PAVLE PAND ŽI Salah Mehdi

Pavle Pandžić
email: pandzic@math.hr

David Vogan

Roger Zierau
email: roger.zierau@okstate.edu

Computing the associated cycles of certain harish-chandra modules

Keywords: 2010 Mathematics Subject Classification. Primary 22E47; Secondary 22E46 (g, K)-module, Dirac cohomology, Dirac index, nilpotent orbit, associated variety, associated cycle, Springer correspondence

Let G R be a simple real linear Lie group with maximal compact subgroup K R and assume that rank(G R ) = rank(K R ). In [MPVZ] we proved that for any representation X of Gelfand-Kirillov dimension 1 2 dim(G R /K R ), the polynomial on the dual of a compact Cartan subalgebra given by the dimension of the Dirac index of members of the coherent family containing X is a linear combination, with integer coefficients, of the multiplicities of the irreducible components occurring in the associated cycle. In this paper we compute these coefficients explicitly.

Introduction

Let G R be a simple real linear Lie group with a Cartan involution θ and maximal compact subgroup K R = G θ R . Let g = k ⊕ p be the Cartan decomposition of the complexified Lie algebra g of G R ; this decomposition is orthogonal with respect to the Killing form B. Let K be the complexification of K R and G a complex Lie group (with Lie algebra g) containing K as the set of fixed points of the complex extension of θ. We assume throughout the paper that g and k have equal rank, i.e., there is a Cartan subalgebra h of g contained in k. We fix such h and write W for the Weyl group of (g, h).

In this paper we are concerned with comparing two important invariants of (g, K R )-modules. One is the Dirac index studied in [MPV]. It is defined using the Dirac operator D ∈ U (g) ⊗ C(p), where U (g) is the universal enveloping algebra of g and C(p) is the Clifford algebra of p with respect to B. If M is a (g, K R )-module, then D acts on M ⊗ S where S is a spin module for C(p). The Dirac cohomology of M is defined as

H D (M ) = Ker(D)/(Im(D) ∩ Ker(D);
it is a module for the spin double cover K of K R (finite-dimensional if M is admissible). This invariant was introduced in [V2], it turned out to be very interesting and also quite computable; see for example [START_REF] Huang | Dirac cohomology, unitary representations and a proof of a conjecture of Vogan[END_REF], [START_REF] Huang | Dirac Operators in Representation Theory[END_REF], [HKP], [HPR], [HPP], [HPZ], [START_REF] Barbasch | Dirac cohomology and unipotent representations of complex groups[END_REF], [START_REF] Barbasch | Dirac cohomology of unipotent representations of Sp(2n, R) and U (p, q)[END_REF], [BPT], [MP], [MZ], [DH].

Decomposing the K-module S as S = S + ⊕ S -induces a decomposition of Dirac cohomology

H D (M ) = H D (M ) + ⊕ H D (M ) -.
The Dirac index of M is then defined as the virtual K-module

DI v (M ) = H D (M ) + -H D (M ) -.
It is proved in [MPV] that Dirac index varies nicely over coherent families of (g, K R )modules. In particular, if {M λ } is such a coherent family, attached to a module M , then the function λ → dim DI v (M λ ) extends to a polynomial on h * , which we denote by DI p (M ).

Another very useful invariant of a Harish-Chandra module M is its associated cycle AC (M ), defined in [V1]. See [MPVZ] for a short review of the definition.

In concrete terms, for irreducible M , AC(M ) can be written as the formal sum

AC(M ) = i m i (M )O i ,
where O i ⊂ p are the real forms of a complex nilpotent G-orbit O C ⊂ g, and the multiplicities m i (M ) are nonnegative integers. The orbit O C is specified by the requirement that O C is the associated variety of the annihilator of M . If M is put into a coherent family {M λ }, then the corresponding multiplicities extend to polynomials m i (M ) on h * . It was conjectured in [MPV], and proved in [MPVZ], that in certain special circumstances these multiplicity polynomials are related to the Dirac index polynomial by

DI p (M ) = i c i m i (M )
for some integers c i . Such a relationship is true when the associated variety of the annihilator of M is contained in O C , with O C corresponding via Springer correspondence to the W -representation generated by the Weyl dimension polynomial P K for K (P K is defined by (3.1)).

The purpose of this paper is to complement [MPVZ] by explicitly computing the constants c i in the classical cases other than SU (p, q). The case G R = SU (p, q) as well as the case of exceptional groups are done in [MPVZ] We start by reviewing some facts about real forms of nilpotent orbits in Section 2, and assembling a few useful general facts about the computations in Section 3. Then we do the case-by-case computations in Sections 4 -8.

Nilpotent orbits and their real forms

We recall that the list of the classical real groups for which the conjecture from [MPV] applies is given in [MPVZ], Section 6, Table 1, along with the relevant explanations. The groups on the list are the connected classical equal rank groups such that the W -representation σ K generated by the Weyl dimension polynomial P K for K is Springer. The list consists of • SU (p, q), q ≥ p ≥ 1;

• SO e (2p, 2q + 1), q ≥ p -1 ≥ 0;

• Sp(2n, R), n ≥ 1;

• SO * (2n), n ≥ 1;

• SO e (2p, 2q), q ≥ p ≥ 1.

The table in [MPVZ] also includes the nilpotent orbits O C corresponding to σ K in each of the cases, as well as the number of real forms of these orbits. Here we explain how to get these real forms, and in particular how to write down the semisimple elements h of the corresponding sl 2 -triples, which we need to begin our computations. We start by recalling that complex nilpotent orbits in classical Lie algebras are in one-to-one correspondence with the set of partitions

[d 1 , • • • , d k ] with d 1 ≥ d 2 ≥ • • • ≥ d k ≥ 1 (if d j
occurs m times, we will simply write d m j ) such that (see [CM], Chapter 5):

• d 1 + d 2 + • • • + d k = n, when g sl(n, C); • d 1 + d 2 + • • • + d k = 2n
+ 1 and the even d j occur with even multiplicity, when g so(2n + 1, C);

• d 1 + d 2 + • • • + d k = 2n
and the odd d j occur with even multiplicity, when g sp(2n, C);

• d 1 + d 2 + • • • + d k = 2n
and the even d j occur with even multiplicity, when g so(2n, C); except that the partitions having all the d j even and occurring with even multiplicity are each associated to two orbits.

We now recall the procedure which attaches sl 2 -triples to complex nilpotent orbits (see Chapter 3 in [CM]). For a positive integer i, define the Jordan block J i to be the i × i matrix Let H = 1 0 0 -1 , X = J 2 = 0 1 0 0 and Y = J t 2 = 0 0 1 0 . Then [H, X] = 2X, [H, Y ] = -2Y and [X, Y ] = H, so that H, X, Y span, over C, the simple Lie algebra sl 2 of 2 × 2 complex matrices with zero trace. For a non-negative integer r, define the linear map ρ r : sl 2 → sl r+1 by

J i =         0 1 0 0 • • • 0 0 0 1 0 • • • 0 • • • • • • • • • • • • • • • • 0 0 0 • • • 0 1 0 0 0 0 • • • 0         For a postive integer n, write [d 1 , d 2 , • • • , d k ] for a partition of n. Define the n × n matrix X [d1,d2,••• ,d k ] =       J d1 0 0 0 • • • 0 0 J d2 0 0 • • • 0 • • • • • • • • • • • • • • • • 0 0 0 0 • • • J d k       Then X [d1,d2
ρ r (H) =         r 0 0 0 • • • 0 0 r -2 0 0 • • • 0 • • • • • • • • • • • • • • • • 0 • • • -r + 2 0 0 • • • 0 -r         ρ r (X) = J r+1 ρ r (Y ) =         0 0 0 0 • • • 0 µ 1 0 0 0 • • • 0 0 µ 2 0 0 • • • 0 • • • • • • • • • • • • • • • • 0 • • • µ r 0         with µ i = i(r + 1 -i) for 1 ≤ i ≤ r
ρ r defines an irreducible representation of sl 2 of dimension r + 1, and any finite dimensional irreducible representation of sl 2 is equivalent to ρ r for some r. The map ρ r induces the homomorphism Φ O : sl 2 → sl n defined by:

Φ O = 1≤j≤k ρ dj -1 so that Φ O (X) = X [d1,d2,••• ,d k ] . The standard sl 2 -triple associated with the com- plex nilpotent orbit O [d1,d2,••• ,d k ] is {H [d1,d2,••• ,d k ] ; X [d1,d2,••• ,d k ] ; Y [d1,d2,••• ,d k ] } where H [d1,d2,••• ,d k ] := Φ O (H), X [d1,d2,••• ,d k ] := Φ O (X) and Y [d1,d2,••• ,d k ] := Φ O (Y ). Choose the Cartan subalgebra consisting of n × n diagonal matrices diag(a 1 , a 2 , • • • , a n )
with zero trace, and fix the positive system of roots { ij | 1 ≤ i < j ≤ n} whose corresponding Borel subalgebra consists of the upper tringular matrices of zero trace. Here i is the complex linear form defined on the Cartan subalgebra

such that i (diag(a 1 , a 2 , • • • , a n )) = a i . Then up to a Weyl group element of sl n , the element H [d1,d2,••• ,d k ] is conjugate to a dominant element (h 1 , h 2 , • • • , h n ) := diag(h 1 , h 2 , • • • , h n ) with h 1 ≥ h 2 ≥ • • • ≥ h n and h 1 + h 2 + • • • + h n = 0. Associated with the orbit O [d1,d2,••• ,d k ] is the weighted Dynkin diagram h1-h2 • h2-h3 • • • • • • • hn-1-hn • Suppose now that G R = SU (p, q), K R = S(U (p) × U (q)
) and g = sl p+q , with q ≥ p ≥ 1 and p + q = n. The dominant h associated with the complex nilpotent orbit

O C = O [2 p ;1 q-p ] is given by h = (1, 1, • • • , 1 p , 0, 0, • • • , 0 q-p , -1, -1, • • • , -1 p ) along with the weighted Dynkin diagram 0 • 1-2 • • • • • • 1 • p-p+1 0 • p+1-p+2 • • • • • • 1 • q -q+1 0 • q+1-q+2 • • • • • • 0 • p+q -p+q-1
Moreover, by Kostant-Sekiguchi, it is known that the nilpotent orbit O C has p + 1 real forms which are in one to one correspondence with nilpotent K-orbits in p. More precisely, for k = 0, 1, 2, • • • , p, let I 0 = 0 and

e k =     0 p I k 0 0 0 0 0 0 I p-k 0 q    
For each k, the element e k belongs to O C . On the other hand, the K-orbit of e k consists of matrices of the form 0 A B 0 with rank(A) = k and rank(B) = p -k

In particular, if k = k then the K-orbits of e k and e k are disjoint. Choosing the positive system

{ i -j | 1 ≤ i < j ≤ p or p + 1 ≤ i < j ≤ p + q}, the (dominant) neutral element of the sl 2 -triple corresponding to the real form K • e k is h k = (1, 1, • • • , 1 k , -1, -1, • • • , -1 p-k , 1, 1, • • • , 1 p-k , 0, 0, • • • , 0 q-p , -1, -1, • • • , -1 k )
The description in terms of Young tableaux of the complex orbit O C and of its real forms is as follows:

O C : :    p :    q -p real forms for O C + - : : + -    k ∈ {0, . . . , p} -+ : : -+    p -k - : -    q -p Consider G R = Sp(2n, R) and K R = U (n). The complexification g = sp 2n of G R is realized as the following set of matrices Z 1 Z 2 Z 3 -Z t 1 | Z 1 n × n complex matrices, Z 2 , Z 3 symmetric complex matrices A Cartan subalgebra in g consists of diagonal complex matrices of the form diag(a 1 , a 2 , • • • , a n , -a 1 , -a 2 , • • • , -a n ). Fix the standard system of positive roots { i ± j | 1 ≤ i < j ≤ n} ∪ {2 k | 1 ≤ k ≤ n}.
As in type A, there is an explicit recipe which attaches an sl 2 -triple to a complex nilpotent orbit (see 5.2.2 in [CM]). We apply this recipe to the nilpotent orbit O C = O [2 n ] , using n chunks coninciding with {2}. We obtain (viewing sp 2n as a subalgebra of sl 2n )

h := diag(1, 1, • • • , 1 n , -1, -1, • • • , -1 n ) which we will simply write h = (1, 1, • • • , 1 n ) along with the weighted Dynkin diagram 0 • 1-2 0 • 2-3 • • • • • • 0 • n-1-n 2 • 2 n
The same argument as in type A shows that O C possesses n + 1 real forms with

h k = (1, 1, • • • , 1 k , -1, -1, • • • , -1 n-k , -1, -1, • • • , -1 k , 1, 1, • • • , 1 n-k )
The description in terms of Young tableaux of the complex orbit O C and of its real forms is as follows:

O C : :          n real forms + - : : + -    k ∈ {0, . . . , n} -+ : : -+    n -k Consider G R = SO e (2p, 2q + 1) and K R = S(O(2p) × O(2q + 1)
). The complexification g = so 2n+1 of G R , with n = p + q and q ≥ p ≥ 1, is realized as the following set of matrices

  0 u v -v t Z 1 Z 2 -u t Z 3 -Z t 1   | u, v ∈ C n , Z 1 n × n complex, Z 2 , Z 3 skew-symmetric A Cartan subalgebra in g consists of diagonal complex matrices of the form diag(0, a 1 , a 2 , • • • , a n , -a 1 , -a 2 , • • • , -a n ) (first
row and column of zeros). Fix the standard system of positive roots

{ i ± j | 2 ≤ i < j ≤ n + 1} ∪ { k | 2 ≤ k ≤ n + 1}.
There is an explicit recipe which attaches an sl 2 -triple to a complex nilpotent orbit (see 5.2.4 in [CM]). We apply this recipe to the nilpotent orbit

O C = O [3,2 2p-2 ,1 2(q-p+1
) ] , using the following chunks : {3}, p-1 {2; 2}'s and q -p+1 {1; 1}'s. We obtain (viewing so 2n+1 as a subalgebra of sl 2n+1 )

h := diag(0, 2, 1, 1, • • • , 1 2p-2 , 0, 0, • • • , 0 2(q-p+1) , -1, -1, • • • , -1 2p-2 , -2)
which we will simply write (dropping the first zero coordinate and shifting indices of i 's)

h = (2, 1, 1, • • • , 1 2p-2 , 0, 0, • • • , 0 q-p+1 ) along with the weighted Dynkin diagram 1 • 1-2 0 • 2-3 • • • • • • 1 • 2p -2p+1 0 • 2p+1-2p+2 • • • • • • 0 • p+q-1-p+q > > 0 • p+q
The nilpotent orbit O C possesses 2 or 3 real forms depending wether q > p -1 or not. The description in terms of Young tableaux of the complex orbit O C and of its real forms is given below. The recipe to produce the real h's from the signed tableau can be stated as follows: the first row of length 3 gives a 2 in the first p coordinates if the row starts with a "+" and a 2 in the p + 1 coordinate if it starts with a "-". For the rows of length two, a "+" (resp. "-") sign in the leftmost box provides +1 in the first p coordinates (resp. in the second group of coordinates p + 1, • • • ); a "+" (resp. "-") sign in the rightmost box provides -1 in the first p coordinates (resp. in the second group of coordinates p + 1, • • • ). In particular, we get

h I 1 = (2, 1, 1, • • • , 1 p-1 , 1, 1, • • • , 1 p-1 , 0, 0, • • • , 0 q-p+1 ) h II 1 = (2, 1, 1, • • • , -1 p-1 , 1, 1, • • • , 1 p-1 , 0, 0, • • • , 0 q-p+1 ) h 2 = (1, 1, • • • , 1 p-1 , 0, 2, 1, 1, • • • , 1 p-1 , 0, 0, • • • , 0 q-p ) only if q > p -1 h II 1 is obtained from h I 1 by the outer automorphism p-1 + p ←→ p-1 -p : • 1-2 • 2-3 • • • • • p-1+ p • p-1-p
The description of the orbit O C and its real forms in terms of Young tableaux is as follows:

O C : :    2p -2 :    2(q -p + 1) + -+ real forms for O C + - : : + -    2p -2 I, II - : -    2(q -p + 1) -+ - + - : : + -    2p -2 only if q > p -1 - : -    2(q -p) + 1
Consider G R = SO e (2p, 2q) and K R = SO(2p) × SO(2q). The complexification g = so 2n of G R , with n = p + q and q ≥ p ≥ 1, is realized as the following set of matrices

Z 1 Z 2 Z 3 -Z t 1 | Z i n × n complex matrices, Z 2 , Z 3 skew-symmetric A Cartan subalgebra in g consists of diagonal complex matrices of the form diag(a 1 , a 2 , • • • , a n , -a 1 , -a 2 , • • • , -a n ). Fix the standard system of positive roots { i ± j | 1 ≤ i < j ≤ n}.
There is an explicit recipe which attaches an sl 2 -triple to a complex nilpotent orbit (see 5.2.6 in [CM]). We apply this recipe to the nilpotent orbit O C = O [3,2 2p-2 ,1 2(q-p)+1 ] , using the following chunks : {3; 1}, p -1 {2; 2}'s and q -p {1; 1}'s. We obtain (viewing so 2n as a subalgebra of sl 2n )

h := diag(2, 1, 1, • • • , 1 2p-2 , 0, 0, • • • , 0 2(q-p+1) , -1, -1, • • • , -1 2p-2 , -2)
which we will simply write

h = (2, 1, 1, • • • , 1 2p-2 , 0, 0, • • • , 0 q-p+1
) along with the weighted Dynkin diagram

1-2 • 2 • 0 • • • 2p-1-2p • 1 2p -2p+1 • 0 • • • • 0 • 0 p+q-1-p+q • 0
The nilpotent orbit O C possesses 3 or 4 real forms depending whether q > p or not. Using a recipe analogous to the one used for type B, we get

h I 1 = (2, 1, 1, • • • , 1 p-1 , 1, 1, • • • , 1 p-1 , 0, 0, • • • , 0 q-p+1 ) h II 1 = (2, 1, 1, • • • , -1 p-1 , 1, 1, • • • , 1 p-1 , 0, 0, • • • , 0 q-p+1 ) h I 2 = (1, 1, • • • , 1 p-1 , 0, 2, 1, 1, • • • , 1 p-1 , 0, 0, • • • , 0 q-p ) h II 2 = (1, 1, • • • , 1 p-1 , 0, 2, 1, 1, • • • , -1 p-1 , 0, 0, • • • , 0 q-p ) only if q = p
As before, h II i is obtained from h I i by the outer automorphism:

• 1-2 • 2-3 • • • • • p-1+ p • p-1-p
The description of the complex orbit O C and its real forms in terms of Young tableaux is as follows:

O C : :    2p -2 :    2(q -p) + 1 + -+ real forms for O C + - : : + -    2p -2 I, II - : -    2(q -p) + 1 -+ - + - : : + -    2p -2 (I, II if q = p) - : -    q -p Consider G R = SO * (2n) = SO(2n, C) ∩ gl(n, H), K R = U (n) and g = so * 2n . For so 2n , a Cartan subalgebra in g consists of diagonal complex matrices of the form diag(a 1 , a 2 , • • • , a n , -a 1 , -a 2 , • • • , -a n ). Fix the standard system of positive roots { i ± j | 1 ≤ i < j ≤ n}.
Using an recipe analogous to that of type D, one can attache an sl 2 -triple to a complex nilpotent orbit. We apply this recipe to the nilpotent orbit O C = O [2 n ] to obtain (viewing so * 2n as a subalgebra of sl 2n )

h := diag(1, 1, • • • , 1 n , -1, -1, • • • , -1 n ) which we will simply write h = (1, 1, • • • , 1 n ) along with the weighted Dynkin diagram 1-2 • 0 • 0 • • • • • • • 0 • 0 n-1-n • 2
The nilpotent orbit O C possesses n 2 + 1 real forms if n is even, and n+1 2 real forms otherwise. Using a recipe analogous to the one used for type D, we get

h k = (1, 1, • • • , 1 2k , -1, -1, • • • , -1 n-2k
) for n even, and

k = 0, • • • , n 2
h k = (1, 1, • • • , 1 2k , 0, -1, -1, • • • , -1 n-2k-1
) for n odd, and

k = 0, • • • , n -1 2
Finally, the description of the complex orbit O C and its real forms in terms of Young tableaux is as follows:

n even:

O C : :          n real forms + - : : + -    k ∈ {0, . . . , n 2 } -+ : : -+    n 2 -k n odd: O C : :    n -1 2 real forms + - : : + -    k ∈ {0, . . . , n-1 2 } -+ : : -+    n-1 2 -k + 3. Some general facts
Let O i be a real form of the orbit O C , and denote the corresponding semisimple element of the (normal) sl(2)-triple by h ∈ h. As in [MPVZ], we attach to h the θ-stable parabolic subalgebra q = l ⊕ u such that l is the centralizer of h in g, and u is the sum of negative eigenspaces for ad h on g. We fix a choice of ∆ + = ∆ + (g, h); in examples, this will always be the standard positive root system. This defines a choice ∆

+ c = ∆ + (k, h) = ∆(k, h)∩∆ + of positive compact roots. Let ∆ + n := ∆ + \∆ + c
be the set of positive noncompact roots. Denote by ρ c (resp. ρ n ) half the sum of positive compact (resp. noncompact) roots. The Weyl dimension polynomial (3.1)

P K (λ) = α∈∆ + c λ, α ρ c , α
will always be defined with respect to this fixed positive root system ∆ + c . If A is a subset of ∆(g, h), write ρ(A) for half the sum of the roots in A.

We choose ∆ + (l) compatibly with ∆ + , i.e., ∆ + (l) = ∆(l) ∩ ∆ + . This also gives a choice for positive roots of l ∩ k, and fixes the Weyl dimension polynomial P L∩K . Denote by ∆ + n (l) the set of noncompact roots in ∆ + (l), and by ∆(p 1 ) the set of noncompact roots that are 1 on h.

The constant c = c i we are going to compute is attached to O i as in [MPVZ]. It is defined by equation ( 6.4) of [MPVZ]; this is up to sign the same equation as (5.9) of [MPVZ], but the sign is made precise using [MPVZ], Remark 3.8, equation (6.1) and the discussion around (6.1). The equation is

(3.2) (-1) N A⊆∆ + n (l) C⊆∆(p 1 ) (-1) #A+#C P K (λ -ρ n (l) + 2ρ(A) -2ρ(C)) = cP L∩K (λ), where (3.3) N = #{α ∈ ∆ + α(h) > 0}.
The computations we are going to make will be easier if equation (3.2) is turned into an analogue of equation (6.5) of [MPVZ]:

Proposition 3.4. Assume that ρ n (l) is orthogonal to all roots of l ∩ k. Then (3.5) (-1) N +#∆ + n (l) A⊆∆ + n (l) C⊆∆(p 1 ) (-1) #A+#C P K (λ -2ρ(A) -2ρ(C)) = cP L∩K (λ).
Proof. This follows by passing from summation over A to summation over the complement of A in ∆ + n (l). For any

A ⊆ ∆ + n (l), -ρ n (l) + 2ρ(A) = ρ n (l) -2ρ(∆ + n (l) \ A) and (-1) #A = (-1) #∆ + n (l) (-1) #(∆ + n (l)\A) , so (3.2) can be rewritten as (-1) N +#∆ + n (l) A⊆∆ + n (l) C⊆∆(p 1 ) (-1) #A+#C P K (λ + ρ n (l) -2ρ(A) -2ρ(C)) = cP L∩K (λ).
We now replace λ by λ -ρ n (l); since ρ n (l) is orthogonal to the roots of l ∩ k, P L∩K (λ -ρ n (l)) = P L∩K (λ), and the statement follows.

In each of the examples we will consider, one can check directly that indeed ρ n (l) is orthogonal to all roots of l ∩ k, and hence we can compute the constant c using (3.5). A little more systematic way of checking this assumption, which will be easy to apply in all cases we consider, is given by the following lemma.

Lemma 3.6. Suppose that all simple factors of l 0 are either compact or noncompact Hermitian. Assume also that ∆ + (l) (induced by ∆ + ) is Borel -de Siebenthal for each noncompact factor of l. Then ρ n (l) is orthogonal to all roots of l ∩ k.

Proof. Since ρ n of any compact factor is 0, it is enough to prove the statement for each of the noncompact factors. Denote by d one of these factors, and let d = c ⊕ s be its Cartan decomposition (so c = d ∩ k and s = d ∩ p). Since d is Hermitian,

s = s + ⊕ s - as a c-module. If ∆ + (d)
is a Borel-de Siebenthal positive root system with respect to a compact Cartan subalgebra of d, then ∆ + n (d) must be equal to ∆(s + ) or ∆(s -), and we can assume ∆ + n (d) = ∆(s + ). It follows that ρ n (d) is the weight of the one-dimensional c-module top s + , and so it must be orthogonal to the roots of c.

Remark 3.7. If l has a simple noncompact factor that is not Hermitian, and if ∆ + (l) is any positive root system for l, then ρ n (l) is not orthogonal to all roots of l ∩ k. Indeed, if d = c ⊕ s is the Cartan decomposition of one such factor, then c is semisimple and hence has no nontrivial one-dimensional modules. So if ρ n (d) were orthogonal to all roots of c, it would have to be 0, but that is not possible since d is noncompact.

The following proposition will enable us to get our constants for some of the real forms of O C without having to do computations.

Proposition 3.8. Let h 1 and h 2 correspond to two real forms of O C . Assume that there is an automorphism σ of g such that

(1) σ preserves the compact Cartan subalgebra h of g;

(2) σ commutes with the Cartan involution, so it preserves k and p;

(3)

σ(∆ + c ) = ∆ + c ; (4) σ(h 1 ) = h 2 .
Then the constants c 1 , c 2 corresponding to h 1 , h 2 are related by

c 2 = (-1) n+N1+N2 c 1 , where n = #[∆ + n (l 2 ) ∩ (-σ(∆ + n (l 1 )))] = #{α ∈ ∆ + n (l 1 ) σα ∈ (-∆ + )}, and N 1 , N 2 are defined as in (3.3), i.e., N i = #{α ∈ ∆ + α(h i ) > 0}, i = 1, 2. Proof. If l i denotes the centralizer of h i in g, then it is clear that σ(l 1 ) = l 2 .
Moreover, the conditions we put on σ ensure that P K •σ = P K , and also

P L2∩K •σ = P L1∩K .
We let σ act on roots by

σ(α) = α • σ -1 .
Then it is clear that σ takes ∆(p 1 ) 1 (the set of noncompact roots that are 1 on h 1 ) to ∆(p 1 ) 2 (the set of noncompact roots that are 1 on h 2 ). Furthermore, σ takes

∆ + n (l 1 ) to ∆ + n (l 2 ), where ∆ + n (l 2
) is a positive root system for l 2 , possibly different from ∆ + n (l 2 ) which is defined using ∆ + . It follows that σ(ρ n (l 1 )) = ρn (l 2 ), where ρn (l 2 ) is the half sum of roots in ∆ + n (l 2 ). Also, for any

A ⊆ ∆ + n (l 1 ), C ⊆ ∆(p 1 ) 1 , we clearly have 2ρ(σ(A)) = σ(2ρ(A)), 2ρ(σ(C)) = σ(2ρ(C)).
Writing the equation (3.2) for c 1 , we get

(-1) N1 A⊆∆ + n (l 1 ) C⊆∆(p 1 ) 1 (-1) #A+#C P K (λ -ρ n (l 1 ) + 2ρ(A) -2ρ(C)) = c 1 P L1∩K (λ).
We now replace λ by σ -1 (λ) and use the equalities P L1∩K • σ -1 = P L2∩K , P K • σ -1 = P K . We also replace summing over A and C by summing over σ(A) and σ(C). We obtain

(3.9) (-1) N1 σ(A)⊆ ∆ + n (l 2 ) σ(C)⊆∆(p 1 ) 2 (-1) #σ(A)+#σ(C) P K (λ -ρn (l 2 ) + 2ρ(σ(A)) -2ρ(σ(C))) = c 1 P L2∩K (λ).
We now want to pass from summing over σ(A) ⊆ ∆ + n (l 2 ) to summing over A ⊆ ∆ + n (l 2 ). To do this, we define

∆ 1 = ∆ + n (l 2 ) ∩ ∆ + n (l 2 ); ∆ 2 = ∆ + n (l 2 ) \ ∆ 1 = ∆ + n (l 2 ) ∩ (-∆ + n (l 2 )); so ∆ + n (l 2 ) = ∆ 1 ∪ ∆ 2 ; ∆ + n (l 2 ) = ∆ 1 ∪ (-∆ 2 ). It follows that for the half sums of roots ρ n (l 2 ), ρn (l 2 ) we have (3.10) ρn (l 2 ) = ρ n (l 2 ) -2ρ(∆ 2 ).
For any

A ⊆ ∆ + n (l 2 ), let A 1 = A ∩ ∆ 1 ; A 2 = A ∩ ∆ 2 ; so A = A 1 ∪ A 2 . To each such A we attach à = A 1 ∪ (-(∆ 2 \ A 2 )) ⊆ ∆ + n (l 2 ).
Then the correspondence A ↔ Ã defines a bijection between the subsets of ∆ + n (l 2 ) and the subsets of ∆ + n (l 2 ). Using (3.10), we see that

2ρ( Ã) -ρn (l 2 ) = 2ρ(A 1 ) -(2ρ(∆ 2 ) -2ρ(A 2 )) -(ρ n (l 2 ) -2ρ(∆ 2 )) = (3.11) 2ρ(A 1 ) + 2ρ(A 2 ) -ρ n (l 2 ) = 2ρ(A ) -ρ n (l 2 ).
It follows that we can rewrite (3.9) into a sum over A instead of a sum over à = σ(A), taking into account that

(-1) # Ã = (-1) n (-1) #A ,
with n as in the statement of the proposition. If we also rename A by A and σ(C) by C, we get (-1)

N1+n A⊆∆ + n (l 2 ) C⊆∆(p 1 ) 2 (-1) #A+#C P K (λ -ρ n (l 2 ) + 2ρ(A) -2ρ(C)) = c 1 P L∩K (λ).
If we compare this with the equation (3.2) written for c 2 , we immediately get the statement of the proposition.

4. The case G R = SU (p, q), p ≤ q This case was treated in [MPVZ] and we just record the results here. The real forms of O C correspond to (4.1)

h k = (1, . . . , 1 k , -1, . . . , -1 p-k | 1, . . . , 1 p-k , 0, . . . , 0 q-p , -1, . . . , -1 k ),
with k = 0, 1, . . . , p. The corresponding constants c = c p,q k can be computed from the formula (3.5).

The set ∆ + n (l) is

∆ + n (l) = {ε i -ε j 1 ≤ i ≤ k, p + 1 ≤ j ≤ 2p -k} ∪ ∪ {ε i -ε j k + 1 ≤ i ≤ p, p + q -k + 1 ≤ j ≤ p + q}.
The set ∆(p 1 ) is empty if q = p, and if q > p, then

∆(p 1 ) = {ε i -ε j 1 ≤ i ≤ k, 2p -k + 1 ≤ j ≤ p + q -k} ∪ ∪ {ε i -ε j 2p -k + 1 ≤ i ≤ p + q -k, k + 1 ≤ j ≤ p}.
We evaluate (3.5) at λ = λ 0 , where λ 0 = (q, q-1, . . . , q-k+1, p, p-1, . . . , k+1 | p-k, . . . , 1, q-k, . . . , p-k+1, k, . . . , 1).

For each choice of

A ⊆ ∆ + n (l) and C ⊆ ∆(p 1 ), we set Λ = λ 0 -2ρ(A) -2ρ(C).
For q > p, we show that there is exactly one C ⊆ ∆(p 1 ) for which P K (Λ) can be nonzero:

C = {ε i -ε j 1 ≤ i ≤ k; 2p -k + 1 ≤ j ≤ p + q -k}, with #C = k(q -p).
Then we show that all Λ as above, with P K (Λ) = 0, are of the form Λ = (i 1 , . . . , i k ; j 1 , . . . , j p-k | j 1 , . . . , j p-k ; q, . . . , p + 1; i 1 , . . . , i k ), with i 1 , . . . , i k ; j 1 , . . . , j p-k a shuffle of p, . . . , 1, i.e.,

i 1 > • • • > i k ; j 1 > • • • > j p-k ; {i 1 , . . . , i k ; j 1 , . . . , j p-k } = {p, . . . , 1}.
For each such Λ there is a unique corresponding A, consisting of roots

α a,b , 1 ≤ a ≤ k, 1 ≤ b ≤ p -k, where α a,b = ε a -ε p+b , if i a < j b ; ε k+a -ε p+q-k+b , if i a > j b . In particular, for each A involved, #A = k(p -k).
This leads to Theorem 4.2. Let G R = SU (p, q), and let k ∈ {0, 1, . . . , p} correspond to the real form of O C given by (4.1). Then c p,q k = (-1) k(p+q-k) p k .

5.

The case G R = SO e (2p, 2q + 1), q ≥ p -1 ≥ 0

There are three real forms of O C if q ≥ p ≥ 1, and two real forms if q = p -1.

5.1. The first real form. This real form exists for all q ≥ p -1 ≥ 0. The corresponding h is

h 1 = (2, 1, . . . , 1 p-1 | 1, . . . , 1 p-1 , 0, . . . , 0 q-p+1
).

Since l = l 1 is built from roots that vanish on h 1 , we see that

∆ + n (l) = {ε i -ε j 2 ≤ i ≤ p, p + 1 ≤ j ≤ 2p -1}. It follows that for any A ⊆ ∆ + n (l), (5.1) 2ρ(A) = (0; a 1 , . . . , a p-1 | -b 1 , . . . , -b p-1 ; 0, . . . , 0), with (5.2) 0 ≤ a i , b j ≤ p -1; i a i = j b j . Furthermore, recall that ∆(p 1 ) consists of noncompact roots that are 1 on h 1 . So ∆(p 1 ) = {ε 1 -ε j p + 1 ≤ j ≤ 2p -1} ∪ ∪ {ε i ± ε j 2 ≤ i ≤ p, 2p ≤ j ≤ p + q} ∪ {ε i 2 ≤ i ≤ p}.
It follows that for any C ⊆ ∆(p 1 ),

(5.3) 2ρ(C) = (c; d 1 , . . . , d p-1 | -c 1 , . . . , -c p-1 ; e 1 , . . . , e q-p+1 ), with 0 ≤ c j ≤ 1; 0 ≤ c ≤ p -1; c = j c j ; (5.4) 0 ≤ d i ≤ 2(q -p + 1) + 1; -(p -1) ≤ e j ≤ p -1.
Note that for q = p -1, there are no coordinates after 2p -1, so there are no zeros at the end of 2ρ(A), and there are no e j . Otherwise, all of the above holds in this special case. By (5.1),

ρ n (l) = (0, p -1, . . . , p -1 | -p + 1, . . . , -p + 1, 0, . . . , 0). This is clearly orthogonal to all roots of l ∩ k, which are equal to

∆(l ∩ k) = {ε i -ε j 2 ≤ i, j ≤ p} ∪ {ε i -ε j p + 1 ≤ i, j ≤ 2p -1}∪ (5.5) ∪{ε i ± ε j 2p ≤ i, j ≤ p + q}.
By Proposition 3.4, this means that we can determine the constant c = c p,q 1 from the equation (3.5). To do this, we take λ = λ 0 , where

λ 0 = ( 1 2 ; q + 1 2 , q - 1 2 . . . , q -p + 5 2 | -1, -2, . . . , -(p -1); q -p + 1, q -p, . . . , 1) (5.6) if q ≥ p ≥ 2; λ 0 = ( 1 2 | q, q -1, . . . , 1) if p = 1, q ≥ 1; λ 0 = ( 1 2 ; p - 1 2 , p - 3 2 . . . , 3 2 | -1, -2, . . . , -(p -1)) if p ≥ 2, q = p -1; λ 0 = ( 1 2 | ) if p = 1, q = 0.
Proposition 5.7. Let Λ = λ 0 -2ρ(A) -2ρ(C), with λ 0 given by (5.6), and with A ⊆ ∆ + n (l) and C ⊆ ∆(p 1 ). If P K (Λ) = 0, then:

(1) If p = 1, then A = C = ∅ and Λ = λ 0 .

(2) If p ≥ 2 and q = p -1, then A = C = ∅ and Λ = λ 0 .

(3) If q ≥ p ≥ 2, then

A = ∅; C = {ε i -ε j 2 ≤ i ≤ p, 2p ≤ j ≤ p + q}; Λ = ( 1 2 ; p - 1 2 , p - 3 2 , . . . , 3 2 | -1, -2, . . . , -(p -1); q, q -1, . . . , p).
Proof. We first note that if p = 1, then

h 1 = (2 | 0, . . . , 0) or h 1 = (2), so ∆ + n (l) = ∆(p 1 ) = ∅
and both A and C are automatically empty. It follows that the only possible Λ is Λ = λ 0 , and this proves the proposition for p = 1. We continue by induction on p. Let us assume that p ≥ 2, that q ≥ p -1 is arbitrary, and that the statement of the proposition is true for G R = SO e (2p -2, 2p -3), i.e., when p, q are replaced by p = p -1, and q = p -2.

By (5.6), (5.1) and ( 5.3), we have

Λ = ( 1 2 -c; q + 1 2 -a 1 -d 1 , q -1 2 -a 2 -d 2 , . . . , q -p + 5 2 -a p-1 -d p-1 | (5.8) | -1 + b 1 + c 1 , -2 + b 2 + c 2 , . . . , -(p -1) + b p-1 + c p-1 ; q -p + 1 -e 1 , q -p -e 2 , . . . , 1 -e q-p+1 ).
(The third row of the above equation is not there if q = p -1.) Using (5.2) and (5.4), we see that the coordinates Λ p+1 , . . . , Λ p+q are in the following intervals:

Λ = (. . . | -1 + b 1 + c 1 [-1,p-1] , -2 + b 2 + c 2 [-2,p-2] , . . . , -(p -1) + b p-1 + c p-1 [-(p-1),1] ; q -p + 1 -e 1 [q-2p+2,q] , q -p -e 2 [q-2p+1,q-1] , . . . , 1 -e q-p+1 [-(p-2),p]
).

(The second row of the above equation is not there if q = p -1.) So Λ p+1 , . . . , Λ p+q are q integers between -(p -1) and q. Moreover, P K (Λ) = 0 implies that these integers are nonzero, different from each other, and no two of them are opposite integers. If q ≥ p, it follows that q, q -1, . . . , p must each be equal to some Λ i , and the only possibility for that is

Λ 2p = q, Λ 2p+1 = q -1, . . . , Λ p+q = p.
So e 1 , . . . , e q-p+1 are all equal to -(p -1), and hence

ε i -ε j ∈ C, ε i + ε j / ∈ C, 2 ≤ i ≤ p, 2p ≤ j ≤ p + q.
(If q = p -1, the above says nothing and should be skipped.) This implies

(5.9)

q -p + 1 ≤ d i ≤ q -p + 2, 1 ≤ i ≤ p -1, with d i = q -p + 1 if ε i+1 / ∈ C, and d i = q -p + 2 if ε i+1 ∈ C. (If q = p -1
, this gives no new information about the d i . The following arguments all work also in case q = p -1 if we delete the last group of coordinates, q, q -1, . . . , p.) Using (5.9) together with the inequalities (5.2), (5.4) for a i and c, we go back to (5.8) and conclude that Λ 1 , . . . , Λ p are in the following intervals:

Λ = ( 1 2 -c [-p+ 3 2 , 1 2 ] ; q + 1 2 -a 1 -d 1 [-1 2 ,p-1 2 ] , q -1 2 -a 2 -d 2 [-3 2 ,p-3 2 ] , . . . , q -p + 5 2 -a p-1 -d p-1 [-p+ 3 2 , 3 2 ] | . . . )
So Λ 1 , . . . , Λ p are p half-integers between -p + 3 2 and p -1 2 . Moreover, P K (Λ) = 0 implies that these half-integers are different from each other, and no two of them are opposite. It follows that one of them must be equal to p -1 2 , and the only possibility is

Λ 2 = p - 1 2 .
So a 1 = 0 and d 1 = q -p + 1. It follows that

ε 2 -ε j / ∈ A, p + 1 ≤ j ≤ 2p -1; ε 2 / ∈ C,
and hence 0 ≤ b j ≤ p -2, 1 ≤ j ≤ p -1.
These improved inequalities for the b j together with inequalities (5.4) for the c j imply

Λ = (. . . | -1 + b 1 + c 1 [-1,p-2] , -2 + b 2 + c 2 [-2,p-3] , . . . , -(p -1) + b p-1 + c p-1 [-(p-1),0]
; q, q -1, . . . , p).

Since Λ p+1 , . . . , Λ 2p-1 are p -1 nonzero integers between -(p -1) and p -2, with no two of them equal or opposite to each other, we conclude that

Λ 2p-1 = -(p -1).
This implies b p-1 = c p-1 = 0, and hence

ε i -ε 2p-1 / ∈ A, 2 ≤ i ≤ p; ε 1 -ε 2p-1 / ∈ C, and
0 ≤ a i ≤ p -2, 2 ≤ i ≤ p -1; 0 ≤ c ≤ p -2.
We see that

Λ = ( 1 2 -c; p -1 2 , q -1 2 -a 2 -d 2 , . . . , q -p + 5 2 -a p-1 -d p-1 | | -1 + b 1 + c 1 , . . . , -(p -2) + b p-2 + c p-2 , -(p -1); q, q -1, . . . , p).
(The third row is not there if q = p -1.)

We now consider the subalgebra g ∼ = so(2p -2, 2p -3) of g built on coordinates ε 1 , ε 3 , . . . , ε p ; ε p+1 , . . . , ε 2p-2 , so the coordinates 2 and 2p -1, 2p, . . . , p + q are deleted. We also consider the real form of O K given by

h 1 = (2, 1, . . . , 1 p-2 | 1, . . . , 1 p-2
),

with centralizer l = l ∩ g . Then

∆ + n (l ) = ∆ + n (l) \ {ε i -ε j i = 2 or j = 2p -1}; ∆(p 1 ) = {ε 1 -ε p+1 , . . . , ε 1 -ε 2p-2 ; ε 3 , . . . , ε p }.
We set

A = A ∩ ∆ + n (l ) = A; C = C ∩ ∆(p 1 ) = C \ {ε i -ε j 2 ≤ i ≤ p, 2p ≤ j ≤ p + q}. Then 2ρ(A ) = (0; a 2 , . . . , a p-1 | -b 1 , . . . , -b p-2 ) = (0; a 1 , . . . , a p-2 | -b 1 , . . . , -b p-2 ); 2ρ(C ) = (c; d 2 -(q -p + 1), . . . , d p-1 -(q -p + 1) | -c 1 , . . . , -c p-2 ) = (c ; d 1 , . . . , d p-1 | -c 1 , . . . , -c p-2 ),
where we define

a i = a i+1 ; b i = b i ; c i = c i ; c = c; d i = d i+1 -(q -p + 1).
The numbers a i , b i , c i , c , d i satisfy analogues of (5.2) and (5.4). We define λ 0 by (5.6), but for G R = SO e (2p -2, 2p -3), i.e.,

λ 0 = ( 1 2 ; p - 3 2 , . . . , 3 2 | -1, -2, . . . , -(p -2)).
Then A , C and Λ = λ 0 -2ρ(A ) -2ρ(C ) satisfy all conditions of the proposition, but p, q are reduced to p = p -1, q = p -2. Moreover, P K (Λ) = 0 is equivalent to P K (Λ ) = 0. Therefore the inductive assumption implies that A = C = ∅, and that Λ = λ 0 . This implies the statement of the proposition for A, C and Λ.

To compute the constant c p,q 1 , we have to compute P L∩K (λ 0 ) where λ 0 is given by (5.6), and P K (Λ) for Λ determined in Proposition 5.7. The main ingredients for this computation are given in the following lemma.

Lemma 5.10. (i) Let P 1 p be the Weyl dimension formula polynomial for so(2p), p ≥ 1, and let λ p = (p -1 2 , p -3 2 , . . . , 1 2 ). Then P 1 p (λ p ) = 2 p-1 . (ii) Let P 2 q be the Weyl dimension formula polynomial for so(2q + 1), q ≥ 1, and let P 2 0 be the constant polynomial 1. Furthermore, let µ q = (q, q -1, . . . , 1) if q ≥ 1, and µ 0 = 0. Then P 2 q (µ q ) = 2 q .

Proof. (i) Let n 1 p be the numerator of P 1 p ; the denominator is then

d 1 p = n 1 p (ρ so(2p) ) = n 1 p (p -1, p -2, . . . , 1, 0).
The factors of n 1 p (λ p ) that correspond to the roots ε i -ε j clearly cancel with the corresponding factors of d 1 p . Denoting by m 1 p (λ p ) respectively e 1 p the product of factors of n 1 p (λ p ) respectively d 1 p corresponding to the roots ε i + ε j , we have

m 1 p (λ p ) = (2p -2)(2p -3) . . . (p + 1)p m 1 p-1 (λ p-1 ); e 1 p = (2p -3)(2p -4) . . . p(p -1) e 1 p-1 . It follows that P 1 p (λ p ) = m 1 p (λ p ) e 1 p = (2p -2)m 1 p-1 (λ p-1 ) (p -1)e 1 p-1 = 2P 1 p-1 (λ p-1 ).
Since P 1 1 is the constant polynomial 1, this proves (i). (ii) There is nothing to prove for q = 0, and it is obvious that

P 2 1 (µ 1 ) = 1 1/2 = 2.
For q ≥ 2, let n 2 q be the numerator of P 2 q ; the denominator is then

d 2 q = n 2 q (ρ so(2q+1) ) = n 2 q (q - 1 2 , q - 3 2 , . . . , 3 2 , 1 2 ).
The factors of n 2 q (µ q ) that correspond to the roots ε i -ε j clearly cancel with the corresponding factors of d 2 q . Denoting by m 2 q (µ q ) respectively e 2 q the product of factors of n 2 q (µ q ) respectively d 2 q corresponding to the roots ε i + ε j and ε i , we have m 2 q (µ q ) = (2q -1)(2q -2) . . . (q + 2)(q + 1)q m 2 q-1 (µ q-1 );

e 2 q = (2q -2)(2q -3) . . . (q + 1)q(q -1 2 ) e 2 q-1 .

It follows that

P 2 q (µ q ) = m 2 q (µ q ) e 2 q = (2q -1)m 2 q-1 (µ q-1 ) (p -1
2 )e 2 q-1 = 2P 2 q-1 (µ q-1 ).

The statement now follows by induction.

To compute P L∩K (λ 0 ), we recall (5.5), which shows that l ∩ k is up to center typically a product of three factors: the u(p -1) on coordinates 2, . . . , p, the u(p -1) on coordinates p + 1, . . . , 2p -1, and the so(2(q -p + 1) + 1) on coordinates 2p, . . . , p + q. (If p = 1, then the first two factors are missing, if q = p -1 then the third factor is missing, and if p = 1 and q = 0, then l ∩ k is one-dimensional. What we say below applies also to these cases with obvious modifications.)

It is clear from the definition (5.6) of λ 0 that for each of the first two factors, the corresponding coordinates of λ 0 differ from the ρ of the factor by a weight orthogonal to the roots of the factor, so in the notation of Lemma 5.10, (5.11) P L∩K (λ 0 ) = P 2 q-p+1 (µ q-p+1 ) = 2 q-p+1 . To compute P K (Λ), we first write Λ = (Λ L | Λ R ) and note that P K (Λ) = P 1 p (Λ L )P 2 q (Λ R ). To use Lemma 5.10, we have to rearrange coordinates of Λ L and Λ R , and use the fact that P 1 p is skew for the Weyl group of so(2p), while P 2 q is skew for the Weyl group of so(2q + 1). To rearrange Λ L to λ p , we only need to bring the 1 2 from the first coordinate to the p-th coordinate, and hence (5.12) P 1 p (Λ L ) = (-1) p-1 2 p-1 . To bring Λ R to µ q = (q, . . . , 1), we need to change p -1 signs, and then bring coordinates p -1, p -2, . . . , 1, in that order, all the way to the right. The sign produced in this way is (-1) (p-1)+(q-p+1)+(q-p+2)+•••+(q-1) = (-1) (p-1)(q-p+1)+ (p-1)p 2 .

Since

(5.13

) (p -1)p 2 ≡ [ p 2
] mod 2, Lemma 5.10 implies that (5.14)

P 2 q (Λ R ) = (-1) (p-1)(q-p+1)+[ p 2 ] 2 q .
Now we substitute (5.11), (5.12) and (5.14) into (3.5). Since

#A + #C = #C = (p -1)(q -p + 1),
and since N from (3.3) is easily checked to satisfy

(5.15) N ≡ p mod 2, we see that the total sign is (-1) [ p 2 ]+1 , and we conclude Theorem 5.16. Let G R = SO e (2p, 2q + 1), q ≥ p -1 ≥ 0, and let c p,q 1 be the constant corresponding to the first real form of O C . Then

c p,q 1 = (-1) [ p 2 ]+1 2 2p-2 .
5.2. The second real form. This real form exists for all q ≥ p -1 ≥ 0. The corresponding h is

h 2 = (2, 1, . . . , 1 p-2 , -1 | 1, . . . , 1 p-1 , 0, . . . , 0 q-p+1
).

This real form is conjugate to the first real form by the automorphism σ = s εp , the reflection with respect to the short noncompact root ε p . The automorphism σ of g clearly satisfies the conditions of Proposition 3.8. Moreover, the number n from Proposition 3.8 is p -1; the roots from ∆ + n (l 1 ) that σ sends to

-∆ + are ε p -ε j , p + 1 ≤ j ≤ 2p -1.
Moreover, by (5.15), N 1 ≡ p mod 2, and another short computation shows that N 2 is always even. The total sign in Proposition 3.8 is thus (-1) n+N1+N2 = -1, so (5.17) c p,q 2 = -c p,q 1 = (-1)

[ p 2 ] 2 2p-2 .
5.3. The third real form. This real form exists for q ≥ p ≥ 1, so we assume this condition in the following. The corresponding h is

h 3 = (1, . . . , 1 p-1 , 0 | 2, 1, . . . , 1 p-1 , 0, . . . , 0 q-p
).

Since l = l 3 is built from roots that vanish on h 3 , we see that

∆ + n (l) = {ε i -ε j 1 ≤ i ≤ p-1, p+2 ≤ j ≤ 2p}∪{ε p ±ε j 2p+1 ≤ j ≤ p+q}∪{ε p }.

It follows that for any

A ⊆ ∆ + n (l), (5.18) 2ρ(A) = (a 1 , . . . , a p-1 ; x | 0; -b 1 , . . . , -b p-1 ; y 1 , . . . , y q-p ), with 0 ≤ a i , b j ≤ p -1; i a i = j b j ; (5.19) 0 ≤ x ≤ 2(q -p) + 1; -1 ≤ y j ≤ 1.
Furthermore, recall that ∆(p 1 ) consists of noncompact roots that are 1 on h 3 . So

∆(p 1 ) = {ε i ± ε j 1 ≤ i ≤ p -1, 2p + 1 ≤ j ≤ p + q} ∪ {ε i 1 ≤ i ≤ p -1} ∪ ∪ {ε j ± ε p p + 2 ≤ j ≤ 2p} ∪ {ε p+1 -ε i 1 ≤ i ≤ p -1}.
It follows that for any C ⊆ ∆(p 1 ),

(5.20) 2ρ(C) = (c 1 , . . . , c p-1 ; u | v; d 1 , . . . , d p-1 ; e 1 , . . . , e q-p ), with

-1 ≤ c i ≤ 2(q -p) + 1; -(p -1) ≤ u ≤ p -1; (5.21) 0 ≤ v ≤ p -1; 0 ≤ d j ≤ 2; -(p -1) ≤ e j ≤ p -1.
If we write (5.18) for A = ∆ + n (l), we get ρ n (l) = (p -1, . . . , p -1; 2(q -p) + 1 | 0; -p + 1, . . . , -p + 1; 0, . . . , 0). This is clearly orthogonal to all roots of l ∩ k, which are equal to

∆(l ∩ k) = {ε i -ε j 1 ≤ i, j ≤ p -1} ∪ {ε i -ε j p + 2 ≤ i, j ≤ 2p}∪ (5.22) ∪{ε i ± ε j 2p + 1 ≤ i, j ≤ p + q} ∪ {ε 2p+1 , . . . , ε p+q }.
By Proposition 3.4, this means that the constant c = c p,q 3 satisfies (3.5) for any λ, and we will compute c p,q 3 by using this for λ = λ 0 , where (5.23) λ 0 = (q-3 2 , q-5 2 . . . , q-p+ 1 2 ; q-p+ 1 2 | p-1; 0, -1, . . . , -(p-2); q-p, q-p-1, . . . , 1).

(Out of the 5 groups of coordinates separated by semicolons and the bar, the first and the fourth group are missing if p = 1, and the fifth group is missing if q = p.)

Proposition 5.24. Let Λ = λ 0 -2ρ(A) -2ρ(C), with λ 0 given by (5.23), and with A ⊆ ∆ + n (l) and C ⊆ ∆(p 1 ). Then Λ p+1 = 0. In particular, P K (Λ) = 0. Proof. By (5.23), (5.18) and (5.20), we have

Λ = (q -3 2 -a 1 -c 1 , . . . , q -p + 1 2 -a p-1 -c p-1 ; q -p + 1 2 -x -u | | p -1 -v; b 1 -d 1 , -1 + b 2 -d 2 , . . . , -(p -2) + b p-1 -d p-1 ;
q -p -y 1 -e 1 , q -p -1 -y 2 -e 2 , . . . , 1 -y q-p -e q-p ).

If p = 1, then the situation is much simpler; in particular, since the coordinates of h 3 are 0 or 2, ∆(p 1 ) is empty, so C = ∅ and 2ρ(C) = 0. It follows that Λ p+1 = Λ 2 = p -1 = 0, and so P K (Λ) = 0 as claimed.

So the proposition is true for p = 1. We continue by induction on p. Let us assume that p ≥ 2, that q ≥ p is arbitrary, and that the statement of the proposition is true for G R = SO e (2p -2, 2p -1), i.e., when p, q are replaced by p = p -1, q = p -1.

By (5.23), (5.19) and (5.21), the coordinates Λ p+1 , . . . , Λ p+q are in the following intervals:

Λ = (. . . | p -1 -v [0,p-1] ; b 1 -d 1 [-2,p-1] , -1 + b 2 -d 2 [-3,p-2] , . . . , -(p -2) + b p-1 -d p-1 [-p,1] ; (5.25) q -p -y 1 -e 1 [q-2p,q] , q -p -1 -y 2 -e 2 [q-2p-1,q-1] , . . . , 1 -y q-p -e q-p [-(p-1),p+1]
).

So Λ p+1 , . . . , Λ p+q are q integers between -p and q. Moreover, P K (Λ) = 0 implies that these integers are nonzero, and no two of them are equal or opposite to each other. It follows that q, q -1, . . . , p + 1 must each be equal to some Λ i , and the only possibility for that is

Λ 2p+1 = q, Λ 2p+2 = q -1, . . . , Λ p+q = p + 1.
It follows that y 1 , . . . , y q-p are all equal to -1, and that e 1 , . . . , e q-p are all equal to -(p -1). So

ε p -ε j ∈ A, ε p + ε j / ∈ A, 2p + 1 ≤ j ≤ p + q; ε i -ε j ∈ C, ε i + ε j / ∈ C, 1 ≤ i ≤ p -1, 2p + 1 ≤ j ≤ p + q.
This implies q -p ≤ x ≤ q -p + 1; (5.26)

q -p -1 ≤ c i ≤ q -p + 1, 1 ≤ i ≤ p -1. Note that x = q -p if ε p / ∈ A and x = q -p + 1 if ε p ∈ A. Similarly, c i = q -p -1 if ε p+1 -ε i ∈ C, ε i / ∈ C; c i = q -p if ε p+1 -ε i ∈ C, ε i ∈ C or ε p+1 -ε i / ∈ C, ε i / ∈ C; and c i = q -p + 1 if ε p+1 -ε i / ∈ C, ε i ∈ C.
Using the same arguments as above, we can also conclude from (5.25) that

Λ 2p = -p.
This implies that b p-1 = 0 and d p-1 = 2. It follows that

ε i -ε 2p / ∈ A, 1 ≤ i ≤ p -1; ε 2p ± ε p ∈ C, so 0 ≤ a i ≤ p -2, 1 ≤ i ≤ p -1; (5.27) -(p -2) ≤ u ≤ p -2.
Using the improved inequalities (5.26) and (5.27), we see that Λ 1 , . . . , Λ p are in the following intervals:

Λ = (q -3 2 -a 1 -c 1 [-1 2 ,p-1 2 ] , q -5 2 -a 2 -c 2 [-3 2 ,p-3 2 ] , . . . , q -p + 1 2 -a p-1 -c p-1 [-(p-3 2 ), 3 2 ] ; q -p + 1 2 -x -u [-(p-3 2 ),p-3 2 ] | . . . So Λ 1 , . . . , Λ p are p half-integers between -(p - 3
2 ) and p -1 2 , such that no two of them are equal or opposite to each other. It follows that

Λ 1 = p - 1 2 ,
and consequently a 1 = 0, c 1 = q -p + 1. Therefore,

ε 1 -ε j / ∈ A, p + 2 ≤ j ≤ 2p; ε p+1 -ε 1 ∈ C, ε 1 / ∈ C,
and we conclude that

0 ≤ b j ≤ p -2, 1 ≤ j ≤ p -1; 1 ≤ v ≤ p -1.
We see that

Λ = (p -1 2 , q -5 2 -a 2 -c 2 , . . . , q -p + 1 2 -a p-1 -c p-1 ; q -p + 1 2 -x -u | | p -1 -v; b 1 -d 1 , . . . , -(p -3) + b p-2 -d p-2 , -p;
q, q -1, . . . , p + 1).

(If q = p, the coordinates q, . . . , p+1 are not there; if p = 2 there are no coordinates involving a i , c i , b i or d i .)

We now consider the subalgebra g ∼ = so(2p -2, 2p -1) of g built on coordinates ε 2 , ε 3 , . . . , ε p ; ε p+1 , . . . , ε 2p-1 , so the coordinates 1 and 2p, 2p + 1, . . . , p + q are deleted. We also consider the real form of O K given by

h 3 = (1, . . . , 1 p-2 , 0 | 2, 1, . . . , 1 p-2
),

with centralizer l = l ∩ g . Then

∆ + n (l ) = {ε i -ε j 2 ≤ i ≤ p -1, p + 2 ≤ j ≤ 2p -1} ∪ {ε p }; ∆(p 1 ) = {ε 2 , . . . , ε p-1 } ∪ {ε j ± ε p p + 2 ≤ j ≤ 2p -1}∪ ∪ {ε p+1 -ε i 2 ≤ i ≤ p -1}.
We set

A = A ∩ ∆ + n (l ) = A \ {ε p -ε j 2p + 1 ≤ j ≤ p + q}; C = C ∩ ∆(p 1 ) = C \ {ε 2p ± ε p ; ε p+1 -ε 1 ; ε i -ε j 1 ≤ i ≤ p -1, 2p + 1 ≤ j ≤ p + q}. Then 2ρ(A ) = (a 2 , . . . , a p-1 ; x -(q -p) | 0; -b 1 , . . . , -b p-2 ) = (a 1 , . . . , a p-2 ; x | 0; -b 1 , . . . , -b p-2 ); 2ρ(C ) = (c 2 -(q -p), . . . , c p-1 -(q -p); u | v -1; d 1 , . . . , d p-2 ) = (c 1 , . . . , c p-2 ; u | v ; d 1 , . . . , d p-2 ),
where we define

a i = a i+1 ; x = x -(q -p); b i = b i ; c i = c i+1 -(q -p); u = u; v = v -1; d i = d i .
The numbers a i , x , b i , c i , u , v , d i satisfy analogues of (5.19) and (5.21).

We define λ 0 by (5.23), but for G R = SO e (2p -2, 2p -1), i.e.,

λ 0 = (p - 5 2 , p - 7 2 , . . . , 1 2 ; 1 2 ; | p -2; 0, -1, . . . , -(p -3)). Then A , C and Λ = λ 0 -2ρ(A ) -2ρ(C )
satisfy all conditions of the proposition, but p, q are reduced to p = p-1, q = p-1.

Therefore the inductive assumption implies that Λ p = 0. So v = p-2, and therefore v = p -1 and Λ p+1 = 0. It follows that P K (Λ) = 0, since Λ is orthogonal to the compact root ε p+1 .

Proposition 5.24 implies that the left hand side of (3.5) is 0 in this case. On the other hand, P L∩K (λ 0 ) = 0 by (5.23) and (5.22). We conclude Theorem 5.28. For G R = SO e (2p, 2q + 1), q ≥ p ≥ 1, the constant corresponding to the third real form is c p,q 3 = 0.

The case

G R = Sp(2n, R), n ≤ 1
The real forms of O C correspond to integers p such that 0 ≤ p ≤ n. We denote n -p by q. The h corresponding to p is

h p = (1, . . . , 1 p , -1, . . . , -1 q ), p = 0, 1, . . . , n.
Since l = l p is built from roots that vanish on h p , we see that

∆ + n (l) = {ε i + ε p+j 1 ≤ i ≤ p, 1 ≤ j ≤ q}. It follows that for any A ⊆ ∆ + n (l), (6.1) 2ρ(A) = (a 1 , . . . , a p | b 1 , . . . , b q ), with 0 ≤ a i ≤ q, 0 ≤ b j ≤ p, i a i = j b j . (6.2)
In particular, ρ n (l) = (q, . . . , q | p, . . . , p), and this is clearly orthogonal to the roots of l ∩ k, which are given by

∆ + (l ∩ k) = {ε i -ε j 1 ≤ i < j ≤ p} ∪ {ε p+i -ε p+j 1 ≤ i < j ≤ q}.
So the constants c = c n p can be calculated from (3.5). Since it is clear that in our present case ∆(p 1 ) = ∅,

(3.5) becomes (6.3)

A⊆∆ + n (l) (-1) #A P K (λ -2ρ(A)) = cP L∩K (λ).
We take λ = λ 0 , where (6.4) λ 0 = (n, n -1, . . . , q + 1 | n, n -1, . . . , p + 1), (If p is 0 or n, then there is only one group of coordinates in the above expression, and λ 0 = (n, n -1, . . . , 1).) Since λ 0 differs from ρ l∩k by a weight orthogonal to all roots of l ∩ k,

P L∩K (λ 0 ) = 1.
So to compute c n p we have to compute the left side of (6.3). The following proposition describes the relevant A and the corresponding Λ. Proposition 6.5. Let Λ = λ 0 -2ρ(A), with λ 0 given by (6.4), and with A ⊆ ∆ + n (l). (i) If p and q are both odd, then P K (Λ) = 0 for all Λ as above. (ii) Suppose that at least one of p, q is even, and suppose that for some A the corresponding Λ satisfies P K (Λ) = 0. Then:

(1) If p = 0 or q = 0, then A = ∅ and Λ = λ 0 = (n, n -1, . . . , 1).

(2) If 0 < p < n, let r = [ p 2 ] and s = [ q 2 ]. Then there is a shuffle

1 ≤ i 1 < • • • < i r ≤ r + s; 1 ≤ j 1 < • • • < j s ≤ r + s of 1, 2, . . . , r + s such that A = {α u,v , β u,v 1 ≤ u ≤ r, 1 ≤ v ≤ s} ∪ B,
where

α u,v = ε p+1-u + ε n+1-v ; β u,v = ε p+1-u + ε p+v , i u < j v ; ε u + ε n+1-v , i u > j v .
and

B =    ∅, p, q even; {ε r+1 + ε p+j | s + 1 ≤ j ≤ q}, p odd; {ε i + ε p+s+1 | r + 1 ≤ i ≤ p}, q odd.
The corresponding Λ has coordinates

Λ 1 = n + 1 -i 1 , . . . , Λ r = n + 1 -i r ; Λ p-r+1 = i r , . . . , Λ p = i 1 ; Λ p+1 = n + 1 -j 1 , . . . , Λ p+s = n + 1 -j s ; Λ n-s+1 = j s , . . . , Λ n = j 1 ,
and possibly in addition

Λ p-r = n -r -s, if p is odd; Λ n-s = n -r -s, if q is odd.
Proof. The statement is obviously true for any n if p = 0 or q = 0. Hence it is true for n = 1. If n = 2 and p = q = 1, there are two cases:

A = ∅, or A = {ε 1 + ε 2 }. If A = ∅, then Λ = Λ 0 = (n | n), so P K (Λ) = 0. If A = {ε 1 + ε 2 }, then Λ = (n -1 | n -1
), and again P K (Λ) = 0. So the proposition is true for n = 2. We proceed by induction on n. Assume that n > 2 and p, q ≥ 1, and assume that the proposition is true for n -2.

Using the definitions and the inequalities (6.2), we see that

Λ = (n -a 1 [p,n] , n -1 -a 2 [p-1,n-1] , . . . , q + 1 -a p [1,q+1] | n -b 1 [q,n] , n -1 -b 2 [q-1,n-1] , . . . , p + 1 -b q [1,p+1]
).

So the coordinates of Λ are n integers between 1 and n, and assuming that P K (Λ) = 0, they have to be different from each other, i.e., Λ has to be a permutation of (n, . . . , 1). In particular, some Λ i must be equal to n and there are two possibilities:

(6.6) Λ 1 = n or Λ p+1 = n.
Assume first that Λ 1 = n. Then a 1 = 0, and it follows that

ε 1 + ε p+j / ∈ A, 1 ≤ j ≤ q.
This implies that 0 ≤ b j ≤ p -1, 1 ≤ j ≤ q, and so

Λ = (n, n -1 -a 2 [p-1,n-1] , . . . , q + 1 -a p [1,q+1] | n -b 1 [q+1,n] , n -1 -b 2 [q,n-1] , . . . , p + 1 -b q [2,p+1]
).

If p = 1, then there is only Λ 1 = n in the left group of coordinates, and we see there is no place to put the coordinate 1. Therefore, if p = 1 then Λ 1 can not be n, hence Λ p+1 = n, so we are in the second case which we treat below. If p > 1, then there is exactly one place where 1 can be, i.e., Λ p = 1. This implies a p = q, and therefore

ε p + ε p+j ∈ A, 1 ≤ j ≤ q. It follows that 1 ≤ b j ≤ p -1, 1 ≤ j ≤ q,
and so

Λ = (n, n -1 -a 2 [p-1,n-1] , . . . , q + 2 -a p-1 [2,q+2] , 1 | n -b 1 [q+1,n-1] , n -1 -b 2 [q,n-2] , . . . , p + 1 -b q [2,p]
).

Let now g ∼ = sp(2(n -2), R) be the subalgebra of g built on coordinates 2, . . . , p -1, p + 1, . . . , n, and let l = l ∩ g . Then

∆ + n (l ) = ∆ + n (l) \ {ε 1 + ε p+j , ε p + ε p+j 1 ≤ j ≤ q}, and we set A = A \ {ε p + ε p+j 1 ≤ j ≤ q}.
We define λ 0 as in (6.4), but with n replaced by n -2 and p replaced by p -2.

Then Λ corresponding to A can be obtained from Λ by deleting coordinates Λ 1 and Λ p , and decreasing all the other coordinates by 1. We now see that Λ is a permutation of (n, . . . , 1) if and only if Λ is a permutation of (n -2, . . . , 1). By inductive assumption, this is equivalent to A and Λ being defined by a shuffle as in the statement of the proposition, and this clearly implies the same statement for A and Λ. The other possibility in (6.6) is handled analogously: Λ p+1 = n implies b 1 = 0, and it follows that

ε i + ε p+1 / ∈ A, 1 ≤ i ≤ p.
This implies that 0 ≤ a i ≤ q -1, 1 ≤ i ≤ p, and so

Λ = (n -a 1 [p+1,n] , n -1 -a 2 [p,n-2] , . . . , q + 1 -a p [2,q] | n, n -1 -b 2 [q-1,n-1] , . . . , p + 1 -b q [1,p+1]
).

If q = 1, then there is only Λ p+1 = n in the right group of coordinates, and we see there is no place to put the coordinate 1. Therefore, if q = 1, Λ p+1 can not be n and we are back to the first case that we already handled. If q > 1, then there is exactly one place where 1 can be, i.e., Λ n = 1.

This implies b q = p, and therefore

ε i + ε n ∈ A, 1 ≤ i ≤ p. It follows that 1 ≤ a i ≤ q -1, 1 ≤ i ≤ p,
and so

Λ = ( n -a 1 [p+1,n-1] n -1 -a 2 [p,n-2] , . . . , q + 1 -a p [2,q] | n, n -1 -b 2 [q-1,n-1] , n -2 -b 3 [q-2,n-2] , . . . , p + 2 -b q-1 [2,p+1]
, 1).

We now reason in the same way as in the first case, and conclude that the proposition follows from the inductive assumption for n -2 with p staying the same and q being replaced by q -2.

To finish the computation of the constant c n p , we first note that for every A described in Proposition 6.5(ii)

(6.7) #A = pq 2 .
Namely, the α u,v and β u,v make for 2rs elements of A. In addition, the set B has 0 elements if p and q are even, s elements if p is odd, and r elements if q is odd. So the total number of elements is 2rs = pq 2 , p, q even; 2rs + s = (2r + 1)s = p q 2 , p odd;

2rs + r = r(2s + 1) = p 2 q, q odd.
On the other hand, since Λ is a permutation of (n, . . . , 1), P K (Λ) is equal to ±1.

To compute the sign, we need to find the parity of the permutation bringing Λ to (n, . . . , 1). This parity can be found by counting the number of inversions in Λ when compared with (n, . . . , 1), i.e., counting the number of pairs (i, j), 1 ≤ i < j ≤ n, such that Λ i < Λ j . We know from Proposition 6.5 that (6.8) Λ = (n + 1 -i 1 , . . . , n + 1 -i r , i r , . . . , i 1 | n + 1 -j 1 , . . . , n + 1 -j s , j s , . . . , j 1 ) if p and q are both even. It is clear that i r , . . . , i 1 are in inversion with n + 1j 1 , . . . , n + 1 -j s ; that is rs inversions. The further inversions are possible only between groups (6.9) n + 1 -i 1 , . . . , n + 1 -i r and n + 1 -j 1 , . . . , n + 1 -j s , and (6.10) i r , . . . , i 1 and j s , . . . , j 1 .

If i u is in inversion with j v , i.e., i u < j v , then n + 1 -i u > n + 1 -j v , i.e., n + 1 -i u is not in inversion with n + 1 -j v .
The converse also holds, and it follows that the total number of inversions in groups (6.9) and (6.10) is again rs. So the total number of inversions in case p and q are even is 2rs = pq 2 .

If p is odd, then Λ is again given by (6.8), except that there is in addition r + s + 1 between n -i r and i r . This coordinate is in inversion with the coordinates n + 1j 1 , . . . , n + 1 -j s , and with no others, so the total number of inversions in this case is 2rs + s = pq 2 .

Similarly, if q is odd, then Λ is given by (6.8), with the addition of r + s + 1 between n + 1 -j s and j s . This coordinate is in inversion with the coordinates i r , . . . , i 1 , and with no others, so the total number of inversions in this case is 2rs + r = pq 2 .

So we have proved that for each Λ from Proposition 6.5,

P K (Λ) = (-1) pq 2 .
Combined with (6.7), and with the fact that N from (3.3) is in this case

N = p 2 + pq + p ≡ [ p + 1 2 ] mod 2,
this tells us that the nonzero contributions to the sum in (6.3) are all equal to (-1)

[ p+1 2 ]
. Since the number of nonzero summands is by Proposition 6.5 equal to the number of (r, s)-shuffles of r + s, i.e., to r+s r , we have proved: Theorem 6.11. Let G R = Sp(2n, R), n ≥ 1, and let p, 0 ≤ p ≤ n, be an integer.

Let r = [ p 2 ] and let s = [ n-p 2 ]. Then the constant c n p for the real form of O C corresponding to p is c n p = 0, if n is even and p is odd; (-1) [ p+1 2 ] r+s r ,
if n is odd, or if n is even and p is even.

7. The case G R = SO e (2p, 2q), q ≥ p ≥ 1

There are three real forms of O C if q > p > 1, four if q = p > 1, and two if p=1.

7.1. The first real form. This real form is defined in all cases; it corresponds to

h 1 = (2, 1, . . . , 1 p-1 | 1, . . . , 1 p-1 , 0, . . . , 0 q-p+1
).

Since l = l 1 is built from roots that vanish on h 1 , we see that other. If q > p, it follows that q -1 2 , q -3 2 , . . . , p + 1 2 must each be equal to some Λ i , and the only possibility for that is

∆ + n (l) = {ε i -ε j 2 ≤ i ≤ p, p + 1 ≤ j ≤ 2p -1}. It follows that for any A ⊆ ∆ + n (l), (7.1) 2ρ(A) = (0; a 1 , . . . , a p-1 | -b 1 , . . . , -b p-1 ; 0, . . . , 0), with (7.2) 0 ≤ a i , b j ≤ p -1; i a i = j b j . Furthermore, recall that ∆(p 1 ) consists of noncompact roots that are 1 on h 1 . So ∆(p 1 ) = {ε 1 -ε j p + 1 ≤ j ≤ 2p -1} ∪ {ε i ± ε j 2 ≤ i ≤ p, 2p ≤ j ≤ p + q}.
Λ 2p = q - 1 2 , Λ 2p+1 = q - 3 2 , . . . , Λ p+q-1 = p + 1 2 .
So e 1 , . . . , e q-p are all equal to -(p -1), and hence

ε i -ε j ∈ C, ε i + ε j / ∈ C, 2 ≤ i ≤ p, 2p ≤ j ≤ p + q -1.
(If q = p, the above says nothing and should be skipped.) This implies

(7.8) q -p ≤ d i ≤ q -p + 2, 1 ≤ i ≤ p -1, with d i being q -p if ε i+1 ± ε p+q / ∈ C, d i = q -p + 2 if ε i+1 ± ε p+q ∈ C, and
d i = q -p + 1 if one of the roots ε i+1 ± ε p+q is in C while the other is not in C.
(If q = p, this gives no new information about the d i . The following arguments all work also in case q = p if we delete the group of coordinates from place 2p to place p + q -1.)

Looking at the bounds for coordinates Λ p+1 , . . . , Λ 2p-1 and Λ p+q , we see that they are p half-integers between -(p -1 2 ) and p -1 2 , such that no two of them are equal or opposite to each other. It follows that some of these Λ j must be equal to ±(p -1 2 ). There are two possibilities:

Λ 2p-1 = -(p - 1 2 ) or Λ p+q = p - 1 2 .
Let us first examine the possibility that Λ p+q = p -1 2 . If this is true, then e p+q = -(p -1), so

ε i+1 -ε p+q ∈ C, ε i+1 + ε p+q / ∈ C, 1 ≤ i ≤ p -1,
and it follows that d i = q -p + 1, 1 ≤ i ≤ p -1. Using this together with the inequalities (7.2), (7.4) for a i and c, we see

Λ = ( 1 2 -c [-(p-3 2 ), 1 2 ] ; p -3 2 -a 1 [-1 2 ,p-3 2 ] , p -5 2 -a 2 [-3 2 ,p-5 2 ] , . . . , 1 2 -a p-1 [-(p-3 2 ), 1 2 ] | . . . ).
So Λ 1 , . . . , Λ p are p half-integers between -(p -3 2 ) and p -3 2 . Moreover, P K (Λ) = 0 implies that no two of these half-integers are equal or opposite to each other. This is impossible, and so Λ p+q can not be p -

1 2 . It follows that Λ 2p-1 = -(p - 1 2 ),
and hence b p-1 = 0; c p-1 = 0. This implies that

ε i -ε 2p-1 / ∈ A, 2 ≤ i ≤ p; ε 1 -ε 2p-1 / ∈ C, and therefore 0 ≤ a i ≤ p -2, 1 ≤ i ≤ p -1; (7.9) 0 ≤ c ≤ p -2.
Then 2ρ(A ) = (0; a 2 , . . . , a p-1 | -b 1 , . . . , -b p-2 ; 0) = (0; a 1 , . . . , a p-2 | -b 1 , . . . , -b p-2 ; 0); 2ρ(C ) = (c; d 2 -(q -p), . . . , d p-1 -(q -p) | -c 1 , . . . , -c p-2 ; e q-p+1 ) = (c ; d 1 , . . . , d p-1 | -c 1 , . . . , -c p-2 ; e q-p+1 ), where we define

a i = a i+1 ; b i = b i ; c i = c i ; c = c; d i = d i+1 -(q -p); e q-p+1 = e q-p+1 .
The numbers a i , b i , c i , c , d i satisfy analogues of (7.2) and (7.4). We define λ 0 by (7.6), but for G R = SO e (2p -2, 2p -1), i.e.,

λ 0 = ( 1 2 ; p - 3 2 , . . . , 3 2 | - 3 2 , - 5 2 , . . . , -(p - 3 2 ); 1 2 ).
Then A , C , and Λ = λ 0 -2ρ(A ) -2ρ(C ) satisfy all conditions of the proposition, but p, q are reduced to p = p -1, q = p -1. Moreover, P K (Λ) = 0 is equivalent to P K (Λ ) = 0. Therefore the inductive assumption implies that A = C = ∅, and that Λ = λ 0 . This implies the statement of the proposition for A, C and Λ.

In view of (3.5), to compute the constant c = c p,q 1 we need to compute P L∩K (λ 0 ) and P K (Λ), where λ 0 is given by (7.6), and Λ is given by Proposition 7.7.

To compute P L∩K (λ 0 ), we note that we described l ∩ k in (7.5); it has up to three factors, two of which are u(p -1), and the third is so(2(q -p + 1)). From the shape of λ 0 it now follows that, in the notation of Lemma 5.10, P L∩K (λ 0 ) = P 1 q-p+1 (λ q-p+1 ), and we see that Lemma 5.10(i) implies that P L∩K (λ 0 ) = 2 q-p .

To compute P K (Λ), with Λ as in Proposition 7.7, we first write Λ = (Λ L | Λ R ) and note that P K (Λ) = P 1 p (Λ L )P 1 q (Λ R ). To use Lemma 5.10, we have to rearrange coordinates of Λ L and Λ R , using the fact that P 1 p is skew for the Weyl group of so(2p), while P 1 q is skew for the Weyl group of so(2q). Moreover, both polynomials are invariant under sign changes of the variables; this follows since the sign change of the j-th coordinate switches roots ε i -ε j and ε i + ε j .

To rearrange Λ L to λ p , we only need to bring the 1 2 from the first coordinate to the p-th coordinate, and hence P 1 p (Λ L ) = (-1) p-1 2 p-1 . To bring Λ R to µ q = (q, . . . , 1), after removing the signs which does not change the expression, we need to bring coordinates p -1 2 , p -3 2 , . . . , 3 2 , in that order, to the right of p + 1 2 , leaving 1 2 at the end. The sign produced in this way is (-1) (q-p)+(q-p+1)+•••+(q-2) = (-1) (p-1)(q-p-1)+ (p-1)p 2 = (-1) (p-1)(q-p-1)+[ p 2 ] ,

and it follows from Lemma 5.10 that

P 1 q (Λ R ) = (-1) (p-1)(q-p-1)+[ p 2 ] 2 q-1 .
Putting this together with the fact that

#A + #C = #C = (p -1)(q -p),
that N of (3.3) satisfies (7.10)

N ≡ p -1 mod 2,
and that

[ p 2 ] + p -1 ≡ [ p -1 2 ] mod 2,
we see that (3.5) implies

Theorem 7.11. For G R = SO e (2p, 2q), q ≥ p ≥ 1, the constant c p,q

1 is c p,q 1 = (-1) [ p-1 2 ] 2 2p-2 .
7.2. The second real form. This real form exists for q ≥ p ≥ 2. It corresponds to

h 2 = (2, 1, . . . , 1 p-2 , -1 | 1, . . . , 1 p-1 , 0, . . . , 0 q-p+1
).

This real form is conjugate to the first real form by the automorphism σ which acts on h by changing the sign of the p-th coordinate, and leaving all other coordinates the same. On the level of g, this is an outer automorphism, which become the standard one if we compose it with the isomorphism so(2p, 2q) ∼ = so(2q, 2p). The automorphism σ satisfies the conditions of Lemma 3.8, and we just have to compute the sign. The number n from Proposition 3.8 is, as in Subsection 5.1, equal to p-1.

The number N 1 is by (7.10) congruent to p -1 modulo 2. Finally, N 2 is easily seen to be always even. The conclusion is that there is no sign in Proposition 3.8, so (7.12) c p,q 2 = c p,q 1 = (-1) [ p-1 2 ] 2 2p-2 .

7.3. The third real form. This real form exists for all q ≥ p ≥ 1. It corresponds to

h 3 = (1, . . . , 1 p-1 , 0 | 2, 1, . . . , 1 p-1 , 0, . . . , 0 q-p
).

Since l = l 3 is built from roots that vanish on h 3 , we see that

∆ + n (l) = {ε i -ε j 1 ≤ i ≤ p -1, p + 2 ≤ j ≤ 2p} ∪ {ε p ± ε j 2p + 1 ≤ j ≤ p + q}. It follows that for any A ⊆ ∆ + n (l), (7.13) 2ρ(A) = (a 1 , . . . , a p-1 ; x | 0; -b 1 , . . . , -b p-1 ; y 1 , . . . , y q-p ), with 0 ≤ a i , b j ≤ p -1; i a i = j b j ; (7.14) 0 ≤ x ≤ 2(q -p); -1 ≤ y j ≤ 1.
Furthermore, recall that ∆(p 1 ) consists of noncompact roots that are 1 on h 3 . So

∆(p 1 ) = {ε i ± ε j 1 ≤ i ≤ p -1, 2p + 1 ≤ j ≤ p + q}∪ ∪ {ε j ± ε p p + 2 ≤ j ≤ 2p} ∪ {ε p+1 -ε i 1 ≤ i ≤ p -1}.
It follows that for any C ⊆ ∆(p 1 ), (7.15) 2ρ(C) = (c 1 , . . . , c p-1 ; u | v; d 1 , . . . , d p-1 ; e 1 , . . . , e q-p ), with

-1 ≤ c i ≤ 2(q -p); -(p -1) ≤ u ≤ p -1; (7.16) 0 ≤ v ≤ p -1; 0 ≤ d j ≤ 2; -(p -1) ≤ e j ≤ p -1.
If we write (7.13) for A = ∆ + n (l), we get ρ n (l) = (p -1, . . . , p -1; 2(q -p) | 0; -p + 1, . . . , -p + 1; 0, . . . , 0). This is clearly orthogonal to all roots of l ∩ k, which are equal to

∆(l ∩ k) = {ε i -ε j 1 ≤ i, j ≤ p -1} ∪ {ε i -ε j p + 2 ≤ i, j ≤ 2p}∪ (7.17) ∪{ε i ± ε j 2p + 1 ≤ i, j ≤ p + q}.
By Proposition 3.4, this means that we can determine the constant c = c p,q 3 from the equation (3.5). We will use this for λ = λ 0 , where

λ 0 = (q - 3 2 , q - 5 2 . . . , q -p + 1 2 ; q -p + 1 2 | p - 3 2 ; 1 2 , - 1 2 , . . . , -p + 5 2 ; (7.18) q -p - 1 2 , q -p - 3 2 , . . . , 1 2 ).
(Out of the 5 groups of coordinates separated by semicolons and the bar, the first and the fourth group are missing if p = 1, and the fifth group is missing if q = p.)

Proposition 7.19. Let Λ = λ 0 -2ρ(A) -2ρ(C), with λ 0 given by (7.18), and with

A ⊆ ∆ + n (l) and C ⊆ ∆(p 1 ). If P K (Λ) = 0, then (1) If q > p ≥ 2, then A = {ε p -ε j 2p + 1 ≤ j ≤ p + q}; C = {ε i -ε j 1 ≤ i ≤ p -1, 2p + 1 ≤ j ≤ p + q}∪ ∪ {ε j ± ε p p + 2 ≤ j ≤ 2p} ∪ {ε p+1 -ε i 1 ≤ i ≤ p -1}; Λ = (p - 1 2 , p - 3 2 , . . . , 3
2 ; 1 2 | - 1 2 ; - 3 2 , - 5 2 , . . . , -(p - 1 2 ); q - 1 2 , q - 3 2 , . . . , p + 1 2 ).
(2) If q > p = 1, then A is as in (1), C = ∅, and

Λ = ( 1 2 | - 1 2 ; q - 1 2 , q - 3 2 , . . . , 3 2 ).
(

) If q = p ≥ 2, then A = ∅; C = {ε j ± ε p p + 2 ≤ j ≤ 2p} ∪ {ε p+1 -ε i 1 ≤ i ≤ p -1}; Λ = (p - 1 2 , p - 3 2 , . . . , 3
2 ; 1 2 | - 1 2 ; - 3 2 , - 5 2 , . . . , -(p - 1 2 )). 3
and consequently b p-1 = 0; d p-1 = 2.

It follows that

ε i -ε 2p / ∈ A, 1 ≤ i ≤ p -1; ε 2p ± ε p ∈ C, and therefore 0 ≤ a i ≤ p -2, 1 ≤ i ≤ p -1; -(p -2) ≤ u ≤ p -2.
We see that

Λ = (p -1 2 , q -5 2 -a 2 -c 2 , . . . , q -p + 1 2 -a p-1 -c p-1 ; q -p + 1 2 -x -u | | p -3 2 -v; 1 2 + b 1 -d 1 , -1 2 + b 2 -d 2 , . . . , -(p -7 2 ) + b p-2 -d p-2 , -(p - 1
2 ); q -1 2 , q -3 2 , . . . , p + 1 2 ). (If q = p, the coordinates q -1 2 , . . . , p + 1 2 are not there; if p = 2 there are no coordinates involving a i , c i , b i or d i .)

We now consider the subalgebra g ∼ = so(2p -2, 2p -1) of g built on coordinates ε 2 , ε 3 , . . . , ε p ; ε p+1 , . . . , ε 2p-1 , so the coordinates 1 and 2p, 2p + 1, . . . , p + q are deleted. We also consider the real form of O K given by h

3 = (1, . . . , 1 p-2 , 0 | 2, 1, . . . , 1 p-2
),

with centralizer l = l ∩ g . Then

∆ + n (l ) = {ε i -ε j 2 ≤ i ≤ p -1, p + 2 ≤ j ≤ 2p -1}; ∆(p 1 ) = {ε j ± ε p p + 2 ≤ j ≤ 2p -1} ∪ {ε p+1 -ε i 2 ≤ i ≤ p -1}. We set A = A ∩ ∆ + n (l ) = A \ {ε p -ε j 2p + 1 ≤ j ≤ p + q}; C = C ∩ ∆(p 1 ) = C \ {ε 2p ± ε p ; ε p+1 -ε 1 ; ε i -ε j 1 ≤ i ≤ p -1, 2p + 1 ≤ j ≤ p + q}. (If q = p, then A = A and C = C \ {ε 2p ± ε p ; ε p+1 -ε 1 }.) Then 2ρ(A ) = (a 2 , . . . , a p-1 ; 0 | 0; -b 1 , . . . , -b p-2 ) = (a 1 , . . . , a p-2 ; 0 | 0; -b 1 , . . . , -b p-2 ); 2ρ(C ) = (c 2 -(q -p), . . . , c p-1 -(q -p); u | v -1; d 1 , . . . , d p-2 ) = (c 1 , . . . , c p-2 ; u | v ; d 1 , . . . , d p-2 ),
where we define

a i = a i+1 ; b i = b i ; c i = c i+1 -(q -p); u = u; v = v -1; d i = d i .
The numbers a i , b i , c i , u , v , d i satisfy analogues of (7.14) and (7.16). and the fact that N of (3.3) satisfies (7.25)

N ≡ p mod 2 we get Theorem 7.26. For G R = SO e (2p, 2q), the constant c p,q 3 corresponding to the third real form of O C is

c p,q 3 = (-1) [ p 2 ]+1 2 2p-1 , q > p; (-1) [ p 2 ]+1 2 2p-2 , q = p.
7.4. The fourth real form. This real form exists if q = p ≥ 2. It corresponds to

h 4 = (1, . . . , 1 p-1 , 0 | 2, 1, . . . , 1 p-2
, -1).

This real form is conjugate to the third real form by the automorphism σ which acts on h by changing the sign of the last coordinate, and leaving all other coordinates the same. On the level of g, this is the standard outer automorphism. It satisfies the conditions of Proposition 3.8, and we just have to compute the sign. The number n from Proposition 3.8 is equal to 0, since σ preserves ∆ + . The number N 1 is by (7.25) congruent to p modulo 2. Finally, N 2 is the same as N 1 because σ preserves ∆ + (or one can do a computation). So there is no sign in Proposition 3.8, and

(7.27) c p,q 4 = (-1) [ p 2 ]+1 2 2p-2
8. The case G R = SO * (2n), n ≥ 1 8.1. The case of even n. If n is even, the real forms of O C correspond to even integers p such that 0 ≤ p ≤ n. We denote n -p by q. The h corresponding to p is

h p = (1, . . . , 1 p | -1, . . . , -1 q ).
Since l = l p is built from roots that vanish on h p , we see that

∆ + n (l) = {ε i + ε p+j 1 ≤ i ≤ p, 1 ≤ j ≤ q}.
It follows that for any A ⊆ ∆ + n (l), (8.1) 2ρ(A) = (a 1 , . . . , a p | b 1 , . . . , b q ), with 0 ≤ a i ≤ q, 0 ≤ b j ≤ p, i a i = j b j . (8.2)

In particular, ρ n (l) = (q, . . . , q | p, . . . , p), and this is clearly orthogonal to the roots of l ∩ k, which are given by (8.3) ∆ + (l ∩ k) = {ε i -ε j 1 ≤ i < j ≤ p} ∪ {ε p+i -ε p+j 1 ≤ i < j ≤ q}.

So the constants c = c n p can be calculated from (3.5). Since it is clear that in the present case ∆(p 1 ) = ∅, 8.2. The case of odd n. For odd n the real forms of O C correspond to even integers p such that 0 ≤ p ≤ n -1. We denote n -1 -p by q, so q is another even integer. The h corresponding to p is Since l = l p is built from roots that vanish on h p , we see that ∆ + n (l) = {ε i + ε p+1+j 1 ≤ i ≤ p, 1 ≤ j ≤ q}. It follows that for any A ⊆ ∆ + n (l), (8.8) 2ρ(A) = (a 1 , . . . , a p | 0; b 1 , . . . , b q ), with 0 ≤ a i ≤ q, 0 ≤ b j ≤ p, i a i = j b j . (8.9)

In particular, ρ n (l) = (q, . . . , q | 0; p, . . . , p), and this is clearly orthogonal to the roots of l ∩ k, which are given by (8.10) ∆ + (l ∩ k) = {ε i -ε j 1 ≤ i < j ≤ p} ∪ {ε p+1+i -ε p+1+j 1 ≤ i < j ≤ q}.

So the constants c = c n p can be calculated from (3.5). The set ∆(p 1 ) consisting of noncompact roots that are 1 on h p is To compute the constant c n p using (3.5), we take λ = λ 0 , where λ 0 = (n, n -1, . . . , q + 2 | p + 1; n -1, n -2, . . . , p + 1), p, q > 0; (8.13) λ 0 = ( | 1; n -1, n -2, . . . , 1), p = 0, q > 0;

∆(p 1 ) = {ε i + ε p+1 1 ≤ i ≤ p} ∪ {-ε p+1 -ε p+1+j 1 ≤ j ≤ q}.
λ 0 = (n, n -1, . . . , 2 | p + 1), p > 0, q = 0; λ 0 = ( | 1), p = q = 0.

Using (8.10), we see that λ 0 differs from ρ l∩k by a weight orthogonal to all roots of l ∩ k, and hence (8.14) P L∩K (λ 0 ) = 1.

So to compute c n p we have to compute the left side of (3.5). The following proposition describes the relevant A and C, and the corresponding Λ.

Proposition 8.15. Let Λ = λ 0 -2ρ(A) -2ρ(C), with λ 0 given by (8.13), and with A ⊆ ∆ + n (l), C ⊆ ∆(p 1 ) as above. Suppose that for some A and C the corresponding Λ satisfies P K (Λ) = 0. Let r = p 2 and s = q 2 . Then there is an (r, s) shuffle

1 ≤ i 1 < • • • < i r ≤ r + s; 1 ≤ j 1 < • • • < j s ≤ r + s
of 1, 2, . . . , r + s such that

A = {α u,v , β u,v 1 ≤ u ≤ r, 1 ≤ v ≤ s},
where

α u,v = ε p+1-u + ε n+1-v ; β u,v = ε p+1-u + ε p+1+v , i u < j v ; ε u + ε n+1-v , i u > j v ,
and

C = {ε i + ε p+1 r + 1 ≤ i ≤ p} ∪ {-ε p+1 -ε p+1+j 1 ≤ j ≤ s}.
The corresponding Λ is Λ = (n+1-i 1 , . . . , n+1-i r , i r , . . . , i 1 | r +s+1; n+1-j 1 , . . . , n+1-j s , j s , . . . , j 1 ).

If p = 0, then the shuffle is necessarily trivial, i.e., there are no i u and (j 1 , . . . , j s ) = (1, . . . , s). This means that

A = ∅; C = {-ε p+1 -ε p+1+j 1 ≤ j ≤ s};
Λ = ( | s + 1; n, . . . , n + 1 -s, s, . . . , 1) = ( | s + 1; 2s + 1, . . . , s + 2, s, . . . , 1).

Similarly, if q = 0 than (i 1 , . . . , i r ) = (1, . . . , r), there are no j v , and

A = ∅; C = {ε i + ε p+1 r + 1 ≤ i ≤ p};
Λ = (n, . . . , n + 1 -r, r, . . . , 1 | r + 1) = (2r + 1, . . . , r + 2, r, . . . , 1 | r + 1).

Finally, if p = q = 0, i.e., n = 1, then the shuffle contains no i u or j v , A = C = ∅, and Λ = λ 0 = ( | 1).

Proof. The statement is obviously true if n = 1, i.e., if p = q = 0. We proceed by induction on n. So let us assume that n ≥ 3 is odd, and let 0 ≤ p ≤ n -1 be an even integer. We assume that the statement is true for n -2 and for any even integer p between 0 and n -3.

Using the definitions and the inequalities (8.9), we see that

Λ = (n -a 1 -c 1 [p,n] , n -1 -a 2 -c 2 [p-1,n-1] , . . . , q + 2 -a p -c p [1,q+2] | p + 1 -d [1,n] ; n -1 -b 1 + e 1 [q,n]
, n -2 -b 2 + e 2 [q-1,n-1] , . . . , p + 1 -b q + e q [1,p+2]

).

So the coordinates of Λ are n integers between 1 and n, and assuming that P K (Λ) = 0, they have to be different from each other, i.e., Λ has to be a permutation of (n, . . . , 1). In particular, some Λ i must be equal to n and there are three possibilities:

(8.16) Λ 1 = n or Λ p+1 = n or Λ p+2 = n.

Assume first that Λ 1 = n; this is only possible if p > 0, i.e., p ≥ 2. Then a 1 = 0, c 1 = 0, and it follows that

ε 1 + ε p+1+j / ∈ A, 1 ≤ j ≤ q; ε 1 + ε p+1 / ∈ C.
This implies that 0 ≤ b j ≤ p -1, 1 ≤ j ≤ q;

-q ≤ d ≤ p -1, , . . . , p + 1 -b q + e q [2,p+2]

).

We see that there is exactly one place where 1 can be, i.e., Λ p = 1. This implies a p = q, c p = 1, and therefore , . . . , p + 1 -b q + e q [2,p+1]

ε p + ε p+1+j ∈ A, 1 ≤ j ≤ q; ε p + ε p+1 ∈ C.
).

Let now g ∼ = so * (2(n -2)) be the subalgebra of g built on coordinates 2, . . . , p -1, p + 1, . . . , n, and let l = l ∩ g . We consider the real form of the corresponding O C given by h = h p-2 . Then ∆ + n (l ) = ∆ + n (l) \ {ε 1 + ε p+1+j , ε p + ε p+1+j 1 ≤ j ≤ q}; ∆(p 1 ) = ∆(p 1 ) \ {ε 1 + ε p+1 , ε p + ε p+1 }, and we set

A = A \ {ε p + ε p+1+j 1 ≤ j ≤ q}; C = C \ {ε p + ε p+1 }.
We define λ 0 as in (8.13), but with n replaced by n-2 and p replaced by p-2. Then Λ corresponding to A and C can be obtained from Λ by deleting coordinates Λ 1 Theorem 8.18. Let G R = SO * (2n), for an odd n ≥ 1, and let p, 0 ≤ p ≤ n -1, be an even integer. Let r = p 2 and let s = n-1-p

2

. Then the constant c n p for the real form of O C corresponding to p is c n p = (-1)

p 2
r + s r .

For the convenience of the reader, below is a table that gives the value of the constant for each real form in every case.

  It follows that for any C ⊆ ∆(p 1 ),(7.3) 2ρ(C) = (c; d 1 , . . . , d p-1 | -c 1 , . . . , -c p-1 ; e 1 , . . . , e q-p+1 ), with 0 ≤ c j ≤ 1; 0 ≤ c ≤ p -1; c = j c j ; (7.4) 0 ≤ d i ≤ 2(q -p + 1);-(p -1) ≤ e j ≤ p -1.

  So for any C ⊆ ∆(p 1 ), (8.11) 2ρ(C) = (c 1 , . . . , c p | d; -e 1 , . . . , -e q ), with 0≤ c i ≤ 1; -q ≤ d ≤ p; 0 ≤ e j ≤ 1; d = i c ij e j .(8.12)

  j ≤ p -1, 1 ≤ j ≤ q, -q + 1 ≤ d ≤ p -1 and so Λ = (n, n -1 -a 2 -c 2 [p-1,n-1] , . . . , q + 3 -a p-1 -c p-1 [2,q+3] , 1 | p + 1 -d [2,n-1] ; n -1 -b 1 + e 1 [q+1,n-1] , n -2 -b 2 + e 2[q,n-2]

  ,••• ,d k ] is a nilpotent element in the complex Lie algebra sl n . Write O [d1,d2,••• ,d k ] for the complex nilpotent orbit under the adjoint group P SL n of sl n . It is convenient to attach to O [d1,d2,••• ,d k ] a Young tableau, i.e a left-justified arrangement of empty boxes of rows with size in the non-increasing order d 1 , d 2 , ..., d k .

The second named author was supported by grant no. 4176 of the Croatian Science Foundation and by the QuantiXLie Centre of Excellence, a project cofinanced by the Croatian Government and European Union through the European Regional Development Fund -the Competitiveness and Cohesion Operational Programme (KK.01.1.1.01.0004). The third named author was supported in part by NSF grant DMS 0967272.

(If p = 1, then h 1 = (2 | 0, . . . , 0), so ∆ + n (l) = ∆(p 1 ) = ∅.) By (7.1), ρ n (l) = (0, p -1, . . . , p -1 | -p + 1, . . . , -p + 1, 0, . . . , 0). This is clearly orthogonal to all roots of l ∩ k, which are equal to ∆(l ∩ k) = {ε i -ε j 2 ≤ i, j ≤ p} ∪ {ε i -ε j p + 1 ≤ i, j ≤ 2p -1}∪ (7.5)

By Proposition 3.4, this means that we can determine the constant c = c p,q 1 from the equation (3.5). We apply (3.5) for λ = λ 0 , where

Proposition 7.7. Let Λ = λ 0 -2ρ(A) -2ρ(C), with λ 0 given by (7.6), and with A ⊆ ∆ + n (l) and C ⊆ ∆(p 1 ). If P K (Λ) = 0, then:

Proof. The proposition is clear if p = 1, since in that case ∆ + n (l) = ∆(p 1 ) = ∅. We continue by induction on p. Let p ≥ 2 and let q ≥ p be arbitrary. We assume that the proposition is true for p = p -1 and q = p -1, and we show it is then also true for p and q. By (7.6), (7.1) and (7.3), we have

). Using (7.2) and (7.4), we see that the coordinates Λ p+1 , . . . , Λ p+q are in the following intervals:

).

So Λ p+1 , . . . , Λ p+q are q half-integers between -(p -1 2 ) and q -1 2 . Moreover, P K (Λ) = 0 implies that no two of these half-integers are equal or opposite to each Using (7.9) and (7.8), we see that Λ 1 , . . . , Λ p are in the following intervals:

So Λ 1 , . . . , Λ p are p half-integers between -(p -3 2 ) and p -1 2 . As before, these half-integers must be different from each other, and no two of them are opposite, so one of them must be equal to p -1 2 , and the only possibility is

So

It follows that

So we see that

2 ); q -1 2 , q -3 2 , . . . , p + 1 2 , 1 2 -e q-p+1 ).

(The coordinates q -1 2 , q -3 2 , . . . , p + 1 2 are not there if q = p.) We now consider the subalgebra g ∼ = so(2p -2, 2p -3) of g built on coordinates ε 1 ; ε 3 , . . . , ε p ; ε p+1 , . . . , ε 2p-2 ; ε p+q , so the coordinates 2 and 2p -1, 2p, . . . , p + q -1 are deleted. We also consider the real form of O K given by

We set

Proof. By (7.18), (7.13) and (7.15), we have

. . , 1 2 -y q-p -e q-p ).

There are five groups of coordinates separated by semicolons and the bar. If p = 1, then the first and the fourth group of coordinates are missing, and if q = p, then the fifth group of coordinates is missing. Using (7.14) and ( 7.16), we see

).

So Λ p+1 , . . . , Λ p+q are q half-integers between -(p-1 2 ) and q -1 2 . Since P K (Λ) = 0, no two of them are equal or opposite to each other. If q > p, it follows that q -1 2 , q -3 2 , . . . , p + 1 2 must each be equal to some Λ i , and the only possibility for that is

It follows that y 1 , . . . , y q-p are all equal to -1, and that e 1 , . . . , e q-p are all equal to -(p -1).

In case q > p = 1, this implies that A is as stated in the proposition, and it is also clear that C = ∅. Moreover, it follows that x = q -p, and so Λ is as stated in the proposition. Since the case q = p = 1 is obvious, this proves the proposition for p = 1 and any q ≥ p.

We proceed by induction on p. Let p ≥ 2 and let q ≥ p be arbitrary. Assuming that the proposition is true for p = q = p -1, we will prove it for p and q.

If q > p, we get back to (7.21) and see that it implies

This implies

, then the above discussion does not apply; in this case (7.22) is true, but this information is already contained in (7.14) and (7.16). The following discussion applies equally to q > p and q = p, but in the latter case the last group of coordinates should be deleted. Using (7.22) together with the inequalities (7.14) for a i and (7.16) for u, we see

2 ) and p -1 2 , such that no two of them are equal or opposite to each other. There are two possibilities:

and it follows that

This implies

So Λ p+1 , . . . , Λ 2p are p half-integers between -(p -3 2 ) and p -3 2 , such that no two of them are equal or opposite to each other, and this is impossible.

We conclude that

and consequently

and therefore

; q -1 2 , q -3 2 , . . . , p + 1 2 ). So Λ p+1 , . . . , Λ 2p are p half-integers between -(p -1 2 ) and p -3 2 , such that no two of them are equal or opposite to each other. It follows that

We define λ 0 by (7.18), but for G R = SO e (2p -2, 2p -1), i.e.,

)).

Then A , C and Λ = λ 0 -2ρ(A ) -2ρ(C ) satisfy all conditions of the proposition, but p, q are reduced to p = p-1, q = p-1. Therefore the inductive assumption implies that A = ∅, that

This implies the statement of the proposition for A, C and Λ.

In view of (3.5), to compute the constant c = c p,q 3 we need to compute P L∩K (λ 0 ) and P K (Λ), where λ 0 is given by (7.18), and Λ is given by Proposition 7.19.

To compute P L∩K (λ 0 ), we note that we described l ∩ k in (7.17); it has up to three factors, two of which are u(p -1), and the third is so(2(q -p)). From the shape of λ 0 it now follows that, in the notation of Lemma 5.10, P L∩K (λ 0 ) = P 1 q-p (λ q-p ), and in case q > p, we see that Lemma 5.10(i) implies that P L∩K (λ 0 ) = 2 q-p-1 .

If q = p, then P L∩K (λ 0 ) = 1, which is not covered by the above formula. (In Lemma 5.10, we could have defined P 1 0 = 1 and λ 0 = 0, but the formula in Lemma 5.10(i) would not work for p = 0.)

To compute P K (Λ) for Λ as in Proposition 7.19, we first write Λ = (Λ L | Λ R ) and note that (7.23) P K (Λ) = P 1 p (Λ L )P 1 q (Λ R ). By Lemma 5.10(i), (7.24) P 1 p (Λ L ) = P 1 p (λ p ) = 2 p-1 . To apply Lemma 5.10 also for Λ R , we must first rearrange coordinates of Λ R , using the fact that P 1 q is skew for the Weyl group of so(2q), and invariant under sign changes of the variables.

To bring Λ R to µ q = (q, . . . , 1), after removing the signs which does not change the expression, we need to bring coordinates

in that order, to the right of p + 1 2 . The sign produced in this way is (-1) (q-p)+(q-p+1)+•••+(q-1) = (-1) p(q-p)+[ p 2 ] ,

and it follows from Lemma 5.10 that

Putting this together with (7.23), (7.24), the fact that #A = q -p; #C = 3(p -1) + (p -1)(q -p),

(3.5) becomes (8.4)

We take λ = λ 0 , where (8.5) λ 0 = (n, n -1, . . . , q + 1 | n, n -1, . . . , p + 1), (If p is 0 or n, then there is only one group of coordinates in the above expression, and λ 0 = (n, n -1, . . . , 1).) Since λ 0 differs from ρ l∩k by a weight orthogonal to all roots of l ∩ k,

So to compute c n p we have to compute the left side of (8.4). The following proposition describes the relevant A and the corresponding Λ.

Proposition 8.6. Let Λ = λ 0 -2ρ(A), with λ 0 given by (8.5), and with A ⊆ ∆ + n (l). Suppose that for some A the corresponding Λ satisfies P K (Λ) = 0. Then:

(1) If p = 0 or q = 0, then A = ∅ and Λ = λ 0 = (n, n -1, . . . , 1).

(2) If p, q > 0, let r = p 2 and s = q 2 . Then there is a shuffle

where

The corresponding Λ is

Proof. The situation is combinatorially exactly the same as for G R = Sp(2n, R), with n, p and q even. Therefore the proof of Proposition 6.5 applies verbatim; the only difference is that the present proof is simpler because p and q are even.

The complete parallel with the case of G R = Sp(2n, R), with n, p and q even extends also to the computation of P K (Λ) for any Λ from Proposition 8.6, and the constant c n p . The only difference is that in the present case, N of (3.3) is

so the sign is now (-1) p 2 . We conclude Theorem 8.7. Let G R = SO * (2n), with n ≥ 2 even. Let p, 0 ≤ p ≤ n, be an even integer. Let r = p 2 and let s = n-p 2 . Then the constant c n p for the real form of O C corresponding to p is

and Λ p , and decreasing all the other coordinates by 1. More precisely, deleting the first and the p-th coordinate, we have

We now see that Λ is a permutation of (n, . . . , 1) if and only if Λ is a permutation of (n -2, . . . , 1). By inductive assumption, this is equivalent to A and Λ being defined by a shuffle as in the statement of the proposition, and this clearly implies the same statement for A and Λ.

The second possibility in (8.16) is Λ p+1 = n, which implies d = -q and hence

This implies

and so

).

We see that there is no place where 1 could be, so this case is impossible if P K (Λ) = 0.

The third possibility in (8.16) is Λ p+2 = n; this is possible only if q > 0, i.e., q ≥ 2. It follows that b 1 = 0, e 1 = 1, and so

and hence

).

We see that there is exactly one place where 1 can be, i.e.,

Λ n = 1.

This implies b q = p, e q = 0 and therefore

and so

, 1).

We now reason in the same way as in the first case, and conclude that the proposition follows from the inductive assumption for n -2 with p staying the same and q being replaced by q -2.

To finish the computation of the constant c n p , we first note that for every A and C described in Proposition 8.15 (8.17) #A = 2rs; #C = r + s.

On the other hand, since Λ is a permutation of (n, . . . , 1), P K (Λ) is equal to ±1.

To compute the sign, we need to find the parity of the permutation bringing Λ to (n, . . . , 1). As in type C, we find this parity by counting the number of inversions in Λ when compared with (n, . . . , 1). We know from Proposition 8.15 that Λ = (n+1-i 1 , . . . , n+1-i r , i r , . . . , i 1 | r +s+1; n+1-j 1 , . . . , n+1-j s , j s , . . . , j 1 ).

Clearly i r , . . . , i 1 are in inversion with n + 1 -j 1 , . . . , n + 1 -j s ; that is rs inversions. Arguing as in type C, we get further rs inversions from the groups n + 1 -i 1 , . . . , n + 1 -i r and n + 1 -j 1 , . . . , n + 1 -j s , and i r , . . . , i 1 and j s , . . . , j 1 . Finally, the coordinate Λ p+1 = r + s + 1 is in inversion with i r , . . . , i 1 and n + 1 -j 1 , . . . , n + 1 -j s .

So the total number of inversions is 2rs + r + s, and combined with (8.17) this implies that the nonzero contributions to the sum in (6.3), which we know come from A, C and Λ as in Proposition 8.15, are all equal to (-1) #A+#C P K (Λ) = 1.

Furthermore, the number N of (3.3) satisfies

Since the number of nonzero summands is by Proposition 8.15 equal to the number of (r, s)-shuffles of r + s, i.e., to r+s r , we have proved:

SOe(2p, 2q + 1) SO(2p) × SO(2q + 1) q ≥ p ≥ 1 [3, 2 2p-2 , 1 2(q-p+1) ]

(2, 1,

) (-1) [(p/2)] 2 2p-2 3rd real form only if q > p -1 SOe(2p, 2q) SO(2p) × SO(2q) q ≥ p ≥ 1 [3, 2 2p-2 , 1 2(q-p)+1 ]

(2, 1,

) (-1) [(p-1)/2] 2 2p-2

(2, 1,

) (-1) [(p-1)/2] 2 2p-2

(1,