1. Introduction

RICHARDSON ORBITS FOR REAL CLASSICAL GROUPS

PETER E. TRAPA

Abstract. For classical real Lie groups, we compute the annihilators and asso ciated vari-

eties of the derived functor mo dules cohomologically induced from the trivial representation.

(Generalizing the standard terminology for complex groups, the nilp otent orbits that arise

as such asso ciated varieties are called Richardson orbits.) We show that every complex

sp ecial orbit has a real form which is Richardson. As a consequence of the annihilator

calculations, we give many new in nite families of simple highest weight mo dules with ir-

reducible asso ciated varieties. Finally we sketch the analogous computations for singular

derived functor mo dules in the weakly fair range and, as an application, outline a metho d

to detect nonnormality of complex nilp otent orbit closures.

1. Introduction

Fix a complex reductive Lie group, and consider its adjoint action on its Lie algebra g.

If q = l u is a parab olic subalgebra, then the G saturation of u admits a unique dense

orbit, and the nilp otent orbits which arise in this way are called Richardson orbits (following

their initial study in [R]). They are the simplest kind of induced orbits, and they play an

imp ortant role in the representation theory of G.

It is natural to extend this construction to the case of a linear real reductive Lie group

G . Let g denote the Lie algebra of G , write g for its complexi cation, and G for the

R R R

complexi cation of G . Let denote the Cartan involution of G , write g = k p for

R R

the complexi ed Cartan decomp osition, and let K denote the corresp onding subgroup of G.

Instead of considering nilp otent orbits of G on g , we work on the other side of the Kostant-

R R

Sekiguchi bijection and consider nilp otent K orbits on p. (As a matter of terminology, we say

that such an orbit O is a K -form of its G saturation.) Fix a -stable parab olic subalgebra

q = l u of g. Then the K saturation of u \ p admits a unique dense orbit, and we call the

orbits that arise in this way Richardson. It is easy to check that this de nition reduces to

the one given ab ove if G is itself complex.

It is convenient to give a slightly more geometric formulation of this de nition. A -stable

parab olic q de nes a closed orbit K q of K on G=Q (where Q is the corresp onding parab olic

subgroup of G). Let denote the pro jection from G=B to G=Q. Then (K q) has a

dense K orbit (say O ) and we may consider its conormal bundle in the cotangent bundle

to G=B . A little retracing of the de nitions shows the image of the conormal bundle to O

under the moment map for T (G=B ) is indeed the K saturation of u \ p, and we thus obtain

a second characterization of Richardson orbits: they arise as dense K orbits in the moment

map image of conormal bundles to orbits of the form O .

The ab ove geometric interpretation is esp ecially relevant in the context of the represen-

tation theory of G . Consider the irreducible Harish-Chandra mo dule (say A ) of trivial

R q

in nitesimal character attached to the trivial lo cal system on O by the Beilinson-Bernstein

This pap er was written while the author was an NSF Postdo ctoral Fellow at Harvard University. 1

2 PETER E. TRAPA

equivalence. Then A is a derived functor mo dule induced from a trivial character and is of

the form considered, for example, in [VZ]; see Section 2.2 for more details. From the preced-

ing discussion (esp ecially the fact that K q is closed), it is easy to see that the D -mo dule

characteristic variety of A is the closure of the conormal bundle to O . (The de nition of

q q

the characteristic variety is recalled in Section 2.3 b elow.) Since the moment map image of

the characteristic variety of a Harish-Chandra mo dule is its asso ciated variety, we arrive at

a third characterization of Richardson orbits: they are the nilp otent orbits of K on p that

arise as dense K orbits in the asso ciated varieties of mo dules of the form A . Note that

since the G saturation of the asso ciated variety of a Harish-Chandra mo dule with integral

in nitesimal character is sp ecial ([BV2 ]), this interpretation implies that the G saturation

of a Richardson orbit is a sp ecial nilp otent orbit for g.

The rst result of the present pap er is an explicit computation of Richardson orbits in

classical real Lie algebras. In typ e A, this is well-known (see [T3] for instance); we give the

answer for other typ es in Sections 3{7. This is not particularly dicult and amounts only

to some elementary linear algebra, but the answer do es have the a p osteriori consequence

that every complex sp ecial orbit has a Richardson K -form.

Theorem 1.1. Fix a special nilpotent orbit O for a complex classical group G. Then there

exists a real form G such that some irreducible component of O \ p is a Richardson orbit

of K on the nilpotent cone of p.

This result has the avor of a corresp onding result for admissible orbits. Mo dulo some

conjectures of Arthur and based on [V4], Vogan gave a simple conceptual pro of that for the

split real form of G, every complex sp ecial orbit has a K -form which is admissible. (Without

relying on the Arthur conjectures, the result has b een established in a case by case manner

for the classical groups by Schwarz [Sc ] and for the exceptional groups by Noel [No] and

Nevins [N].) It would b e worthwhile to check Theorem 1.1 for the exceptional groups. A

conceptual argument might b e very enlightening.

Our second main result concerns the annihilators of the mo dules A (). Using the ex-

plicit form of the computation of Richardson orbits in the classical case, one may adapt an

argument from [T2 ] to establish the following result.

Theorem 1.2. For the classical groups, the annihilator of any module of the form A is

explicitly computable.

The computation, which is carried out in Section 8.6 and is relatively clean, is made in

terms of the tableau classi cation of the primitive sp ectrum of U(g) due to Barbasch-Vogan

([BV1 ]) and Gar nkle ([G1 ]{[G4 ]). Using these calculations, one can immediately apply the

main techniques of [T2 ] to compute the annihilators and vanishing of many (and p ossibly

all) weakly fair A () mo dules of the classical groups. It is imp ortant to recall that these

highly singular mo dules can b e reducible, and implicit in the previous sentence is a metho d

to detect cases of such reducibility. In turn, a theorem of Vogan's (see [V3]) asserts that the

reducibility of a weakly fair A () is sucient to deduce the nonnormality of the complex

orbit closure that arises as the asso ciated variety of Ann (A ()). Together with the

U(g)

reducibility computations, this allows one in principle to deduce the nonnormality of certain

orbit closures. It may b e interesting to pursue these ideas in the still op en case of the very

even orbits in typ e D .

In this context, it is imp ortant to note that this metho d can b e used to work directly with SO (n; C ) (and

not O (n; C )) orbits; see Remark 7.3 b elow.

RICHARDSON ORBITS FOR REAL CLASSICAL GROUPS 3

The computations of annihilators of weakly fair A () mo dules may still seem rather tech-

nical to those unfamiliar with real groups. Yet they are imp ortant, even for applications to

simple highest weight mo dules. We prove the following result, which is logically indep endent

from the rest of the pap er, in Section 8.2; the notation is as in Section 2.1.

Theorem 1.3. Fix G complex semisimple (not necessarily classical). Suppose I is the

annihilator of an A module for some real form G of G. If Ann (L(w )) = I , then

q R

U(g)

AV(L(w )) is irreducible, i.e. is the closure of a unique orbital variety for g. If G is classical,

this orbital variety is e ectively computable.

This result gives new examples of simple highest weight mo dules with irreducible asso-

ciated varieties; using more re ned ideas (which will pursued elsewhere), it leads to many

more examples. The pro of of Theorem 1.3 is based on an interesting (but indirect) interac-

tion b etween the highest weight category and the category of Harish Chandra mo dules for a

real reductive group. It would b e very useful to understand this interaction in a more direct

manner.

2. Background and Notation

Throughout we retain the notation established in the intro duction for a real reductive Lie

group G .

2.1. Highest weight mo dules. Let h b e a subalgebra of g. In general, we write ind for

the change of rings functor U(g) .

U(h)

Fix a Borel b = h n in g, and write for the corresp onding half-sum of p ositive ro ots.

Let w denote the long element in W = W (h; g). For w 2 W , let M (w ) denote the Verma

mo dule ind (C ); here C is the one dimensional U(b) mo dule corresp onding to

w w w w

o o

the indicated weight. We write L(w ) for the unique simple quotient of M (w ).

j j

Given a highest weight mo dule, let X denote the subspace obtained by applying U (b)

applied to the highest weight vector. The asso ciated graded ob ject is a C [n ] mo dule; let

AV(X ) denote its supp ort.

2.2. The mo dules A . Fix G and let q = l u b e a -stable parab olic subalgebra of g. Let

q R

q, where the bar notation indicates L b e the analytic subgroup of G corresp onding to q \

R R

complex conjugation with resp ect to g . Consider the one dimensional (q ;L \ K ) mo dule

top

u, and set S = dim(u \ k). De ne

g;L\K g;K

top

( ) (ind A = ( u));

S q

q;L\K g;L\K

here is the S th derived Bernstein functor. A more detailed account of this notation can

b e found in [KV, Chapter 5].

2.3. Asso ciated varieties and characteristic cycles of Harish-Chandra mo dules.

Fix G , and let X b e a nite length (g;K ) mo dule. Fix a K -stable go o d ltration of X , and

consider the S (g) mo dule obtained by passing to the asso ciated graded ob ject gr(X ). By

identifying (g=k) with p (and noting the K -invariance of the ltration), we can consider the

supp ort of gr(X ) as a subvariety of p. This subvariety, denoted AV(X ), is a ( nite) union

of closures of nilp otent K orbits on p, and is called the asso ciated variety of X .

Let D denote the sheaf of algebraic di erential op erators on B, the variety of Borel sub-

algebras in g. If X has trivial in nitesimal character, we can rep eat the ab ove construction

4 PETER E. TRAPA

for the (D ;K ) mo dule D X . This de nes a subvariety CV (X ) of T B called the char-

U(g)

acteristic variety of X . It is a union of closures of conormal bundles to K orbits on B. The

moment map image of CV (X ) is AV(X ) (once we identify p with p ).

Both invariants may b e re ned by considering the rank of the asso ciated graded ob ject

along the irreducible comp onents of its supp ort. In the former case we obtain the asso ciated

cycle of X , a linear combination (with natural numb er co ecients) of closures of nilp otent

K orbits on p. In the latter case we obtain the characteristic cycle of X , a linear combination

of closures of conormal bundles to K orbits on B.

2.4. Tableaux. We adopt the standard (English) notation for Young diagrams and standard

Young tableaux of size n. We let YD (n) denote the set of Young diagrams of size n, and

SYT(n) the set of standard Young tableaux of size n.

A standard domino tableau of size n is a Young diagram of size 2n which is tiled by

two-by-one and one-by-two dominos lab eled in a standard con guration; that is, the tiles

are lab eled with distinct entries 1;::: ; n so that the entries increase across rows and down

columns. We let SDT (n) (resp. SDT (n)) denote the set of standard domino tableau of size

C D

n whose shap e is that of a nilp otent orbit for S p(2n; C ) (resp. O (2n; C ); see Prop osition 2.1.

Finally, we de ne SDT (n) to b e the set of Young diagrams of size 2n + 1 and shap e the

form of a nilp otent orbit for O (2n + 1); C ) (Prop osition 2.1), whose upp er left b ox is lab eled

0, and whose remaining 2n b oxes are tiled by dominos lab eled in a standard con guration.

A signed Young tableau of signature (p; q ) is an arrangement of p plus signs and q minus

signs in a Young diagram of size p + q so that the signs alternate across rows, mo dulo the

equivalence of interchanging rows of equal length. We denote the set of signature (p; q )

signed tableau by YT (p; q ). We let c(p; q ) denote the unique element of YT (p; q ) whose

shap e consists of a single column.

Given T 2 YT (p; q ), we write n for the numb er of rows of T of length j b eginning with

the sign , and set n = n + n .

j j

2.5. Adding a column to a signed tableau. The key combinatorial op eration in the

computation of asso ciated varieties is that of adding the column c(r; s) (notation as in

Section 2.4), either from the left or the right, to an existing T 2 YT (p; q ) to obtain new

tableaux

c(r; s) T ; T c(r; s) 2 YT (p + r; q + s):

We rst describ e T c(r; s). This signed tableau is obtained by by adding r pluses and

s minuses, from top to b ottom, to the row-ends of T so that

(1) at most one sign is added to each row-end; and

(2) the signs of the resulting diagram must alternate across rows.

(3) each sign is added to as high a row as p ossible, sub ject to requirements (1) and (2),

p ossibly after interchanging rows of equal length.

RICHARDSON ORBITS FOR REAL CLASSICAL GROUPS 5

For example,

+ + +

+ + + +

+ + + + +

+ + + +

+ + +

+ +

= :

The signed tableau c(r; s) T is obtained by exactly the same pro cedure, except that the

signs are added to the beginnings of each row of T . For example,

+ + + +

+ + + + +

+ + + + + +

+ + + +

+ + +

= :

The preceding discussion evidently gives two ways of de ning the addition of two columns

0 0

c(r ; s ) c(r ; s ). A little checking shows that they coincide. In general, we want to make

1 1

sense of successive addition of column,

c c c ;

1 2 n

but one must check that the op eration asso ciates in a suitable sense. For instance, by

c c c c , we could mean either c (c (c c )), c ((c c ) c ), or two other

1 2 3 4 1 2 3 4 1 2 3 4

p ossibilities. (Note that (c c ) (c c ) is not a p ossibility since the middle is not

1 2 3 4

the sum of a single column and a tableau.) We leave it to the reader to supply the details

of proving that the notation c c c is indeed well-de ned.

1 2 n

2.6. Orbits of G on N (g ). We recall the standard partition classi cation of nilp otent

orbits and sp ecial nilp otent orbits for complex classical Lie groups. (We state the result for

the disconnected even orthogonal group for applications b elow).

Prop osition 2.1 ([CMc, Chapter 6]). Recal l the notation of Section 2.4

(1) Orbits of SL(n; C ) on N (sl(n; C )) are parametrized by partitions of n. Al l orbits are

special.

(2) Orbits of S p(2n; C ) on N (sp(2n; C )) are parametrized by partitions of 2n in which

odd parts occur with even multiplicity. Such an orbit is special if and only if the

number of even rows between consecutive odd rows or greater than the largest odd

row is even.

(3) Orbits of O (n; C ) on N (so(n; C )) are parametrized by partitions in which even parts

occur with even multiplicity. If n is even (resp. odd), such an orbit is special if and

only if the number of odd rows between consecutive even rows is even and the number

of odd rows greater than the largest even row is even (resp. odd).

6 PETER E. TRAPA

2.7. Orbits of K on N (p ). We recall the following parametrization of K nN (p ) for various

classical real groups G . (These parametrizations di er slightly from the p erhaps more

standard one given in [CMc, Chapter 9], but the corresp ondence b etween the two is obvious.)

Recall that since O (p; q ) is disconnected, the complexi cation of O 2 K nN (p ) need not

b e a single orbit of SO (n; C ) on N (so(n; C )) , though it is of course single orbit under the

action of O (n; C ).

Prop osition 2.2. Recal l the notation of 2.4.

(1) For G = U (p; q ), K nN (p ) is parametrized by YT (p; q ). (As a matter of notation,

we set YT (SU (p; q )) = YT (p; q ).)

(2) For G = S p(2n; R ), K nN (p ) is parametrized by the subset

YT (S p(2n; R )) YT (n; n)

of elements such that for each xed odd part, the number of rows beginning with +

coincides with the number beginning with .

(3) For G = S p(p; q ), K nN (p ) is parametrized by the subset

YT (S p(p; q )) YT (2p; 2q )

consisting of signed tableaux such that

(a) For each xed even part, the number of rows beginning with + coincides with

the number beginning with ; and

(b) The multiplicity of each odd part beginning with + (respectively ) is even.

In particular, al l parts occur with even multiplicity, and the complexi cation of any

O 2 K nN (p ) is special.

(4) For G = SO (2n), K nN (p ) is parametrized by the subset

YT (SO (2n)) YT (2n; 2n)

consisting of signed tableaux such that

(a) For each xed odd part, the number of rows beginning with + coincides with the

number beginning with ; and

(b) The multiplicity of each even part beginning with + (respectively ) is even.

In particular, al l parts occur with even multiplicity, and the complexi cation of any

O 2 K nN (p ) is special.

(5) For G = O (p; q ), K nN (p ) is parametrized by the subset

YT (O (p; q )) YT (p; q )

consisting of signed tableaux such that for each xed even part, the number of rows

beginning with + equals the number beginning with .

Pro of. Although this is very well known, we will need some details of the parametrization

b elow. Consider rst the case of U (p; q ). Let E (resp. E ) b e a complex vector space of

dimension p (resp. q ). Then p = Hom(E ;E ) Hom(E ;E ), K = GL(E ) GL(E )

+ + +

acts in the natural way, and an element (A; B ) of p is nilp otent if and only if AB and BA are

nilp otent endomorphisms of E and E . Fix a basis e ;::: ; e for E and e ; : : : ; e for

+ +

p q

1 1

E . A mild generalization of the argument leading to the Jordan normal form of a nilp otent

endomorphism of C shows that any nilp otent (A; B ) 2 N (p) is a direct sum (in an obvious

sense) of terms of the form

+ + +

(1) e 7! e 7! e 7! e 7! e 7! 0;

i i i+1 i+1 i+j

RICHARDSON ORBITS FOR REAL CLASSICAL GROUPS 7

+ +

(2) e 7! e 7! e 7! e 7! e 7! 0;

i i i+1 i+1 i+j

or such a term with the e 's and e 's interchanged. We represent the rst term as a single

row + + + the second as + + and likewise for the other terms (with + and

signs inverted). This gives the parametrization in (1).

Parts (2){(5) follow easily along the same lines. For instance, consider part (2). Here

S p(2n; R ) = U (n; n) \ S p(2n; C ), and N (p) consists (as ab ove) of pairs (A; B ) sub ject to the

additional requirement that

X X

+ +

Ae = a e () Ae = a e ;

ij ij

i j n+1j n+1i

j i

and similarly for B . This symmetry requirement implies that the `Jordan blo cks' are now

either of the form

+ + + +

7! 0 7! e 7! e 7! e 7! e 7! e 7! e 7! e e

n+1i i+1 i i

k n+1k n+2k n+2k

(i.e. an even row b eginning with +), the ab ove element with the e 's and e 's interchanged

(i.e. an even row b eginning with ), or the pair

+ +

7! 0 (with k n) 7! e 7! e 7! e e

i+1 i i

+ + +

e 7! e 7! e 7! e 7! 0;

n+1i n+1i ni

n+1k

(i.e. a pair of o dd rows b eginning with opp osite signs). This gives the parametrization in

(2). The argument is nearly identical for (3){(5).

2.8. A collapse algorithm for S p(p; q ) and SO (2n). Let p = p +p and q = q +q . Fix

1 2 1 2

0 0

a tableau T 2 YT (S p(p ; q )). Then the tableau T = c(p ; q ) T c(p ; q ) need not

1 1 2 2 2 2

b elong to YT (S p(p; q )). We now de ne a combinatorial manipulation of T to pro duce a

new tableau T 2 YT (S p(p; q )) called the collapse of T (or c-collapse to distinguish it from

the d-collapse b elow). This is needed in the statement of Prop osition 5.1 b elow.

If T 2 YT (S p(p; q )), then set T = T . One can check that the de nition of T implies

that the numb er of rows of a xed even length b eginning with + coincides with the numb er

b eginning with . So if T 2= YT (S p(p; q )), there exists an o dd numb er of rows of a xed

o dd length (say 2k + 1) ending with sign . Cho ose k maximal with this prop erty, and x

such a row (say R ). The de nition of T implies that there is a row (say S ) of length 2k 1

ending with sign . Then move the terminal b ox lab eled in R to the end of the row

S . Rearrange the resulting diagram to obtain T 2 YT (2p; 2q ). If T 2 YT (S p(p; q )),

1 1

set T = T . Otherwise rep eat this pro cedure to obtain T . After a nite numb er of steps,

c 1 2

necessarily T 2 YT (S p(p; q )), and we set T = T .

l l

We need to develop an analogous pro cedure for SO (2n). Fix T 2 YT (SO (2n)). Then

the tableau T = c(p ; q ) T c(q ; p ) need not b elong YT (SO (2n)). If it do es, set

2 2 2 2

T = T . One can check from the de nition of T that the numb er of rows with a xed

o dd length b eginning with the sign is the same as the numb er b eginning with . So if

T 2= YT (SO (2n)), there exists some even row (of length, say, 2k ) and some sign such

that the numb er of rows of length 2k ending with the sign is o dd. Cho ose k maximal with

resp ect to this condition, and x such a row (say R ). The de nition of T implies that there

is some row (say S ) of length 2k 2 ending with the sign . (Here S may have length

0; in this case, we interpret S as ending with b oth + and .) Move the terminal b ox of

R to the end of S , and rearrange the resulting diagram to obtain T 2 YT (2n; 2n). If

8 PETER E. TRAPA

T 2 YT (SO (2n)), set T = T . Otherwise rep eat this pro cedure to obtain T . After a

1 1 2

nite numb er of steps, necessarily T 2 YT (SO (2n)), and we set T = T .

l d l

These algorithm can b e characterized as follows. (For later use, we also include the

analogous statements for the other relevant real forms.)

0 0

Prop osition 2.3. (1) Set G = U (p; q ), and x positive integers p ; p ; q ; q such that

R 1 1

0 0 0 0 0

p = p + p and q = q + q . Fix T 2 YT (p ; q ), and set

1 1

T = T c(p ; q ):

1 1

Then T parametrizes the largest orbit among those parametrized by tableaux obtained

from T by adding p plus signs (resp. q minus signs) to the row ends of the resulting

diagram.

(2) Set G = S p(2n; R ) and x positive integers m; p, and q such that n = m + p + q . Fix

T 2 YT (S p(2m; R )) and set

T = c(p; q ) T c(q ; p):

Then T 2 YT (S p(2n; R )), and T parametrizes the largest orbit among those parametrized

by tableaux obtained by adding p plus signs (resp. q minus signs) to the beginnings of

rows of T and q plus signs (resp. p minus signs) to the ends of rows of the resulting

diagram.

0 0 0

(3) Set G = S p(p; q ) and x positive integers p ; p ; q and q such that p = p + p and

R 1 1 1

0 0

q = q + q . Fix T 2 YT (S p(p; q )) and set

T = c(p ; q ) T c(p ; q ):

1 1 1 1

Then T (the c-col lapse of T de ned above) parametrizes the largest orbit among

those parametrized by tableaux obtained by adding p plus signs (resp. q minus signs)

1 1

to the beginnings of rows of T and p plus signs (resp. q minus signs) to the ends

1 1

of rows of the resulting diagram.

(4) Set G = SO (2n), and x positive integers m; p and q such that n = m + p + q . Fix

T 2 YT (SO (2m)) and set

T = c(p; q ) T c(q ; p):

Then T (the d-col lapse of T de ned above) parametrizes the largest orbit among

those parametrized by tableaux obtained by adding p plus signs (resp. q minus signs)

to the beginnings of rows of T and q plus signs (resp. p minus signs) to the ends of

rows of the resulting diagram.

0 0 0

(5) Set G = O (p; q ), and x positive integers p ; p ; q ; and q such that: p = p + p ;

R 1 1 1

0 0 0 00 0 0

q = q + q ; and p + q p + q mo d (2). Fix T 2 YT (O (p ; q )) and set

0 00

T = c(p; q ) T c(p; q ):

Then T 2 YT (O (p; q )), and T parametrizes the largest orbit among those parametrized

by tableaux obtained by adding p plus signs (resp. q minus signs) to the beginnings of

rows of T and p plus signs (resp. q minus signs) to the ends of rows of the resulting

diagram.

Sketch. This essentially follows from the parametrizations in Prop osition 2.2. The only

p oint that we have not made explicit is the closure order on nilp otent K orbits on p. Consider

the partial order on YT (2p; 2q ) corresp onding to closure order for U (p; q ) It is relatively

easy to check from the pro of of Prop osition 2.2 that this partial order is generated by the

RICHARDSON ORBITS FOR REAL CLASSICAL GROUPS 9

covering relations A A de ned as follows. Fix A 2 YT (p; q ), and cho ose a row (say R )

of length d in A. Supp ose this row ends in the sign , and cho ose a row (say R ) of maximal

0 0 0 0

length (say d ) sub ject to the condition that d d 2 and that R ends in . Here R may

have length 0, in which case we adopt the convention that its terminal sign is b oth + and

. Remove the terminal b ox of R lab eled with a + (thus changing R into a row of length

0 0 0

d 1 ending in ) and move it to the end of R (thus changing R to a row of length d to

one of length d + 1 ending in ). Rearrange the rows to have decreasing length. This de nes

0 0

a new element A 2 YT (p; q ) and we declare A A to b e a covering relation. Now the

statement in (1) is clear.

Consider the statements for (2),(3), and (5). In each of these cases we can emb ed G

e e

into an appropriate G = U (p; q ) and the K saturation of the K orbit parametrized by a

signed tableau T 2 YT (G ) is simply the orbit parametrized by T 2 YT (p; q ). So the

assertions in (2), (3), and (5) follow. The case of SO (2n) is slightly di erent, since it is not

a subgroup of U (p; q ), but it isn't much more dicult.

Remark 2.4. From the pro of of Prop osition 2.3, we obtain a more natural characterization

of the c-collapse algorithm. Set G = S p(p; q ). Then G is a subgroup of G = U (2p; 2q ).

R R R

Assume that the Cartan involutions of the two groups are arranged compatibly. Let p =

0 0

p + p and q = q + q , x T 2 YT (S p(p ; q )), and set T = c(p ; q ) T c(p ; q ). Let

1 2 1 2 1 1 2 2 2 2

e e e e

O denote the K orbit for G parametrized by T . Then the intersection of the closure of O

with p has a unique dense K orbit and it is parametrized by T .

e e

More generally, if O is any K orbit for G = U (p; q ), we have given enough details to

compute the intersection of the closure of O with p. We leave the formulation of this result

(and its analogs for O (p; q ) and S p(2n; R )) to the reader.

3. U (p; q )

The K conjugacy classes of -stable parab olic subalgebras for U (p; q ) are parametrized by

P P

an ordered sequence of pairs (p ; q );::: ; (p ; q ) such that p = p and q = q . The Levi

1 1 r r i i

i i

subgroup of U (p; q ) corresp onding to such a parab olic subalgebra is U (p ; q ) U (p ; q ).

1 1 r r

Prop osition 3.1. In the above notation, let q be parametrized by (p ; q );::: ; (p ; q ). Then

1 1 r r

AV(A ()) is the closure of the orbit parametrized by

c(p ; q ) c(p ; q ) c(p ; q )

1 1 2 2 r r

with notation as in Sections 2.5 and 2.7.

r 1

0 0 0 0 0

Pro of. Set p = p , and likewise for q . Let q denote the -stable parab olic for G =

i=1 R

0 0

U (p ; q ) parametrized by (p ; q );::: ; (p ; q ). Recall the pro cedure outlined in the pro of

1 1 r 1 r 1

p q q p

) ;E ) Hom(E ;E of Prop osition 2.2 that asso ciates to each nilp otent element of Hom(E

+ +

q p p p

. Implicitly this , and likewise for E E = E a signature (p; q ) signed tableau. Write E

+ + +

into G = U (p; q ), and with this in mind it is easy to check that de nes an inclusion of G

0 0

p q q p

0 0

r r

); ;E ) Hom (E ;E q \ u = q \ u Hom(E

+ +

where

0 0 0 0

p q q p

0 0

): ;E ) Hom(E ;E q \ u Hom (E

+ +

0 0 0 0 0

Fix a K orbit through 2 q \ u parametrized by T . From the description of the

parametrization in the pro of of Prop osition 2.2, it is clear that the K orbit through any

10 PETER E. TRAPA

0 0

= is parametrized by a tableau obtained from T by augmenting p 2 q \ u with j

q \u

0 0

rows of T ending with by a plus sign and q rows of T ending with + by a minus sign.

(An empty row is interpreted as ending with b oth + and .) Using the closure ordering

outlined in the pro of of Prop osition 2.3, we conclude that T c(p ; q ) parametrizes the

r r

0 0

= . The prop osition now follows by an easy largest K orbit through 2 q \ u with j

q \u

induction.

Corollary 3.2 (Barbasch-Vogan). Every K orbit for U (p; q ) is Richardson.

Pro of. If O is parametrized by T 2 YT (p; q ), let p (resp. q ) b e the numb er of plus

K i i

(resp. minus) signs in the ith column of T , and let q b e the -stable parab olic corresp onding

to the sequence of pairs (p ; q ); (p ; q );::: . Using Prop osition 3.1, it is easy to check that

1 1 2 2

O = AV(A ).

K q

4. S p(2n; R )

The K conjugacy classes of -stable parab olic subalgebras for S p(2n; R ) are parametrized

by a tuple consisting of a p ositive integer m n and an ordered sequence of pairs (p ; q );::: ; (p ; q )

1 1 r r

(p + q ) = n. The Levi subgroup of S p(2n; R ) corresp onding the such a such that m +

i i

parab olic subalgebra is S p(2m; R ) U (p ; q ) U (p ; q ).

1 1 r r

Prop osition 4.1. Retain the above notation, and let q = l u be parametrized by the

sequence m; (p ; q );::: ; (p ; q ). Then AV(A ()) is the closure of the orbit parametrized by

1 1 r r q

c(p ; q ) c(p ; q ) c(p ; q ) c(m; m) c(q ; p ) c(q ; p ) c(q ; p );

r r r 1 r 1 1 1 1 1 r 1 r 1 r r

with notation as in Sections 2.5 and 2.7 .

Pro of. The current prop osition follows in the same way as Prop osition 3.1 with only minor

r 1

0 0 0 00

mo di cations. Let G = S p(2n ; R ) where n = m + (p + q ), set n = p + q , and let

i i r r

R i=1

0 0

q b e parametrized by m; (p ; q );::: ; (p ; q ). Fix an inclusion G G = S p(2n; R ),

1 1 r 1 r 1 R

00 0

n n 0 0 n 0

. E = E U (n ; n ), and G U (n; n). As in the pro of of Prop osition 3.1, write E G

0 0

This de nes an inclusion U (n ; n ) U (n; n), and assume it is restricts to the inclusion of

G G . Then one checks directly that

0 0 0 0

q p q p

n n 0 0 n n

r r r r

); ) Hom(E ;E ;E ) u \ p Hom(E ;E ;E ) Hom(E u \ p Hom(E

+ +

where

0 0 0 0

0 0 n n n n

q \ u Hom(E ;E ) Hom(E ;E ):

+ +

0 0 0 0 0

Fix a K orbit through 2 q \ u parametrized by T . From the description of the

0 0

= is parametrized parametrization in the pro of of Prop osition 2.2, any 2 q \ u with j

q \u

by a tableau obtained from T by

(1) adding p plus signs (resp. q minus signs) to the beginnings of the rows of T that

r r

b egin with (resp. +); and

(2) to the resulting tableau, adding p minus signs (resp. q plus signs) to the ends of

r r

the rows of T that b egin with (resp. +),

with the usual convention that empty rows b egin and end with b oth plus and minus signs.

Now the prop osition follows from Prop osition 2.3 and an inductive argument.

Corollary 4.2. Let O 2 K nN (p ) be parametrized by a signed tableaux T . Then O is

Richardson if and only if T satis es the fol lowing conditions

RICHARDSON ORBITS FOR REAL CLASSICAL GROUPS 11

(1) The number of even rows between consecutive odd row or greater than the largest odd

row is even;

(2) Fix a maximal set (say S ) of even rows either between consecutive odd rows or greater

than the largest odd row. Then al l rows of a xed length (say 2k ) in S begin with

the same sign (say (k )). Moreover if rows of length 2k and 2l appear in S , then

k +l

(k )(l ) = (1) .

In particular, given a complex special orbit O , there exists some O 2 Irr (O \ p ) such that

O is Richardson.

Pro of. Given an orbit O of the form app earing in the corollary, we rst inductively

O . So x O and T as ab ove, and let c b e construct a -stable q such that AV(A ) =

K K q

the numb er of columns of T . If c = 1, O is the zero orbit and obviously Richardson.

Inductively we can assume that any orbit parametrized by a tableau T with less than c

columns (and satisfying the condition in the statement of the corollary) is Richardson. Fix

k maximal such that n + n 6= 0 (so, in particular, n or n may b e zero). There

2k +1 2k 2k +1 2k

are several cases to consider.

Case (I): There exists k < k such that n 6= 0.

2k +1

(a): n is even (p ossibly zero); the hyp othesis of the corollary then implies that n n =

2k 2k

0. De ne a new tableau T obtained by mo difying the rows of length greater than 2k 2 in

T as follows:

n +(n )=2+n n n

2k +1 2k 2k 1 2k 2 2k 2

; T = (2k 1) (2k 2) (2k 2) (1)

, but we maintain the ab ove notation for emphasis. One can = n here, of course, n

odd odd

0 0 0

check directly that T 2 YT (S p(2n ; R )) for 2n = 2n 2n n and, moreover, that it

2k +1 2k

satis es the conditions in the statement of the corollary. Since the numb er of columns of T is

0 0

strictly less than the numb er of columns of T , inductively we can nd a q for S p(2n ; R ) such

0 0 0

). Supp ose q is parametrized O = AV(A that the orbit O parametrized by T satis es

by the sequence . Let q b e parametrized by the sequence

+ +

; n + (n =2); n + (n =2) :

2k +1 2k 2k +1 2k

= 0, one can check directly that n Using the hyp othesis that n

2k 2k

+ + + +

c c n + (n =2); n + (n =2) T n + (n =2); n + (n =2)

2k +1 2k 2k +1 2k 2k +1 2k 2k +1 2k

O , as we wished to show. and Prop osition 4.1 implies AV(A ) =

K q

(b) n is o dd. (So, in particular, n = 0.) De ne T by mo difying the rows of length

2k 2k 1

greater than 2k 3 in T as follows

n n +n n

0 2k +1 2k 2k 2 2k 3

: T = (2k 1) (2k 2) (2k 3) (1)

0 0 0 0

Again one can check that T 2 YT (S p(2n ; R )) for n = nn n and that T satis es

2k +1 2k

0 0

the conditions of the corollary. Let O b e the orbit corresp onding to T , and by induction

0 0

) with q parametrized by . Let q b e parametrized by O = AV(A write

+ +

; (n + n ; n + n ):

2k +1 2k 2k +1 2k

O using Prop osition 4.1. Then one checks as in case (a) that AV(A ) =

K q

12 PETER E. TRAPA

Case (I I): n 6= 0 and there is no k < k such that n 6= 0. Mo dify the rows of

2k +1 2k +1

length greater than 2k 4 in T as follows

n +n n n

2k 4 2k 2 2k 2k +1

(2) : (2k 4) (2k 2) T = (2k 1)

0 0 0

Inductively, we can nd q for S p(2n ; R ) with n = n n n such that the closure

2k +1 2k

0 0 0

). Supp ose q is parametrized by . Let q for of the orbit parametrized by T is AV(A

S p(2n; R ) b e parametrized by

+ +

; (n + n ; n + n )

2k +1 2k 2k +1 2k

Then AV(A ) = O .

q K

The discussion in Case (I) and Case (I I) prove that every orbit app earing in the corollary

are Richardson. We also need to prove that every Richardson orbit is of this form. Supp ose

that q is parametrized by the sequence m; (p ; q );::: ; (p ; q ). Let q b e parametrized by

1 1 r r

the sequence m; (p ; q );::: ; (p ; q ). Again using zero as the base case, we can assume

1 1 r 1 r 1

) is of the form describ ed in that the tableau T parametrizing the dense orbit in AV(A

the Theorem. According to Prop osition 4.1, to check that the dense orbit in AV(A ) is of

the form describ ed in the corollary (and hence complete the pro of of the corollary), we need

to prove that c(p ; q ) T c(q ; p ) is of the required form. This is a straightforward

r r r r

combinatorial check whose details we omit

5. S p(p; q )

The K conjugacy classes of -stable parab olic subalgebras for S p(p; q ) are parametrized by

0 0

a tuple of a pair of integers (p ; q ) together with an ordered sequence of pairs (p ; q );::: ; (p ; q )

1 1 r r

P P

0 0

q = q . The Levi subgroup of S p(p; q ) corresp onding p = p and q + such that p +

i i

0 0

to this parab olic subalgebra is S p(p ; q ) U (p ; q ) U (p ; q ).

1 1 r r

Prop osition 5.1. In the above notation let q be parametrized by

0 0

(p ; q ); (p ; q );::: ; (p ; q );

1 1 r r

and recal l the col lapsing algorithm of Proposition 2.3(2). Then AV(A ()) is the closure of

the orbit parametrized by

0 0

c(p ; q ) c(p ; q ) [c(p ; q )c(p ; q )c(p ; q )] c(p ; q ) c(p ; q ) ;

r r r 1 r 1 1 1 1 1 c r 1 r 1 r r

c c

with notation as in Sections 2.5 and 2.7.

Sketch. This is very similar to Prop osition 4.1. The inductive analysis shows that p plus

signs must b e added to b oth the b eginning and ends of T , and likewise for q minus signs.

The app earance of the collapse algorithm is explained by Prop osition 2.3.

Corollary 5.2. Let O 2 K nN (p) be parametrized by T 2 YT (S p(p; q )). Then O is

K K

Richardson if and only if there exists and integer N such that

(1) For each j < N , al l rows of length 2j + 1 begin with the same sign; and

(2) For each j > N , the number of rows of length 2j is less than or equal to 4.

Pro of. As in the case of S p(2n; R ) it a detailed but elementary combinatorial check to

verify that every Richardson orbit is of the indicated form. We omit the details.

We now show that every orbit O app earing in the corollary is indeed Richardson by

O . As in the case of Corollary 4.2, the con- constructing a -stable q such that AV(A ) =

K q

struction is inductive, reducing the corollary to the case that O is the zero orbit and hence K

RICHARDSON ORBITS FOR REAL CLASSICAL GROUPS 13

obviously Richardson. Fix O as in the corollary, and let T b e the tableau parametrizing

it (Prop osition 2.2), and retain the notation for the index N as ab ove. Write

2k +1

(2k ) ; T = (2k + 1)

and assume that n + n is nonzero. There are a numb er of cases to consider.

2k +1 2k

Case (I) N = k , so all o dd rows of length less than 2k+1 b egin with the same sign . De ne

n n

2k +1 2k 1

0 n n p

2k 2

: (1) T = (2k 1) (2k 2) (2k 3) (2k 4)

0 0

Then T satis es the condition of the corollary (with N = N 1), and inductively there

0 0 0

exists a q parametrized by the sequence such that T parametrizes the orbit dense in

). Let q b e parametrized by AV(A

2k +1 2k +1

X X

+ 0 +

n : n ; n =2 + ; n =2 +

1 1

i i

i=1 i=2

Then T parametrizes the orbit dense in AV(A ).

Case (I I) N < k .

(a): n = 0. Mo dify the rows of length greater than 2k 2 in T as follows

+n n n

2k 1 2k 2 2k +1

(2k 2) : T = (2k 1)

0 0 0

Then T satis es the condition of the corollary so we can nd q parametrized by so that

). Let q b e parametrized by T parametrizes the dense orbit in AV(A

): ; n ; n

2k +1 2k +1

Then T parametrizes the dense orbit in AV(A ).

(b): n 6= 0. The conditions of the corollary imply that n = 2 or 4. In the former

2k 2k

+ +

case, cho ose (a ; a ) 2 f(0; 1); (1; 0)g; in the latter case, set (a ; a ) = (1; 1). Mo dify the

rows of length greater than 2k 2 in T as follows,

n +2a +n n

0 2k +1 2k 1 2k 2

T = (2k 1) (2k 2) :

0 0 0

Inductively we can assume there exists q parametrized by such that T parametrizes the

). Let q b e parametrized by dense orbit in AV(A

+ 0 +

+ a ): + a ; n ; n

2k +1 2k +1

Then T parametrizes the dense orbit in AV(A )

6. SO (2n).

The K conjugacy classes of -stable parab olic subalgebras for SO (2n) are parametrized

by a tuple consisting of a p ositive integer m n and an ordered sequence of pairs (p ; q );::: ; (p ; q )

1 1 r r

such that m + (p + q ) = n. The corresp onding Levi subgroup of SO (2n) is isomorphic

i i

to SO (2m) U (p ; q ) U (p ; q ).

1 1 r r

Prop osition 6.1. In the above notation let q be parametrized by

m; (p ; q );::: ; (p ; q );

1 1 r r

14 PETER E. TRAPA

and recal l the d-col lapsing algorithm of Proposition 2.3(3). Then AV(A ()) is the closure

of the orbit parametrized by

c(p ; q ) c(p ; q ) [c(p ; q )c(m; m)c(q ; p )] c(q ; p ) c(q ; p ) ;

r r r 1 r 1 1 1 1 1 r 1 r 1 r r

d d

with notation as in Sections 2.5 and 2.7.

Pro of. This is very similar to the cases already treated. We omit the details.

Corollary 6.2. Let O 2 K nN (p ) be parametrized by a signed tableaux T . Then O is

K K

Richardson if and only if there exists an integer N such that

(1) For each j < N , the number of rows of length 2j + 1 is less than or equal to 4; and

(2) For each j > N , al l rows of length 2j begin with the same sign.

Sketch. This is very similar to the S p(p; q ) case treated ab ove. We omit the details.

7. O (p; q )

The K conjugacy classes of -stable parab olic subalgebras for O (p; q ) are parametrized

0 0

by a tuple consisting of a pair of p ositive integers (p ; q ) and an ordered sequence of pairs

(p ; q );::: ; (p ; q ) such that

1 1 r r

(1) p + 2 p = p;

(2) q + 2 q = q ; and

0 0

(3) p + q p + q (2).

0 0

The corresp onding Levi subgroup of O (p; q ) corresp onding is isomorphic to O (p ; q )

U (p ; q ) U (p ; q ).

1 1 r r

Prop osition 7.1. In the above notation let q be parametrized by

0 0

(p ; q ); (p ; q );::: ; (p ; q ):

1 1 r r

Then AV(A ()) is the closure of the orbit parametrized by

0 0

c(p ; q ) c(p ; q ) c(p ; q ) c(p ; q ) c(p ; q ) c(p ; q ) c(p ; q );

r r r 1 r 1 1 1 1 1 r 1 r 1 r r

with notation as in Sections 2.5 and 2.7 .

Pro of. This is once again very similar to the preceding cases. We omit the details.

Corollary 7.2. Let O 2 K nN (p ) be parametrized by a signed tableaux T . Then O is

K K

Richardson if and only if T satis es the fol lowing conditions

(1) If p + q is even (resp. odd), the number of odd rows between consecutive even rows is

even and the number of odd rows greater than the largest even row is even (resp. odd).

(2) Fix a maximal set (say S ) of odd rows between consecutive even rows. Then al l rows

of a xed length (say 2k + 1) in S begin with the same sign (say (2k + 1)). Moreover

k +l

if rows of length 2k + 1 and 2l + 1 appear in S , then (2k + 1)(2l + 1) = (1) .

In particular, given a complex special orbit O , there exists some O 2 Irr (O \ p ) such that

O is Richardson.

Sketch. This is very similar to the case of S p(2n; R ). We omit the details.

RICHARDSON ORBITS FOR REAL CLASSICAL GROUPS 15

Remark 7.3. For simplicity, we have thus far restricted ourselves to the disconnected group

G = O (p; q ). We now discuss the case of G = SO (p; q ). Supp ose O is a Richardson orbit

R K

for O (p; q ) corresp onding to a -stable parab olic q. It is well-known how O splits into (at

I II

most two) orbits for SO (p; q ). Supp ose this is indeed happ ens, and write O = O [ O .

K K

A simple argument using the equivariance of the moment map shows that the K orbit O

(notation as in the intro duction) splits into two K orbits O [ O ; i.e. the K conjugacy

q q

I II

class of q splits into two K conjugacy classes represented by q and q . We conclude

I II

that b oth O and O are Richardson for SO (p; q ). The identical argument applies to the

K K

connected group SO (p; q ), where a single K orbit may split into two or four orbits for

SO (p; C ) SO (q ; C ). Hence we obtain the following result.

Prop osition 7.4. Suppose G = SO (p; q ) or SO (p; q ), and O is a nilpotent K orbit on

R K

p which is a union of irreducible components of a Richardson orbit for O (p; q ). Then O is

Richardson.

8. Annihilators of A modules

In this section, we compute the annihilators of the A mo dules for classical groups. Some

motivation for these calculations is provided by Theorem 1.3 from the intro duction, which

is proved in 8.2 b elow.

; 8.1. Coincidences among A mo dules. For completeness, we identify when A = A

q q

see [T2, Prop osition 3.10], for instance.

Prop osition 8.1. Suppose G is one of the groups discussed above, and let q be a -stable

parabolic of g parametrized by a sequence

; (p ; q );::: ; (p ; q );

1 1 r +1 r +1

0 0

here is empty if G = U (p; q ), = m as above for G = S p(2n; R ) , and = (p ; q ) as

R R

above for O (p; q ). Suppose there is an index j , 1 j r , such that q = q = 0. De ne a

j j +1

new sequence of pairs

0 0 0 0

; (p ; q );::: ; (p ; q );

1 1 r r

by combining the j th and (j + 1)st entries,

0 0

(p ; q ) = (p ; q ) if i < j ;

i i

0 0

(p ; q ) =

(p ; q ) = (p + p ; 0) if i = j ; and

j j

i i

0 0

(p ; q ) = (p ; q ) if i > j + 1,

i1 i1

i i

. The analogous statement and let q denote the corresponding parabolic. Then A ' A

holds if p = p = 0. Moreover, these conditions describe al l coincidences among the A

j j +1 q

modules.

De nition 8.2. Consider a sequence of pairs app earing in Prop osition 8.1. We say the

sequence is saturated if there are no adjacent terms with p = p = 0 and no adjacent

j j +1

terms with q = q = 0. (Thus with this terminology, Prop osition 8.1 says that the A

j j +1 q

mo dules are parametrized by saturated sequences of the form app earing in the prop osition.)

16 PETER E. TRAPA

8.2. Pro of of Theorem 1.3. Fix G . Let B b e a -stable Borel in G corresp onding to a

+ +

choice of p ositive ro ots , write = ( ), and b = h n. Let q b e a -stable parab olic

containing b. Let AV(A ) = O and recall the orbit O of K on B := G=B de ned in the

q K q

intro duction. Fix a generic p oint N 2 AV(A ) and consider the variety of Borel subalgebras

in g containing N , B = fb j N 2 bg. For a dominant weight 2 h , consider the integer

p() de ned to b e the Euler characteristic of the Borel-Weil line bundle G C restricted

B, the conormal bundle to O . Then it is known (see, to the intersection of B with T

e.g., [J2 ]) that 7! p() extends to a harmonic p olynomial in S (h ). In more classical

language, p is a Joseph p olynomial.

Consider the coherent family X ( + ) based at X () = A . A simple argument shows

that for dominant AV(X ( + )) = AV(X ()). Consider the function q that takes 2 h

dominant and maps it to the multiplicity of O in the asso ciated cycle of X ( + ). Then

q extends to a harmonic p olynomial on S (h ). It is known ([Ch]) that q is prop ortional to

the Goldie rank p olynomial q of I := Ann (X ()), the annihilator of A .

I q

U(g)

In the intro duction we sketched the computation of the characteristic variety of A . This

argument in fact shows that the characteristic cycle of A is the closure of the conormal

bundle to O with multiplicity one. A result of Chang ([Ch]) implies that for dominant,

p() = q (). In other words, as harmonic p olynomials on h , p coincides (up to a constant)

with the Goldie rank p olynomial of I . Now Theorem 1.3 follows from the main theorem

of [J1 ].

8.3. The -invariant. Fix a Borel b in g, let X b e a simple U(g) mo dule, and let b e

a b-dominant representative of its in nitesimal character. Let b e a p ositive simple ro ot,

and supp ose h ; i is integral and nonzero. Then is said to b e in the -invariant of X if

the translation functor from in nitesimal character to the wall de ned by is zero when

applied to X (see [V2 ], for example).

Prop osition 8.3. Let q = l u be a -stable parabolic containing b. Then the set of simple

roots contained in l is contained in (A ). Moreover, in the setting of Proposition 8.1, if

we exclude adjacent compact factors of the same signature (i.e. if q is parametrized by a

saturated sequence in the terminology of De nition 8.2), then this containment is in fact

equality.

Pro of. The prop osition certainly follows from the Langlands parameter computations

of [VZ] and the -invariant calculations of [V1 ]. A more direct argument is contained in [T2,

Lemma 3.12] and its pro of.

8.4. Primitive ideals and tableaux. Supp ose g is complex and reductive. Consider the

set of primitive ideals Prim(U(g)) of primitive ideals in U(g) containing the maximal ideal in

Z(g) corresp onding to under the Harish-Chandra isomorphism. We recall the parametriza-

tion of this set for g classical.

If g is of typ e A , Joseph parametrized Prim(U(g)) in terms of SYT(n). (For precise

details of exactly how we want to arrange this parametrization, see [T2, Ssection 3].) Implicit

in this parametrization is a choice of p ositive ro ots. As in [T2, Section 3], we make the

standard choice of p ositive ro ots

= f := e e j 2 j ng:

j j 1 j

If g is classical of typ e X = B ;C ; or D , Barbasch-Vogan and Gar nkle attached a prim-

n n n

itive ideal I (T ) 2 Prim(U(g)) to each T 2 SDT (n), and showed that this assignment is

RICHARDSON ORBITS FOR REAL CLASSICAL GROUPS 17

bijective when restricted to SDT (n) , the subset of SDT (n) of sp ecial shap e (Prop osi-

X X

tion 2.1). In this way, we may sp eak of the primitive ideal attached to a domino tableau. We

follow Gar nkle's conventions for this assignment. In particular, there is an implicit choice

of simple ro ots which, in the resp ective three cases, is as follows:

= f := e ; := e e j 2 j ng

1 1 j j 1 j

= f := 2e ; := e e j 2 j ng

1 1 j j 1 j

= f := e + e ; := e e j 2 j ng

1 1 2 j j 1 j

When we discuss the -invariant of I 2 Prim(U(g)) (Section 8.3), we will always implicitly

make the choice of p ositive ro ots indicated ab ove.

8.5. invariants on the level of tableaux.

Lemma 8.4. Let g be a classical reductive Lie algebra of type A , B ,C , or D . Fix

n1 n n n

I 2 Prim(U(g)) , and let T denote the standard Young tableau of size n (if X = A ) or

standard domino tableau of size n (if X 6= A ) parametrizing I as in Section 8.4, and recal l

the notation established there. Then

(1) If X = B or C , 2 (I ) if and only if the domino labeled 1 in T is vertical.

n n 1

(2) For j 2, 2 (I ) if and only if the box (or domino) labeled j 1 in T lies strictly

above the box (or domino) labeled j in T . More precisely, counting the topmost row

as the rst row of T , let r denote the largest number so that that there appears a

label j 1 in the r th row. Similarly de ne the index s to be the smal lest number so

that the sth row contains the label j . Then 2 (I ) if and only if r > s.

Pro of. In typ e A, this is a well-known feature of the Robinson-Schensted algorithm. The

assertion for other classical typ es follows from the discussion in [G2 , Section 1]

Lemma 8.5. Let G G be two groups of the form discussed in Sections 3{7. Let r

1;R 2;R i

denote the rank of g . In the notation of Proposition 8.1, let q be the -stable parabolic of

i 1 1

g parametrized by the saturated sequence (De nition 8.2)

; (p ; q );::: ; (p ; q );

1 1 r r

and let q be the -stable parabolic of g parametrized by the saturated sequence

2 2 2

; (p ; q );::: ; (p ; q ); (p ; q ):

1 1 r r r +1 r +1

Let T denote the special-shape tableau parametrizing Ann(A ). Then Ann(A ) is the

2 q q

2 1

primitive ideal attached (via the discussion in Section 8.4) to the subtableau T consisting of

the rst r boxes (or dominos) of T

1 2

Sketch. The results of [V2 ] (in typ e A), [G3 ] (in typ es B and C ), and [G4 ] (in typ e D )

imply that the primitive ideal attached to the rst r b oxes of T (resp. T ) is completely

1 1 2

characterized by the action of certain wall-crossing translation functors in the simple ro ots

;::: ; and (outside of typ e A) on A (resp. A ). Since translation functors

2 r 1 1 q q

1 1 2

commute with derived Zuckerman (or Bernstein) functors, it is easy to see that the relevant

wall-crossing information is identical for b oth A and A . The lemma follows.

q q

1 2

The following two results are crucial observations ab out the combinatorial algorithms of

Sections 3{7.

Lemma 8.6. Retain the setting and notation of Lemma 8.5. Then the shape of T coincides

with that of AV(A ). In particular, T has special shape.

q 1 1

18 PETER E. TRAPA

Sketch. Obviously we may assume that we are not in Typ e A. The other cases are a little

more delicate. They are treated in Section 8.7.

Lemma 8.7. Retain the notation of Lemma 8.5. Write S for the skew-shape obtained by

). Then there is at most one way ) from the shape of AV(A removing the shape of AV(A

q q

2 1

to tile S by boxes (in type A) or dominos (otherwise) labeled r + 1;::: ; r such that each

1 2

index j lies strictly above j + 1 (in the sense of Lemma 8.4).

Pro of. Lemmas 8.5 and 8.6 imply that S can b e tiled by dominos. It is easy to see from

the form of the algorithms that each row of S has length at most 2. This immediately gives

the lemma.

8.6. An inductive computation of annihilators of A mo dules. At last we are in a

p osition to compute the tableau T parametrizing Ann(A ). We may argue as in [T2, Section

5]. Let s b e the rank of q. Supp ose q is parametrized by a saturated sequence

; (p ; q );::: ; (p ; q ); (p ; q ):

1 1 r r r +1 r +1

If this sequence has a single term, A is the trivial representation, whose annihilator is of

course known. Inductively we may assume that we have computed the sp ecial-shap e tableau

0 0

, where q is parametrized by the saturated sequence T parametrizing the annihilator of A

; (p ; q );::: ; (p ; q ):

1 1 r r

0 0

Let s b e the numb er of b oxes in T . Lemmas 8.5 and 8.6 imply that we know the p osition

0 0

of the rst s b oxes (or dominos) in T : they coincide with T . It remains to sp ecify the

remaining b oxes (or dominos) s + 1, ..., s. Prop osition 8.3 and Lemma 8.4 implies that

each index j must b e entered ab ove j + 1. Since we have computed AV(A ) in Sections 3{7,

we know the shap e of T , and Lemma 8.7 thus implies that there is a unique way to p osition

the indices s + 1, ..., s in T sub ject to the ab ove restrictions. This pro cedure explicitly

computes the annihilators of the A mo dules.

Example 8.8. Let (i = 1;::: ; 3) b e the sequence consisting of the rst i entries of

2; (4; 0); (1; 1). According to the Section 4, parametrizes a -stable parab olic for S p(4; R ),

parametrized q for S p(10; R ), and parametrizes q for S p(14; R ). Of course A is

2 2 3 3 q

the trivial representation whose asso ciated variety is the zero orbit and whose annihilator is

given by

Prop osition 7.1 computes the asso ciated variety of A as

+ :

RICHARDSON ORBITS FOR REAL CLASSICAL GROUPS 19

Lemma 8.5 implies that the domino tableau parametrizing Ann(A ) lo oks like

with the empty entries remaining to b e sp eci ed. The -invariant considerations of Lemma 8.4

and 8.7 imply that indeed Ann(A ) is parametrized by

1 3

2 4

Theorem 1.3 implies that if the ab ove tableau is the left tableau that Gar nkle's algorithm

([G1 ]) attached to w 2 W (C ), then the asso ciated variety of the simple highest weight

mo dule L(w ) for sp(12; C ) is irreducible. (More precisely, it is the closure of the orbital

variety corresp onding to the ab ove tableau in the parametrization of [T3].) Continuing

inductively, Prop osition 4.1 computes AV(A ) as

+ +

Arguing as ab ove, we deduce that Ann(A ) is parametrized by

1 3

2 4

8 :

20 PETER E. TRAPA

Again Theorem 1.3 now implies that if w 2 W (C ) has the indicated left Gar nkle tableau,

the asso ciated variety of L(w ) is irreducible. This completes the example.

8.7. Pro of of Lemma 8.6. Retain the notation of Lemma 8.6. Let s denote the numb er

of dominos in T . Inductively we may assume that the algorithm of Section 8.6 computes

for this tableau. (Since the sp ecial-shap e domino tableau parametrizing Ann(A ); write T

the algorithm of Section 8.6 apparently made use of Lemma 8.6, one must b e a little careful

that no circularity is involved in the induction. None is.) We can thus explicitly enumerate

through certain op en cycles ([G1 , the p ossible shap es of T : they are obtained by moving T

Section 5]). Write S for the skew shap e obtained by removing the shap e of T from that of

T . Prop osition 8.3 and Lemma 8.4 imply S can b e tiled by the indices s + 1; : : : ; s that

2 1 2

each index j is entered ab ove j + 1. Using the enumeration of the p ossible shap es for T

and the e ect of the relevant op en cycles ([G1 , Prop osition 1.5.24(i)]), it is a detailed (but

relatively straightforward) check that the only way such a tiling is p ossible is if T indeed has

. (The saturated hyp othesis is required here.) The precise details the (sp ecial) shap e of T

are left to the reader. (All of the essential features app ear at the rst stage of the induction,

and there the ab ove argument is transparent.)

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Department of Mathematics, University of Utah, Salt Lake City, UT 84112

E-mail address : [email protected]