorem 3.2.
Each of the following virtual ( g , K ) module has zero Dirac index: a) X ǫ + η ≡ δ (mod 2) X (2 j − , k + 1 − j ; ǫ, η ) , for any ≤ j ≤ k − and δ ∈ { , } ; b) X ǫ X (2 k, ǫ,
0) + X η X (0 , k ; 0 , η ) + k − X j =1 X ǫ + η ≡ δ (mod 2) X (2 j, k − j ; ǫ, η ) for δ ∈{ , } . In each case above, both ǫ and η run over { , } .Proof. Fix any special e K -type E τ := w Λ − ρ c with w Λ given by (19). We will show that E τ has coefficient zero in the Dirac index of each of the above virtual ( g , K ) module. One candraw the same conclusion for each non-special E τ . Thus the desired conclusion follows.(a) By Theorem 3.7(2II) of [3], when δ ≡ n + 1 (mod 2), each summand has zero Diraccohomology. Thus the conclusion is trivial. Now assume that δ ≡ n (mod 2). By Theorem3.7(2I) of [3], the coefficient of E τ in DI( X (2 j − , k + 1 − j ; ǫ, η )) is( − M ( τ ) (cid:18) k − j − (cid:19) = ( − j ′ + ··· + j ′ v + v + j − ǫ (cid:18) k − j − (cid:19) . The total sum is zero when ǫ and η run over { , } such that ǫ + η ≡ δ (mod 2).(b) By Proposition 3.4 of [3], the coefficient of E τ in DI( X (2 k,
0; 0 ,
0) + X (2 k,
0; 1 , − j ′ + ··· + j ′ v + k (2 k − = ( − j ′ + ··· + j ′ v + k . Similarly, the coefficient of E τ in DI( X (0 , k ; 0 ,
0) + X (0 , k ; 0 , − j ′ + ··· + j ′ v . Fix 1 ≤ j ≤ k −
1, by Theorem 3.7(1) of [3], the coefficient of E τ inDI( X ǫ + η ≡ δ (mod 2) X (2 j, k − j ; ǫ, η ))is equal to ( − j ′ + ··· + j ′ v + j (2 j − (cid:18) kj (cid:19) = ( − j ′ + ··· + j ′ v + j (cid:18) kj (cid:19) . Note that (cid:0) k − j (cid:1) + (cid:0) k − j − (cid:1) = (cid:0) kj (cid:1) . Therefore, the total sum is( − j ′ + ··· + j ′ v k − X j =1 ( − j (cid:18) kj (cid:19) + ( − k = ( − j ′ + ··· + j ′ v (1 − k = 0 . (cid:3) Remark 3.3.
In [6], Barbasch and Trapa studied the number of stable combinations ofspecial unipotent representations whose annihilator is equal to the closure of a complexspecial nilpotent orbit O .For π u = X ( r, s ; ǫ, η ), O corresponds to the partition [2 k n − k ], and its Lusztig Spal-tenstein dual O ∨ corresponds to the partition [2 n − k + 1 , k − ,
1] in SO (2 n + 1 , C ). Thenthe main theorem of [6] implies that there are 3 (resp., 4) such stable combinations attachedto O ∨ if k = 1 (resp., if k > all nilpotent orbits in the same special piece as O ∨ into account.Since a necessary condition for a linear combination for some (non-trivial) special unipo-tent representations to be stable is that its Dirac index is zero, we conjecture that the sum ofthe linear combinations in Theorem 3.2(a), and the linear combinations in Theorem 3.2(b)are the 3 (if k = 1) or 4 (if k >
1) stable combinations specified in the above paragraph.
IRAC INDEX OF SOME UNITARY REPRESENTATIONS OF Sp (2 n, R ) AND SO ∗ (2 n ) 11 In particular, for O = [2 ] in Sp (4 , R ), we obtain the 3 stable combinations attached to O ∨ = [3 1 ] given in [6, Example 2.2].3.2. Dirac index of π u = X ′ ( r, s ; ǫ, η ) . Now consider Sp (2 n, R ) with n odd. Let r + s =2 k = n + 1, where r, s are non-negative integers. Then there is another family of unipotentrepresentations X ′ ( r, s ; ǫ, η ) studied in [3]. All of them have infinitesimal character ρ c = ( k − , k − , . . . , − k + 1) . Moreover, one has that X ′ (2 k,
0; 0 , ∼ = X ′ (2 k − ,
1; 1 , , X ′ (0 , k ; 0 , ∼ = X ′ (1 , k −
1; 0 , . Now we assume that both p and q are positive. Then ( ǫ, η ) can be (0 , ,
1) or (1 , e K -type E in H D ( X ′ ( r, s ; ǫ, η )): • ( ǫ, η ) = (1 , σρ = (2 b + s − , . . . , b s − + 1 , − a r − , . . . , − a − r ) . Thus h ρ − σρ, ζ i = (2 a r + 1) + · · · + (2 a + r ) . Therefore, the sign is ( − r ( r +1)2 . • ( ǫ, η ) = (0 , σρ = (2 b + s − , . . . , b s − + 1 , − c, − a r − − , . . . , − a − r + 1) , where c is either a r or − b s . Thus h ρ − σρ, ζ i ≡ (2 a r − + 1) + · · · + (2 a + r −
1) (mod 2) . Therefore, the sign is ( − r ( r − . • ( ǫ, η ) = (0 , σρ = (2 b + s, . . . , b s + 1 , − a r − − , . . . , − a − r + 1) . Thus h ρ − σρ, ζ i = (2 a r − + 1) + · · · + (2 a + r − . Therefore, the sign is ( − r ( r − .As in Theorem 3.2, we give a linear combination of ( g , K )-modules that is conjectually astable combination: Theorem 3.4.
Let G = Sp (2 n, R ) with n odd. The following virtual ( g , K ) module has zeroDirac index: X ′ (0 , k ; 0 ,
0) + k − X j =1 X ′ (2 j, k − j ; 0 ,
0) + X ′ (2 k,
0; 0 , . Proof.
Let us figure out the coefficient of E in the Dirac index of each X ′ (2 j, k − j ; 0 , • For X ′ (0 , k ; 0 , ∼ = X ′ (1 , k −
1; 0 , − (cid:0) k − (cid:1) = ( − (cid:0) k (cid:1) . • For X ′ (2 j, k − j ; 0 , ≤ j ≤ k −
1, the coefficient is ( − j (2 j − (cid:0) kj (cid:1) = ( − j (cid:0) kj (cid:1) . • For X ′ (2 k,
0; 0 , ∼ = X ′ (2 k − ,
1; 1 , − k (2 k − (cid:0) k − k − (cid:1) = ( − k (cid:0) kk (cid:1) .Therefore, the total sum is( − (cid:18) k (cid:19) + k − X j =1 ( − j (cid:18) kj (cid:19) + ( − k (cid:18) kk (cid:19) = (1 − k = 0 . (cid:3) Dirac index of unipotent representations of SO ∗ (2 n )Let G be SO ∗ (2 n ). We fix a Vogan diagram for so ∗ (2 n ) as in Fig. 2, where the simpleroots are e − e , e − e , . . . , e n − − e n − , e n − − e n − , e n − − e n and e n − + e n whenenumerated from left to right, with e n − + e n being dotted. Therefore, ζ = ( , . . . , ). Figure 2.
The Vogan diagram for so ∗ (2 n, R )By fixing this Vogan diagram, we have actually fixed∆ + ( k , t ) = { e i − e j | ≤ i < j ≤ n } , and chosen ∆ + ( p , t ) = { e i + e j | ≤ i < j ≤ n } . Unipotent representations of SO ∗ (2 n ) . This subsection aims to classify all theunipotent representations of SO ∗ (2 n ) whose infinitesimal characters satisfy Theorem 1.1.Similar to [3], our basic tool is theta correspondence [13, 14, 15]. Theorem 4.1.
Let G = SO ∗ (2 n ) . Then the unipotent representations of G whose infinites-imal characters satisfy Theorem 1.1 are obtained from theta lifts of the trivial representationin Sp ( r, s ) with r + s ) ≤ n − .Proof. By the condition of the infinitesimal character Λ of π u given by Theorem 1.1, thecoordinates of Λ must be integers. Therefore, the associated variety of Ann U ( g ) ( π u ) mustbe a special nilpotent orbit, and π u must be a special unipotent representation . Therefore,Λ = h ∨ /
2, where h ∨ is the semisimple element of a Jacobson-Morozov triple of a special(more precisely, even) nilpotent orbit.Putting in the extra condition that w Λ is ∆ + ( k , t )-dominant, w Λ can only be of the formΛ ′ k = ( n − k − , . . . , k, . . . , , , − , . . . , − k )for 0 ≤ k < n . IRAC INDEX OF SOME UNITARY REPRESENTATIONS OF Sp (2 n, R ) AND SO ∗ (2 n ) 13 Suppose now the infinitesimal character of π u is equal to Λ = Λ ′ k . ThenAV( Ann U ( g ) ( π u )) = O k , where O k is the nilpotent orbit in so (2 n, C ) corresponding to the partition [2 k n − k ]. By[21], the number of such special unipotent representations is equal to k + 1.For each r + s = k , the theta lift of the trivial representation of Sp ( r, s ) to SO ∗ (2 n ) hasinfinitesimal character Λ k [27]. Moreover, its associated variety is equal to the closure of the K -nilpotent orbit O r,s with partition [(+ − ) r ( − +) s + n − k − n − k ]. See [20]. Therefore, the k +1 representations constructed in this way are distinct, and satisfy AV( Ann U ( g ) ( π u )) = O k .This exhausts all possibilities of unipotent representations. (cid:3) We denote the representation obtained by theta lift of the trivial representation of Sp ( r, s )by X ( r, s ). In the next subsection, we will study the Dirac cohomology and Dirac index of X ( r, s ). It is worth noting that for n ≤
6, all X ( r, s ) can be realized as A q ( λ )-modules. Thefirst example of X ( r, s ) not being an A q ( λ )-module is X (1 ,
1) in SO ∗ (14).4.2. Dirac index of X ( r, s ) . We will proceed as in [3] and the previous section to find theDirac index of X ( r, s ) for 2 k = 2( r + s ) ≤ n −
1. Firstly, recall that X ( r, s ) has infinitesimalcharacter Λ k = ( n − k − , . . . , , , − , . . . , − k ) and the highest weights of its K -types are(24) ( r − s, . . . , r − s ) + ( a , a , . . . , a r , a r , n − k z }| { , . . . , , − b s , − b s , . . . , − b , − b )with a ≥ · · · ≥ a r and b ≥ · · · ≥ b s all being non-negative integers. Note that these K -types appear in X ( r, s ) with multiplicity one.Now let us compute H D ( X ( r, s )). Note that Λ ′ k is conjugate toΛ = ( n − k − , . . . , k + 1 , k, . . . , , , − , . . . , − k ) . By Theorem 1.1, any e K -type E τ occurring in H D ( X ( r, s )) must bear the form τ = x Λ − ρ c , for some x ∈ W ( g , t ) . Let u, v be two non-negative integers such that u + v = n − k −
1. Then we must have(25) x Λ = ( i , . . . , i u , k, . . . , , , − , . . . , − k, − j v , . . . , − j ) , where i > · · · > i u , j > · · · > j v and { i , . . . , i u , j , . . . , j v } = { k + 1 , . . . , n − k − } . We put i ′ t = i t − k and j ′ t = j t − k . Then { i ′ , . . . , i ′ u , j ′ , . . . , j ′ v } = { , . . . , n − k − } . Now it boils down to solve the equation(26) σρ − ρ c + w K µ = wτ, w ∈ W ( k , t ) , σ ∈ W ( g , t ) , where µ is a K -type of X ( r, s ) as described in (24). Similar to Section 3 of [3], we have(27) σρ =( b + u + 2 s, b + u + 2 s − , . . . , b s + u + 2 , b s + u + 1 , i ′ , . . . , i ′ u , , − j ′ v ,. . . , − j ′ , − a r − v − , − a r − v − , . . . , − a − v − r + 1 , − a − v − r ) . Let us arrange n − k, . . . , n − k pairs of consecutive integers:(28) n − k, n − k + 1 || n − k + 2 , n − k + 3 || · · · || n − , n − . Fix τ = x Λ − ρ c with x Λ being given by (25). Note that v + 2 r ≤ ( n − k −
1) + 2 k ≤ n − , u + 2 s ≤ ( n − k −
1) + 2 k ≤ n − . Thus for any choice of r pairs of consecutive integers from (28), there is a unique solution to(26) in terms of a , . . . , a r . Therefore, the multiplicity of E τ in H D ( X ( r, s )) is (cid:0) kr (cid:1) . Moreover,it follows from (27) that h ρ − σρ, ζ i = j ′ + · · · + j ′ v + ( a r + v + 1) + ( a r + v + 2) + · · · + ( a + v + 2 r −
1) + ( a + v + 2 r ) . Thus the sign of E τ in H D ( X ( r, s )) is(29) ( − j ′ + ··· + j ′ v + r . The above discussion leads to the following.
Theorem 4.2.
Let G be SO ∗ (2 n ) . Let k be a positive integer such that k + 1 ≤ n . TheDirac cohomology of X ( r, k − r ) is (cid:18) kr (cid:19) n − k − X v =0 X j ′ > ··· >j ′ v E ( i ′ ,...,i ′ u ,k,..., , , − ,..., − k, − j ′ v ,..., − j ′ ) − ρ c , where { j ′ , . . . , j ′ v } run over the subsets of { , , . . . , n − k − } with cardinality v , and { i ′ , . . . , i ′ u } is the complementary set. Using (29), one easily deduces the following.
Corollary 4.3.
Let k be a positive integer such that k + 1 ≤ n . The virtual ( g , K ) module P kr =0 X ( r, k − r ) of SO ∗ (2 n ) has zero Dirac index. Dirac index for general modules
We now study the Dirac index for all unitary representation given in Theorem 2.5 for G = Sp (2 n, R ) and SO ∗ (2 n ). Firstly, we describe the θ -stable parabolic subalgebras q = l + u of g using H ∈ t ∗ of the form (c.f. Section 3 of [26])(30) H = ( p l z }| { l, . . . , l, p l − z }| { l − , . . . , l − , · · · , p z }| { , . . . , , m z }| { , . . . , , − , . . . , − | {z } q , · · · , − l + 1 , . . . , − l + 1 | {z } q l − , − l, . . . , − l | {z } q l )such that l = u ( p l , q l ) ⊕ u ( p l − , q l − ) ⊕ · · · ⊕ u ( p , q ) ⊕ g ′ m , IRAC INDEX OF SOME UNITARY REPRESENTATIONS OF Sp (2 n, R ) AND SO ∗ (2 n ) 15 where g ′ m is of the same type as g with rank m . And we choose (∆ + ) ′ ( g , t ) such that ρ ′ sp := ( p l terms z }| { n, n − , . . . , n − p l + 1 , p l − terms z }| { n − ( p l + q l ) , . . . , n − ( p l + q l ) − p l − + 1 , · · · , p terms z }| { m + p + q , . . . , m + q + 1 , m, m − , . . . , , − ( m + 1) , − ( m + 2) , . . . , − ( m + q ) | {z } q terms , · · · , − ( n − ( p l + q l ) − ( p l − + q l − ) + 1) , . . . , − ( n − ( p l + q l ) − p l − ) | {z } q l − terms , − ( n − ( p l + q l ) + 1) , . . . , − ( n − p l ) | {z } q l terms )for G = Sp (2 n, R ), and ρ ′ so := ρ ′ sp − ( p l + ··· + p z }| { , . . . , , m z }| { , . . . , , q + ··· + q l z }| { − , . . . , − G = SO ∗ (2 n ).We now describe the ( l , L ∩ K )-modules Z appearing in Theorem 2.5. They are all of theform(31) Z ( λ l , . . . , λ , π u ) := C λ l ⊠ · · · ⊠ C λ ⊠ π u , where each C λ i is a unitary character of u ( p i , q i ), and π u is a weakly unipotent representation.It suffices to focus on those π u covered in Sections 3 and 4 for the study of DI( L S ( Z )). Definition 5.1.
Let G = Sp (2 n, R ) or SO ∗ (2 n ), and q = l + u be the θ -stable parabolic subal-gebra of g defined by the element H in (30). For each ( l , L ∩ K ) module Z = Z ( λ l , . . . , λ , π u )given by (31), the chains attached to Z are defined by Z ( λ l , . . . , λ , π u ) ! ( C λ l l , . . . , C λ , C )where C λi := ( µ i + λ, µ i − λ, . . . , µ i − ( p i + q i ) + 1 + λ ) p i ,q i , µ i = ρ − l X t = i +1 ( p t + q t )for 1 ≤ i ≤ l , and C := Λ u is the g ′ m -dominant infinitesimal character of the unipotentrepresentation π u .One reason for introducing the above notation is that the chains of Z gives the infinitesimalcharacter of L S ( Z ). Moreover, it gives us an easy way to determine the (weakly) goodnessor (weakly) fairness of L S ( Z ). Lemma 5.2.
Let Z ( λ l , . . . , λ , π u ) be an ( l , L ∩ K ) -module corresponding to the chains ( C λ l l , . . . , C λ , C ) given in Definition 5.1. Then L S ( Z ( λ l , . . . , λ , π u )) is in the good range ifand only if λ i − λ i − > − for i = l, l − , . . . , and λ + ( m + 1) > u m In other words, L S ( Z ( λ l , . . . , λ , π u )) is good if and only if (smallest entry of the i th -chain) > (largest entry of the ( i − th -chain), i = l, . . . , .And L S ( Z ( λ l , . . . , λ , π u )) is weakly good if we replace the above strict inequalities with ≥ .Moreover, L S ( Z ( λ l , . . . , λ , π u )) is in the fair range if and only if µ l − ( p l + q l )( p l + q l − > · · · > µ − ( p + q )( p + q − > . In other words, L S ( Z ( λ l , . . . , λ , π u )) is fair if and only if(average value of the entries of C λ l l ) > · · · > (average value of the entries of C λ ) > And L S ( Z ( λ l , . . . , λ , π u )) is weakly fair if we replace the above strict inequalities with ≥ . Example 5.3.
Let G = SO ∗ (14) and q = l + u is determined by H = (2 , , , , , − , − l = u (1 ,
1) + u (1 ,
1) + so ∗ (6). The chains corresponding to Z (0 , , π u ) with π u = X (1 , C = (6 , , , C = (4 , , , C = (1 , , . So L S ( Z (0 , , π u )) is in the good range by Lemma 5.2.The chains corresponding to Z ( − , − , π u ) are C − = (4 , , , C − = (3 , , , C = (1 , , . Thus L S ( Z ( − , − , π u )) is in the weakly good range. (cid:3) Example 5.4.
Let G = Sp (22 , R ), and let q be the θ -stable parabolic subalgebra of g definedby H = (1 , , , , , , , , , − , − L S ( Z ( − , X (4 ,
2; 0 , C − = (5 , , , , C = (5 , , , , , , − , − . The module L S ( Z ( − , X (4 ,
2; 0 , (cid:3) We begin our study of the Dirac index of L S ( Z ( λ l , . . . , λ , π u )) by looking at the spe-cial case when λ l = · · · = λ = 0. Note that for all π u in Section 3 and Section 4, L S ( Z (0 , . . . , , π u )) is always in good range by Lemma 5.2. Using Proposition 4.1 of [12],one easily deduce the following. Theorem 5.5.
Let G be Sp (2 n, R ) or SO ∗ (2 n ) , with θ -stable parabolic q = l + u and Z (0 , . . . , , π u ) be defined as above. Then DI( L S ( Z (0 , . . . , , π u ))) is nonzero if and only if π u has nonzero Dirac index.More precisely, let g A := u ( p, q ) where p := P lt =1 p t , q := P lt =1 q t , and q A = l A + u A be a θ -stable parabolic subalgbera of g A with ( l A ) := u ( p l , q l ) ⊕ · · · ⊕ u ( p , q ) . Suppose DI( A q A ( C n − p + q +12 )) = X u,v ǫ u,v e E ( κ u | κ v ) , DI( π u ) = X w δ w e E κ w . Then
DI( L S ( Z (0 , . . . , , π u ))) = X v,w ǫ u,v δ w e E ( κ u ; κ w ; κ v ) , where ( v , . . . , v ℓ ) is defined to be ( v , . . . , v ℓ ) := ( − v ℓ , . . . , − v ) . IRAC INDEX OF SOME UNITARY REPRESENTATIONS OF Sp (2 n, R ) AND SO ∗ (2 n ) 17 Example 5.6.
We continue with Example 5.3, where G = SO ∗ (14), and Z = Z (0 , , X (1 , n − p + q +12 = 7 − = , and by Theorem 2.4,DI( A q A ( C )) = + e E ( , − | , − )+( , | , ) − e E ( , − | , − )+( , | , ) − e E ( , − | , − )+( , | , ) + e E ( , − | , − )+( , | , ) = + e E (6 , | , − e E (5 , | , − e E (6 , | , + e E (5 , | , . Applying Theorem 4.2 to the unipotent representation π u = X (1 ,
0) of SO ∗ (6) givesDI( π u ) = − e E (1 , , − . So Theorem 5.5 implies thatDI( L S ( Z (0 , , π u ))) = − e E (6 , | , , − |− , − + e E (5 , | , , − |− , − + e E (6 , | , , − |− , − − e E (5 , | , , − |− , − . (cid:3) As for the general case when L S ( Z ( λ l , . . . , λ , π u )) is in weakly fair range, we begin bystudying all possible λ l , . . . , λ (or equivalently, C λ l l , . . . , C λ , C ) such that the infinitesimalcharacter satisfies Theorem 1.1.Indeed, one only needs to focus on the chains C λ i i that are interlaced with the ‘unipotent’chain C , since the case when C λ i i and C λ j j are interlaced is identical to the situation of U ( p, q )given in [12]. To start with, recall the chain (i.e., the infinitesimal character) correspondingto the unipotent representation X ( r, s ; ǫ, η ) in Sp (2 n, R ) is equal to C = ( n − k, . . . , k + 1 , k ; k − , k − , . . . , , , , where 2 k = r + s ≤ n ;and the chain for X ′ ( r, s ; ǫ, η ) in Sp (2 n, R ) for odd n is of the form C = ( n − , n − , . . . , , , X ( r, s ) in SO ∗ (2 n ) is of the form C = ( n − k − , . . . , k + 1; k, k, . . . , , , , where 2 k = 2( r + s ) < n. It is obvious that for any weakly fair L S ( Z ( λ l , . . . , λ , X ′ ( r, s ; ǫ, η ))) such that C λ t t and C are interlaced, the infinitesimal character of L S ( Z ( λ l , . . . , λ , X ′ ( r, s ; ǫ, η ))) violates Theorem1.1, and hence it must have zero Dirac cohomology and Dirac index.As for the other two types of unipotent representations, in order for the infinitesimalcharacter of L S ( Z ( λ l , . . . , λ , π u )) to satisfy Theorem 1.1, its chains C λ l l , . . . , C λ , C mustbe of the form:(32) ( A ; R l ) p l ,q l ( R l − ) p l − ,q l − . . . . . . ( R ) p ,q ( R l ; B l ; R l − ; B l − ; . . . ; R ; B ; K ) = C ,where K := ( ( k − , k − , . . . , , ,
0) if G = Sp (2 n, R )( k, k, . . . , , ,
0) if G = SO ∗ (2 n ) (we also allow p l = q l = 0, so that C l does not exist in (32)). For the rest of this section,we study DI( L S ( Z ( λ l , . . . , λ , π u ))) for π u = X ( r, s ; ǫ, η ) or X ( r, s ), whose chains are of theform (32).The following result strengthens Lemma 7.2.18(b) of [31]. Lemma 5.7.
Let λ ∈ h ∗ be dominant integral for ∆ + ( g , h ) which may be singular. Let F ν be a finite-dimensional ( g , K ) module with extreme weight ν . Assume that λ + ν is dominantfor ∆ + ( g , h ) , and that (33) w ( λ + ν ) = λ + µ, where µ is a weight of F ν and w ∈ W ( g , h ) . Assume moreover that (34) h ν, α i = 0 , where α is any root in ∆ + ( g , h ) such that h λ , α i = 0 . Then we must have µ = ν .Proof. By (33), we have µ = w ( λ + ν ) − λ . Thus(35) h µ, µ i = h λ + ν, λ + ν i − h w ( λ + ν ) , λ i + h λ , λ i . On the other hand, by Lemma 6.3.28 of [31],(36) w ( λ + ν ) = λ + ν − X α ∈ ∆ + ( g , h ) n α α, where n α are non-negative real numbers. Therefore, by dominance of λ ,(37) h w ( λ + ν ) , λ i ≤ h λ + ν, λ i . Substituting (37) into (35), we get h µ, µ i ≥ h λ + ν, λ + ν i − h λ + ν, λ i + h λ , λ i = h ν, ν i . We must have h µ, µ i = h ν, ν i since µ is a weight of F ν . Thus (37) must be an equality,and we conclude that λ is perpendicular to any α in (36) whose coefficient n α is positive.Combining (33) and (36), we have that µ = ν − X α ∈ ∆ + ( g , h ) , n α > n α α. Thus h ν, ν i = h µ, µ i = h ν, ν i − X α ∈ ∆ + ( g , h ) , n α > h ν, n α α i + k X α ∈ ∆ + ( g , h ) , n α > n α α k = h ν, ν i + k X α ∈ ∆ + ( g , h ) , n α > n α α k , where the last step uses (34). We conclude that X α ∈ ∆ + ( g , h ) , n α > n α α = 0 . Thus µ = ν as desired. (cid:3) IRAC INDEX OF SOME UNITARY REPRESENTATIONS OF Sp (2 n, R ) AND SO ∗ (2 n ) 19 Proposition 5.8.
Let L S ( Z ( λ l , . . . , λ , π u )) be a ( g , K ) module given by Theorem 2.5 suchthat its chains C λ l l , . . . , C λ , C satisfy (32) . Suppose the Dirac index of L S ( Z (0 , . . . , , π u )) is given in Theorem 5.5 by DI( L S ( Z (0 , . . . , , π u ))) = X z ǫ z e E κ z . Then
DI( L S ( Z ( λ l , . . . , λ , π u ))) = X z ǫ z e E κ z − ( λ ) , where ( λ ) := ( p l z }| { λ l , . . . , λ l , p l − z }| { λ l − , . . . , λ l − , . . . , p z }| { , . . . , , m z }| { , . . . , , − , . . . , − | {z } q , · · · , − λ l − , . . . , − λ l − | {z } q l − , − λ l , . . . , − λ l | {z } q l ) Proof.
For a fixed q = l + u and a fixed unipotent representation π u , Z ( λ l , . . . , λ , π u ) are inthe same coherent family [31, Definition 7.2.5] of virtual ( l , L ∩ K ) modules. Therefore, by[31, Corollary 7.2.10], the vitural ( g , K ) modules X C λll ; ... ; C λ ; C := X i ( − i R S − i ( Z ( λ l , . . . , λ , π u ))are in a coherent family. Recall that C λt is obtained by adding λ to each coordinate of C t . If Z ( λ l , . . . , λ , π u ) corresponds to chains of the form (32), it is in the weakly fair range. Thusby Theorem 2.5, X C λll ; ... ; C λ ; C = R S ( Z ( λ l , . . . , λ , π u )) = L S ( Z ( λ l , . . . , λ , π u )) . Since C is always singular, in order to obtain the Dirac index of L S ( Z ( λ l , . . . , λ , π u ))from that of L S ( Z (0 , . . . , , π u )), one needs to modify [22, Theorem 4.7] to the startinginfinitesimal character ( C l ; . . . ; C ; C ) which is singular .Indeed, by Lemma 5.7, Step 1 of [22, Theorem 4.7] remains true for all X µ such that µ isof the form(38) µ = ( µ , µ , . . . , µ x ; K ); µ ≥ µ ≥ · · · ≥ µ x , i.e., µ is in the same dominant Weyl chamber as X C l ; ... ; C ; C := L S ( Z (0 , . . . , , π u )). Inparticular, this includes all X C λll ; ... ; C λ ; C = L S ( Z ( λ l , . . . , λ , π u )) in the weakly good range.Therefore, the proposition holds for all L S ( Z ( λ l , . . . , λ , π u )) in the weakly good range.As for X C λll ; ... ; C λ ; C in weakly fair range but outside the weakly good range, the param-eter ( C λ l l ; . . . ; C λ ; C ) is no longer dominant, i.e., it is not in the same Weyl chamber as( C l ; . . . ; C ; C ). So we need to study the change of Dirac index by crossing the wall betweentwo chambers as in Step 2 – 3 of [22, Theorem 4.7].Let w ∈ W be the shortest Weyl group element mapping µ dom in the dominant Weylchamber to wµ dom = ( C λ l l ; . . . ; C λ ; C ). Then w is a permutation of the first x coordinatesof µ dom , where x is given in (38). Since the first x coordinates of µ dom and ( C λ l l ; . . . ; C λ ; C ) are strictly greater than the coordinates of K , the same arguments in Step 2 – 3 of [22,Theorem 4.7] remain valid, and the result follows. (cid:3)
Example 5.9.
We continue with Examples 5.3 and 5.6. The module L S ( Z ( − , − , π u )) haschains C − = (4 , , , C − = (3 , , , C = (1 , , L S ( Z ( − , − , π u ))) = − e E (4 , | , , − |− , − + e E (3 , | , , − |− , − + e E (4 , | , , − |− , − − e E (3 , | , , − |− , − = − e E (4 , | , , − |− , − − e E (3 , | , , − |− , − . (cid:3) Example 5.10.
We continue with Example 5.4. By Theorem 2.4,DI( A q A ( C n − p + q +12 )) = e E ( | , ) − e E ( | , ) + e E ( | , ) . Moreover, by Section 3.1, the Dirac index of X (4 ,
2; 0 ,
0) is equal to+ e E (5 , , | , , , − , − − e E (3 | , , , − , − |− , − − e E (5 , | , , , − , − |− + 2 e E (4 , | , , , − , − |− + 2 e E (5 , | , , , − , − |− − e E (2 , , , − , − |− , − , − − e E (5 | , , , − , − |− , − + e E (4 | , , , − , − |− , − . Therefore, by Proposition 5.8, the Dirac index of L S ( Z ( − , X (4 ,
2; 0 , − − e E ( , , | , , , − , − |− , − , − ) + (+1)(+2) e E ( , , | , , , − , − |− , − , − ) +(+2)(+1) e E ( , , | , , , − , − |− , − , − ) = +6 e E (5 , , | , , , − , − |− , − , − . (cid:3) Based on Proposition 5.8, we will give explicit formulas on the multiplicities of e K -typesin DI( L S ( Z ( λ l , . . . , λ , π u ))) in the following subsections.5.1. Dirac index for L S ( Z ( λ l , . . . , λ , X ( r, s ; ǫ, η ))) . We begin by studying the signs ofdifferent e K -types in DI( X ( r, s ; ǫ, η )): Lemma 5.11.
Consider the unipotent representation X ( r, s ; ǫ, η ) of Sp (2 n, R ) . Let τ = w Λ k − ρ c , τ ′ = w ′ Λ k − ρ c be two special e K -types such that w Λ k = ( i , . . . , i u , k, k − , . . . , − k + 1 , − j v , . . . , − j ) ,w ∗ Λ k = ( i ∗ , . . . , i ∗ u , k, k − , . . . , − k + 1 , − j ∗ v , . . . , − j ∗ ) , where { i , . . . , i u , j , . . . , j v } = { i ∗ , . . . , i ∗ u , j ∗ , . . . , j ∗ v } = { k + 1 , . . . , n − k − , n − k } . Let ξ ∈ S n − k denote the permutation ( i , . . . , i u | j , . . . , j v ) ( i ∗ , . . . , i ∗ u | j ∗ , . . . , j ∗ v ) . Then the sign of E τ and the sign of E τ ∗ in DI( X ( r, s ; ǫ, η )) differ by det( ξ ) . IRAC INDEX OF SOME UNITARY REPRESENTATIONS OF Sp (2 n, R ) AND SO ∗ (2 n ) 21 Proof.
Recall that the sign of E τ in DI( X ( r, s ; ǫ, η )) is ( − M ( τ ) with M ( τ ) being given in(21). Note that j ′ + · · · + j ′ v − v is equal to the length of the following permutation ζ : ( n − k, n − k − , . . . | . . . , k + 2 , k + 1) ( i , . . . , i u | j , . . . , j v ) . Similar things apply to E τ ∗ once we put ζ ∗ : ( n − k, n − k − , . . . | . . . , k + 2 , k + 1) ( i ∗ , . . . , i ∗ u | j ∗ , . . . , j ∗ v ) . Since ξ = ζ ∗ ζ − ∈ S n − k , we have that l ( ξ ) ≡ l ( ζ ∗ ) + l ( ζ − ) ≡ ( j ∗ ) ′ + · · · + ( j ∗ v ) ′ − v + j ′ + · · · + j ′ v − v (mod 2) ≡ ( j ∗ ) ′ + · · · + ( j ∗ v ) ′ − j ′ − · · · − j ′ v (mod 2)= M ( τ ∗ ) − M ( τ ) (mod 2)Therefore, ( − M ( τ ∗ ) = ( − M ( τ ) det( ξ ). (cid:3) Remark 5.12.
The same result holds if τ and τ ∗ are both non-special.Now let us compare the signs of DI( X ( r, s ; ǫ, η )) of a special τ and a non-special τ ′′ . Lemma 5.13.
Consider the unipotent representation X ( r, s ; ǫ, η ) of Sp (2 n, R ) . Let τ = w Λ k − ρ c be special with w Λ k = ( i , . . . , i u ; k, k − , . . . , − k + 1; − j v . . . , − j ) . Let τ ′′ = w ′′ Λ k − ρ c be non-special with w ′′ Λ k = ( i , . . . , i u ; k − , . . . , − k + 1 , − k ; − j v , . . . , − j ) where { i , . . . , i u , j , . . . , j v } = { k + 1 , . . . , n − k − , n − k } . Then the signs of E τ and E τ ′′ in DI( X ( r, s ; ǫ, η )) differ by ( − r = ( − s .Proof. As before, the sign of E τ in DI( X ( r, s ; ǫ, η )) is ( − M ( τ ) . On the other hand, asdiscussed in Section 3.1, the sign of the non-special E τ ′′ is ( − N ( − w K τ ′′ ) , where N ( − w K τ ′′ ) = i ′ + · · · + i ′ u + s ( η + u ) + s ( s − n ( n + 1)2 . It remains to verify that M ( τ ) + N ( − w K τ ′′ ) has the same parity as r (and s ). Indeed, M ( τ ) + N ( − w K τ ′′ )= ( n − k )( n − k + 1)2 + r ( ǫ + v ) + s ( η + u ) + r ( r − s ( s − n ( n + 1)2 ≡ n ( n + 1)2 − k (2 n + 1) + 2 k + r ( ǫ + v ) + r ( η + u ) + r ( r − s ( s − n ( n + 1)2= k + r ( ǫ + η + n − k ) + r ( r − s ( s − ≡ k + r ( ǫ + η + n ) + r ( r − s ( s − ≡ ( k + 0 + r + s = k + k ≡ r, s are even k + ǫ + η + n + r − + s − = k + ( ǫ + η + n ) + k − ≡ r, s are oddThe last step need a bit explanation. Assume that r and s are both odd. Then Theorem3.7(2) of [3] says that E τ and E − w K τ ′′ appear in H D ( r, s ; ǫ, η ) if and only if ǫ + η ≡ n (mod2).That is, if and only if ǫ + η + n is even. (cid:3) Combining the above two lemmas, we have the following result.
Proposition 5.14.
Let τ = w Λ k − ρ c and τ ∗ = w ∗ Λ k − ρ c be such that w Λ k = ( i , . . . , i u , i u +1 ; k − , . . . , − ( k − − j v , . . . , − j ); w ∗ Λ k = ( i ∗ , . . . , i ∗ u , i ∗ u +1 ; k − , . . . , − ( k − − j ∗ v , . . . , − j ∗ ) , where { i , . . . , i u , i u +1 , j , . . . , j v } = { i ∗ , . . . , i ∗ u , i ∗ u +1 , j ∗ , . . . , j ∗ v } = { k, k + 1 , . . . , n − k } . Then the signs of E τ and E τ ∗ in DI( X ( r, s ; ǫ, η )) differ by det( ξ ) , where ξ ∈ S n − k +1 is thepermutation defined by ( i , . . . , i u , i u +1 | j , . . . , j v ) ( i ∗ , . . . , i ∗ u , i ∗ u +1 | j ∗ , . . . , j ∗ v ) . Proof.
By Lemma 5.11 and Remark 5.12, the proposition holds if i u +1 = i ∗ u +1 = k (that is,if τ and τ ∗ are both special) and j v = j ∗ v = k (that is, if τ and τ ∗ are both non-special). Itremains to study the case when τ is special while τ ∗ is non-special. Thus we assume that i u +1 = k and j ∗ v = k . In particular, it suffices to check that the statement holds for w Λ k = ( i , . . . , i u , k ; k − , . . . , − k + 1; − ( k + 1) , − j v − , . . . , − j ); w ∗ Λ k = ( i , . . . , i u , k + 1; k − , . . . , − k + 1; − k, − j v − , . . . , − j ) . Let τ ♯ = w ♯ Λ k − ρ c be given by w ♯ Λ k = ( i , . . . , i u , k + 1 , k ; k − , . . . , − k + 1; − j v − , . . . , − j ) . By Lemma 5.13, in DI( X ( r, s ; ǫ, η )), we havesign( e E τ ♯ ) sign( e E τ ∗ ) = ( − r . Since M ( τ ♯ ) = j ′ + · · · + j ′ v − + r ( ǫ + v −
1) + r ( r + 1)2 , IRAC INDEX OF SOME UNITARY REPRESENTATIONS OF Sp (2 n, R ) AND SO ∗ (2 n ) 23 while M ( τ ) = j ′ + · · · + j ′ v − + 1 + r ( ǫ + v ) + r ( r + 1)2 = M ( τ ♯ ) + r + 1 , we have that sign( e E τ ) sign( e E τ ♯ ) = ( − r +1 . Therefore, sign( e E τ ) sign( e E τ ∗ ) = ( − r ( − r +1 = − . On the other hand, the permutation ξ in S n − k +1 moving w Λ k to w ∗ Λ k is( i , . . . , i u , k, j , . . . , j v − , k + 1) ( i , . . . , i u , k + 1 , j , . . . , j v − , k )which has determinant −
1. Hence the result follows. (cid:3)
From now on, given a ( g , K ) module π and a e K -type E τ , we will use | τ : DI( π ) | (resp.,[ τ : DI( π )]) for the multiplicity (resp., signed multiplicity) of E τ in the Dirac index of π . Theorem 5.15.
Let G = Sp (2 n, R ) . Consider L S ( Z ( λ l , . . . , λ , X ( r, s ; ǫ, η ))) such that itschains are of the form (32) . Let A + ∐ A − (resp., B + t ∐ B − t ) be any partition of A (resp., B t ). Then the multiplicity (cid:12)(cid:12)(cid:12)(cid:12) e E ( A + ; R l ; B + l ; ... ; R ; B +1 ; K ; A − ; R l ; B − l ; ... ; R ; B − ) : DI( L S ( Z ( λ l , . . . , λ , X ( r, s ; ǫ, η )))) (cid:12)(cid:12)(cid:12)(cid:12) (where ( j , . . . , j v ) := ( − j v , . . . , − j ) , K = ( k − , . . . , , , − , . . . , − k + 1) ) is equal to X M t ∐ N t = R t ; |M l | = q l −|A − | , |N l | = p l −|A + | ; |M t | = q t , |N t | = p t for t 1. By (39), we must have |N | = p − |A + | ≥ , |M| = q − |A − | ≥ as stated in the last part of the theorem.Fix any choice of M ∐ N = R . Put N := |N | = p − |A + | , M := |M| = q − |A − | . By Proposition 5.8, for each possibility of the e K -types of the form (39), they contribute toDI( L S ( Z ( λ, X ( r, s ; ǫ, η )))) with signed multiplicity: a M , N b M , N e E ( A + ; N ; M ; B +1 ; K ; A − ; M ; N ; B − ) = ( − MN a M , N b M , N e E ( A + ; M ; N ; B +1 ; K ; A − ; M ; N ; B − ) = ( − MN a M , N b M , N e E ( A + ; R ; B +1 ; K ; A − ; R ; B − ) . We claim that a M , N b M , N has a constant sign. Indeed, let M ∐ N = M ∗ ∐ N ∗ = R , |M| = |M ∗ | = M, |N | = |N ∗ | = N. By the knowledge of Dirac index from [16] for one-dimensional modules, a M ∗ , N ∗ = det( ξ ) a M , N , where ξ : ( A + ; N |A − ; M ) ( A + ; N ∗ |A − ; M ∗ ) . On the other hand, by Proposition 5.14, b M ∗ , N ∗ = det( ζ ) b M , N , where ζ : ( M ; B +1 |N ; B − ) ( M ∗ ; B +1 |N ∗ ; B − )It is obvious that det( ξ ) det( ζ ) = 1. Thus the claim holds, and the desired multiplicity is X M ∐ N = R ; |M| = M, |N | = N | a M , N || b M , N | = X M ∐ N = R ; |M| = M, |N | = N | b M , N | , which finishes the proof. (cid:3) Corollary 5.16. Retain the setting in Theorem 5.15. The multiplicity (cid:12)(cid:12)(cid:12)(cid:12) e E ( A + ; R l ; B + l ; ... ; R ; B +1 ; K ; A − ; R l ; B − l ; ... ; R ; B − ) : DI( L S ( Z ( λ l , . . . , λ , X ( r, s ; ǫ, η )))) (cid:12)(cid:12)(cid:12)(cid:12) is equal to (cid:18) |R l | p l − |A + | (cid:19) l − Y t =1 (cid:18) p t + q t p t (cid:19) (cid:12)(cid:12)(cid:12) e E ( m − k,m − k − ,...,m − k − w +1; K ; m − k − w,...,k +1 ,k ) : DI( X ( r, s ; ǫ, η )) (cid:12)(cid:12)(cid:12) , where w := P lt =1 ( q t + |B + t | ) − |A − | .Proof. By Proposition 3.4 and Theorem 3.7 of [3], the multiplicities in each summand ofTheorem 5.15 are all equal. So the result follows from counting the number of terms in thesummation. (cid:3) Example 5.17. We continue with the module L S ( Z ( − , X (4 , , , C − = (5 , , , C = (5 , , , , , , − , − . Thus R = { , , } , K = (2 , , , − , − , A = B = ∅ . IRAC INDEX OF SOME UNITARY REPRESENTATIONS OF Sp (2 n, R ) AND SO ∗ (2 n ) 25 By Corollary 5.16, the Dirac index of L S ( Z ( − , X (4 , 2; 0 , e K -type e E ( R ;2 , , , − , − R ) = e E (5 , , , , , − , − − , − , − = E (0 , , , , , , , , , , with w = ( q + |B +1 | ) − |A − | = 2 + 0 − (cid:18) − (cid:19) (cid:12)(cid:12)(cid:12) e E (5 , , , , − , − − : DI( X (4 , , , (cid:12)(cid:12)(cid:12) = 3 × . (cid:3) Dirac index for L S ( Z ( λ l , . . . , λ , X ( r, s ))) . Now we set G = SO ∗ (2 n ) and π u = X ( r, s ). To obtain an result on the Dirac index of L S ( Z ( λ l , . . . , λ , X ( r, s ))) analogousto Theorem 5.15, one needs to know the signs of e E τ in DI( X ( r, s )) (c.f. Proposition 5.14).Indeed, (29) gives the sign of each e K -type appearing in DI( X ( r, s )), which immediatelyimplies the following. Proposition 5.18. Let τ = w Λ k − ρ c , τ ∗ = w ∗ Λ k − ρ c such that w Λ k = ( i , . . . , i u ; k, . . . , − k ; − j v , . . . , − j ) ,w ∗ Λ k = ( i ∗ , . . . , i ∗ u ; k, . . . , − k ; − j ∗ v , . . . , − j ∗ ) , where { i , . . . , i u , j , . . . , j v } = { i ∗ , . . . , i ∗ u , j ∗ , . . . , j ∗ v } = { k + 1 , . . . , n − k − } . Then the sign of E τ and E τ ∗ appearing in DI( X ( r, s )) differ by det( ξ ) , where ξ ∈ S n − k − is the permutation defined by ( i , . . . , i u | j , . . . , j v ) ( i ∗ , . . . , i ∗ u | j ∗ , . . . , j ∗ v ) . Now we can state the following result for DI( L S ( Z ( λ l , . . . , λ , X ( r, s )))). Theorem 5.19. Let G be SO ∗ (2 n ) . Consider L S ( Z ( λ l , . . . , λ , X ( r, s ))) such that its chainsare of the form (32) . Let A + ∐ A − (resp., B + t ∐ B − t ) be any partition of A (resp., B t ). Thenthe multiplicity (cid:12)(cid:12)(cid:12)(cid:12) e E ( A + ; R l ; B + l ; ... ; R ; B +1 ; K ; A − ; R l ; B − l ; ... ; R ; B − ) : DI( L S ( Z ( λ l , . . . , λ , X ( r, s )))) (cid:12)(cid:12)(cid:12)(cid:12) (here K = ( k, . . . , , , − , . . . , − k ) ) is equal to (cid:18) r + sr (cid:19)(cid:18) |R l | p l − |A + | (cid:19) l − Y t =1 (cid:18) p t + q t p t (cid:19) if p l ≥ |A + | and q l ≥ |A − | . Otherwise, the multiplicity is equal to zero.Proof. The proof is analogous to that of Theorem 5.15 and Corollary 5.16. Namely, themultiplicity is equal to X M t ∐ N t = R t ; |M l | = q l −|A − | , |N l | = p l −|A + | ; |M t | = q t , |N t | = p t for t An analogous statement for Corollary 5.16 and Theorem 5.19 holds alsowhen π u = triv is the trivial representation of Sp (2 m, R ) or SO ∗ (2 m ). Under this setting, C = ( ( m, . . . , , 1) for G = Sp (2 m, R );( m − , . . . , , 0) for G = SO ∗ (2 m )and L S ( Z ( λ l , . . . , λ , triv)) is a weakly fair A q ( λ )-module since triv is one-dimensional, andhence Theorem 2.4 applies. Note that the infinitesimal character of L S ( Z ( λ l , . . . , λ , triv))satisfies Theorem 1.1 if its chains satisfy(40) ( A ; R l ) p l ,q l ( R l − ) p l − ,q l − . . . . . . ( R ) p ,q ( R l ; B l ; R l − ; B l − ; . . . ; R ; B ) = C , , or when G = Sp (2 m, R ) and 0 ∈ C , we may have(41) ( A ; R l ) p l ,q l ( R l − ) p l − ,q l − . . . . . . ( R ; 0) p ,q ( R l ; B l ; R l − ; B l − ; . . . ; R ) = C , , or(42) C = ( A ; R ; 0) p ,q ( R ) = C , . It turns out that in all cases, the multiplicity (cid:12)(cid:12)(cid:12)(cid:12) e E ( A + ; R l ; B + l ; ... ; R ; B +1 ; A − ; R l ; B − l ; ... ; R ; B − ) : DI( L S ( Z ( λ l , . . . , λ , triv))) (cid:12)(cid:12)(cid:12)(cid:12) in (40) , or (cid:12)(cid:12)(cid:12)(cid:12) e E ( A + ; R l ; B + l ; ... ; R ; 0; A − ; R l ; B − l ; ... ; R ) : DI( L S ( Z ( λ l , . . . , λ , triv))) (cid:12)(cid:12)(cid:12)(cid:12) in (41) , or (cid:12)(cid:12)(cid:12) e E ( A + ; R ; 0; A − ; R l ) : DI( L S ( Z ( λ l , . . . , λ , triv))) (cid:12)(cid:12)(cid:12) in (42)are equal to (cid:18) |R l | p l − |A + | (cid:19) l − Y t =1 (cid:18) p t + q t p t (cid:19) (in the third case, we have l = 1 and the second term above will not show up).6. Parity of spin-lowest K -types and cancellation in the Dirac index Let π be an irreducible unitary ( g , K ) module. Fix a K -type µ of π . Let µ be any K -typeof π . We assume that(43) h µ − µ , ζ i ∈ Z We call the parity of this integer the parity of µ , and denote it by p ( µ ). Definition 6.1. We say that the K -type E µ of π is related to the e K -type E ν of H D ( π ) if E ν is a PRV-component [25] of E µ ⊗ Spin G .The following result aims to clarify the link between the possible cancellations of e K -typesin H D ( π ) and the parities of the spin-lowest K -types of π . IRAC INDEX OF SOME UNITARY REPRESENTATIONS OF Sp (2 n, R ) AND SO ∗ (2 n ) 27 Theorem 6.2. Let π be an irreducible unitary ( g , K ) module such that (43) holds. Then Hom e K ( H + D ( π ) , H − D ( π )) = 0 if and only if for each e K -type E ν of H D ( π ) (if exists), all thespin lowest K -types of π which are related to E ν have the same parity.Proof. It suffices to consider the case that H D ( π ) = 0. Then Hom e K ( H + D ( π ) , H − D ( π )) = 0 ifand only if the occurrences of E ν either all live in H + D ( π ), or all live in H − D ( π ). Here E ν runs over all the distinct e K -components of H D ( π ).Take two arbitrary K -types E µ and E µ of π which are related to the e K -type E ν . Thenthere exist w , w ∈ W ( g , t ) such that { ( w ρ − ρ c ) + w K µ } = { ( w ρ − ρ c ) + w K µ } = ν. Here w K stands for the unique longest element of W ( k , t ). Removing the two brackets, wehave(44) ( w ρ − ρ c ) + µ − ( µ − w K µ ) + γ = ( w ρ − ρ c ) + µ − ( µ − w K µ ) + γ , where each γ i is a non-negative integer combination of roots in ∆ + ( k , t ), and so is each µ i − w K µ i . Therefore, µ − ( ρ − w ρ ) − ( µ − w K µ ) + γ = µ − ( ρ − w ρ ) − ( µ − w K µ ) + γ . In other words, µ − h Φ w i − ( µ − w K µ ) + γ = µ − h Φ w i − ( µ − w K µ ) + γ . Taking inner products with ζ and passing to mod 2, we have that(45) p ( µ ) + l ( w ) ≡ p ( µ ) + l ( w ) (mod 2)by using (7). Now the desired conclusion follows from (45) and Lemma 2.3. (cid:3) Example 6.3. Let us consider the following irreducible unitary representation π of Sp (10 , R ). G:Sp(10,R)set p=parameter (KGB (G)[444],[4,2,4,0,1]/1,[1,0,2,-1,1]/1)is_unitary(p)Value: trueprint_branch_irr_long (p,KGB (G,31), 65)m x lambda hw dim height1 179 [ 2, 2, 2, 2, 1 ]/1 [ 2, -1, -1, -3, -3 ] 1200 421 32 [ 3, 2, 2, 0, 0 ]/1 [ 3, 0, -1, -3, -3 ] 5400 501 4 [ 3, 2, 2, 0, 0 ]/1 [ 2, -1, -2, -3, -4 ] 5120 511 91 [ 3, 3, 2, 0, 1 ]/1 [ 2, -2, -2, -4, -4 ] 2250 571 180 [ 3, 3, 2, 2, 1 ]/1 [ 4, -1, -1, -3, -3 ] 3850 581 179 [ 3, 3, 2, 2, 1 ]/1 [ 3, -1, -1, -3, -4 ] 7425 581 179 [ 4, 2, 2, 2, 1 ]/1 [ 2, -1, -1, -3, -5 ] 9240 601 56 [ 3, 3, 2, 1, 0 ]/1 [ 3, 0, -2, -3, -4 ] 16170 601 1 [ 3, 3, 2, 1, 0 ]/1 [ 4, 1, -1, -3, -3 ] 16170 611 4 [ 4, 2, 2, 1, 0 ]/1 [ 2, -1, -3, -3, -5 ] 8624 631 4 [ 3, 3, 3, 0, 0 ]/1 [ 3, -1, -2, -4, -4 ] 9625 63 By Theorem 1.1, the atlas height of any spin-lowest K -type of π is less than or equal to55. It turns out that the first three K -types are exactly all the spin lowest K -types of π : µ = (2 , − , − , − , − , µ = (3 , , − , − , − , µ = (2 , − , − , − , − . Recall that ζ = ( , , , , ). Thus µ and µ have the same parity, which is opposite tothat of µ . Moreover, µ contributes E (1 , , , , to H D ( π ), while µ and µ both contribute E (0 , , − , − , − . We conclude that there is no cancellation when passing from H D ( π ) to DI( π ).We now use results in the previous sections to compute DI( π ). Indeed, π = L S ( − , triv)corresponds to the chains C − = (3 , , , , C = (2 , A = { } , R = { , } , and | e E ( ;2 , , : DI( π ) | = | e E (3 , , , − , − : DI( π ) | = | E (1 , , , , : DI( π ) | = (cid:18) − (cid:19) = 1 , | e E (2 , , , : DI( π ) | = | e E (2 , , − , − , − : DI( π ) | = | E (0 , , − , − , − : DI( π ) | = (cid:18) − (cid:19) = 2 . (cid:3) Example 6.4. Let us revisit Example 4.4 of [12]. Adopt the setting there, and let ̟ , . . . , ̟ be the fundamental weights for ∆ + ( k , t ). We have that ζ = ̟ = (1 , , , . We use [ a, b, c, d ] to stand for the k -type a̟ + b̟ + c̟ + d̟ . Note that w K = − 1. Forthe A q ( λ ) module, we note that∆( u , t ) ⊆ ∆ + ( k , t ) ∪ w (1) ∆ + ( p , t ) , where w (1) = s α ∈ W ( g , t ) . Then one can figure out that ∆( u ∩ p , t ) consists of the followingeight roots:(0 , − , , , (0 , , , , ( 12 , , , − 12 ) , ( 12 , , , 12 ) , (1 , , , − , (1 , , , , (1 , , , , (1 , , , . Here the first root (0 , − , , 0) = − α , and it is the unique one in ∆ − ( p , t ).One identifies that λ + 2 ρ ( u ∩ p ) = [0 , , , K -type of the A q ( λ ). As computed in Example 6.3 of [8], there is only one e K -type in H D ( A q ( λ )), i.e., E ν with ν = [0 , , , K -types of A q ( λ ) in total, both of which are related to the e K -type E ν : µ := [0 , , , 4] = λ + 2 ρ ( u ∩ p ) + (1 , , , , µ := [0 , , , 1] = λ + 2 ρ ( u ∩ p ) + 2( − α ) . Therefore, µ and µ have distinct parities. By (45), it must happen that one E ν lives in H + D ( A q ( λ )), while the other E ν lives in H − D ( A q ( λ )). Thus they cancel in DI( A q ( λ )), whichthen vanishes. Indeed, as been explicitly obtained in Example 6.3 of [8], w = w (2) = s s , w = w (10) = s s s s s s s . Thus the E ν from E µ ⊗ Spin G lives in H + D ( A q ( λ )), while the E ν from E µ ⊗ Spin G lives in H − D ( A q ( λ )). (cid:3) IRAC INDEX OF SOME UNITARY REPRESENTATIONS OF Sp (2 n, R ) AND SO ∗ (2 n ) 29 Funding Dong was supported by the National Natural Science Foundation of China (grant 11571097,2016-2019). Wong is supported by the National Natural Science Foundation of China (grant11901491) and the Presidential Fund of CUHK(SZ). Acknowledgements We thank Professor Vogan sincerely for guiding us through coherent families. References [1] J. Adams, M. van Leeuwen, P. Trapa and D. Vogan, Unitary representations of real reductive groups ,Ast´erisque (2020).[2] D. 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(2) (1984), no. 1,141–187.[33] D. Vogan, Dirac operators and unitary representations School of Mathematical Sciences, Soochow University, Suzhou 215006, P. R. China Email address : [email protected] (Wong) School of Science and Engineering, The Chinese University of Hong Kong, Shenzhen,Guangdong 518172, P. R. China Email address ::