INVARIANT DIFFERENTIAL OPERATORS ON A REDUCTIVE LIE

JOURNAL OF THE AMERICAN MATHEMATICAL SOCIETY Volume 6, Number 4, Oetober 1993

INVARIANT DIFFERENTIAL OPERATORS ON A REDUCTIVE LIE ALGEBRA AND WEYL GROUP REPRESENTATIONS NOLAN R. WALLACH

INTRODUCTION

Let V be a finite-dimensional vector space, and let G be a subgroup of GL( V). Set D( V) equal to the algebra of differential operators on V with polynomial coefficients and D( V) G equal to the G invariants in D( V). If 9 is a reductive Lie algebra over C then ~ egis a Cartan subgroup of g, and if G is the adjoint group of 9 then W is the Weyl group of (g, ~) , Harish-Chandra introduced an algebra homomorphism, J, of D(g)G to D(~)w [H3]. J isgiven by the obvious restriction mapping on the subalgebra of invariant polynomials and on the invariant constant coefficient differential operators, and ker J is the ideal, ..Y, of D(g)G consisting of elements that annihilate all G invariant polynomials. In this paper we prove that if 9 has no factor of type E then J is surjective. We also prove that for general g, the homomorphism is surjective after localizing by the discriminant of g. If go is a real form of 9 and if Go is the adjoint group of go then Harish-Chandra has shown that ..Y is precisely the ideal in D(g)G of operators that annihilate all Go invariant distributions on "completely invariant" open subsets of go [H2]. Our first application of our analysis of J is to give a new proof of this important theorem. In light of this theorem the space of Go -invariant distributions on a completely invariant open subset of go is a D(g)G-module that "pushes down" to w w a D(~) -module. To analyze these D(~) -modules we develop a theory analogous to Howe's formalism of dual pairs, proving an equivalence of categories between an appropriate category of D(~)w-modules and the category of all W-modules over C. We show that the D(g)G-module of distributions on go supported in the nilpotent cone of go is (as a D(~)w module) in our category. Thus, to each distribution supported on the nilpotent cone we can associate a (finite dimensional) representation of W. If the distribution is the orbital integral corresponding to a fixed nilpotent element of go then we prove that the representation of W is irreducible and derive a formula for the Fourier transform of the orbital integral in terms of W -harmonic polynomials corresponding Received by the editors April 17, 1991 and, in revised fonn, June 5, 1992. 1991 Mathematics Subject Classification. Primary 22E30, 22E45. Key words and phrases. Research partially supported by an NSF summer grant. © 1993 American Mathematical Society 0894-0347/93 $1.00 + S.25 per page

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to this representation of W. In [BV2, BV3, HK] results of this nature were proved in the case when 90 is a Lie algebra over C looked upon as a Lie algebra over R. They prove that the indicated Fourier transform is given in terms of a harmonic polynomial transforming according to the Springer representation associated to the corresponding nilpotent G-orbit in 9 [S]. We use this theorem to prove that our general correspondence between nilpotent Go -orbits of 90 is given by the Springer correspondence for the corresponding G-orbits in 9. In particular, our theory yields a new approach to the Springer correspondence. We say that 9 is "nice" if d is surjective. (As indicated above one can prove "niceness" for all 9 without ideals of type E.) For nice 9 we show that the surjectivity of d can be used to give a new proof of Harish-Chandra's famous local L 1 theorem for invariant eigendistributions on completely invariant subsets of 90 • The question of whether or not 9 is nice is related to a long standing problem concerning Weyl group invariants in two copies of a Cartan subalgebra. We consider the contragradient action of W on ~... and are interested in the invariants of W in 9'(~ x ~ ... ) (the polynomials on ~ x f), 9'(~ x ~"')w , under the action s/(x, A) = /(S-l X, S-l A). We choose a basis of ~ and the dual basis in ~... and thereby have linear coordinates Xl ' ••• ,xl on ~ and dual linear coordinates {I' ... ,{/ on ~ .... Set P = E{/J/8Xi on 9'(~ x ~ ... ) (P is the usual polarization operator). P stabilizes 9'(~ x f)w , and it has been suggested that 9'(~ x ~"')w is the algebra generated by E pk 9'(~)w . For lack of a name let us call this the "polarization hypothesis". If this were true then it is a simple matter to prove that all 9 are good. For 9 of type An the polarization hypothesis can be found in [W]. By a modification of the argument of Weyl it is easy to show that the hypothesis is also true for types Bn and en and G2 • However, the hypothesis is false for Dn for n ~ 4. In the first appendix to this paper we give a counterexample for D4 and introduce what we call the "revised polarization hypothesis". This revision is sufficient to prove "niceness", and it is true for Dn. For F4 even this is false. However, one can prove a result for F4 which is sufficient to prove that it is also nice. The proof for F4 will appear elsewhere. For E 6 , E 7 , and E8 the question of niceness will most likely be testable using the next generation of computers. However, we hope that there is an elegant theorem on Weyl group invariants (in the spirit of Chevalley's proof that 9'(~)w is a polynomial ring) that will give a uniform argument. The author began his work on this chain of ideas after a conversation he had with Roger Howe (walking in the Torrey Pines Reserve). In this conversation Howe described his work on the action of the algebra generated by the Casimir polynomial and the Laplacian on 9"'(.5((2, R»Go • An outgrowth of this conversation was that it seemed quite likely that D(~)w that is generated by the Weyl group invariant polynomials and the Weyl group invariant differential operators (indeed, Howe sketched a proof of the result for An using a theorem of Weyl alluded to above). We also thank T. Enright, B. Kostant, and D. Vogan for helpful conversations. 1.

POLYNOMIAL DIFFERENTIAL OPERATORS INVARIANT UNDER A FINITE GROUP

We begin this section with a simple result that will play an important role in this paper.


DIFFERENTIAL OPERATORS ON A REDUCI1VE LIE ALGEBRA

781

Lemma 1.1. Let,9J' be an algebra over C with unit, and let

~ be a subalgebra of ,9J' containing 1 such that there exists a linear map P of ,9J' onto ~ such that P(I) = 1 and P(ab) = P(a)b for a E,9J', b E ~ . If V is a ~-module then the map V -+ ,9J' ®9J V given by v 1-+ 1 ®9J V is injective.

Proof. We denote by ,9J' ® V the tensor product over C. Suppose that v E V and 1 ®9Jv = 0 in ,9J' ®9J V. Then there exist ai' ... , am E,9J', VI' ... , vn E V ,and bjj E ~ such that

(1.1)

1 ®V

= Eaj(bij ®Vj -1 ®bjjv). jj

If we apply P ® I to both sides of (1.1) then we have (P( 1) 1®v =

E P(aj)(b jj ® Vj -

1 ® bjjv) E ~ ®

= 1) v.

ij

Thus 1 ®9J V = 0 in

~ ®9J

V . Hence v = 0 .

Note. Let ,9J' be an algebra over C with a filtration ,9J'j c ,9J'i+l, Uj.sat i = ,9J' , dim,9J'i < 00. Let G be a compact Lie group acting on .sat byautomorphisms such that g,9J'j c ,9J'j for all i and g E G. If the corresponding representation of G on ,9J'j is continuous for all i and if ~ = ,9J'G = {a E ,9J'lga = a, g E G} then set Pa =

fo g(a) dg

with d g normalized invariant measure on G. The conclusion of Lemma 1 is therefore true for ~. We will apply Lemma I to this context without further comment. Let V be a finite-dimensional vector space over C. We will use the notation in Appendix 1. Let G be a subgroup of GL(V). Then G acts on .9(V) by g'f(x)=f(g-I X ) for fE.9(V), XE V, gEG. If DED(V) then we set g. D = gDg- 1 • We note that if g E G then g. Dk(V) C Dk(V). If M is a G-module then we set MG

= {m E Mig· m = m,

g E G}.

Then .9(V)G, S(V)G ,and D(V)G are respectively subalgebras of .9(V), S(V) , and D(V). We include the following observation since its proof involves one of the basic ideas in the paper.

Lemma 1.2. If G is afmite group then D(V)G is a simple algebra over C. Proof. If D E D(V) then we use the notation ord(D) for the usual order of D as a differential operator. Let I be a nonzero two sided ideal in D( V) G • Let DEI be a nonzero element with ord(D) minimal. If f E .9(V)G then ord[f, D) < ord(D). Thus, [f, D) = 0 for all f E .9(V)G. This implies that D(fh) = fDh for all f, g E .9(V)G. Hence D acts on .9(V)G by


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multiplication by D· 1 E 9'(V)G. Since 9'(V) is finitely generated as a 9'(V)G-module under multiplication, there exist u I ' •.• ' un E 9'(V)G such that dU I A··· A dUn =F O. This implies that there is an open nonempty subset, U, of V such that u I ' .•. ,un defines a system of holomorphic local coordinates on U. Thus if DI E D(V) then DI = L al(u l ' ... ,un) i I au l!

alII i

•••

aU n"

on U. Since qUI' ... , un] is contained in 9'(V)G , we see that if D I 9'(V)G = DI = O. Hence D is given by multiplication by f = D· 1. Since D =F 0, f =F O. As a right D( V)G -module under multiplication D( V) is finitely generated. Hence there exist D I , ••• ,Dp such that D(V) = EDiD(V)G. Set

o then

There exists k such that (adf)k Dj = 0 for all 1 ::; j ::; p. Thus I acts by 0 on D( V) ®D( V)G M. Since D( V) is simple, this implies that D( V) ®D(V)G M = O. So Lemma 1.1 implies that M = O. Thus 1= D(V)G. M

= D(V)G / I.

Then f· M

r+

= O.

We look upon 9'(V) as a D(V)-module under the usual action as differential operators, and we look upon S(V) as a D(V)-module under the obvious identification with D(V)/D(V)9'+(V) (notation as in Appendix 1).

Proposition 1.3. Assume that G isfmite. Let M be a D(V)G-module such that if mE M then dimS(V)G m < 00 (resp. dim9'(V)G m < 00). If mE M, p E 9'(V)G - {O} (resp. S(V)G - {O}) is such that pm = 0 then m = o. If M is finitely generated as a D( V) G -module then M is offinite length. Proof. We assume that if m E M then dimS(V)G m < 00. Let m E M be such that pm = 0 for some p E 9'(V)G - {O}. Set N = D(V)G m . We show that N = {O}. Set NI = D(V) ®D(V)G N. If q E S(V) (resp. f E 9'(V)) and

if D E D(V) then there exists k such that ad(q)k D = 0 (resp. ad(f)k D = 0). Since S(V) is finitely generated as a S(V)G-module under multiplication, this implies that if m l = 1 ® m then dimS(V)m l < 00. Set F = S(V)ml . Then there exists k such that pk F = O. Furthermore, NI = 9'(V)F . We filter NI by setting g-j NI = Ei~j9'j(V)F . Then Di(V)g-j NI

c

g-i+ j N I •

Let d be the degree of p. Our assumptions imply that dimg- j NI ::; dimF L(dimSi(V) - dimSi_dk(V)) ::;

C/- I .

i~j

Thus Bernstein's theorem (see Appendix 1, Theorem 2) implies that NI = {O}. We now assume that M is finitely generated as a D(V)G-module. Set MI = D(V)

®D(V)G

M.


DIFFERENTIAL OPERATORS ON A REDUCTIVE LIE ALGEBRA

783

If Z}' ••• , Zd are such that D(V) =Ei ziD(V)G and if M = D(V)GF with F a finite-dimensional subspace of M then, if F} = Ei zi ®D(V)G F, F} generates Ml as a D(V)-module. As above, dimS(V)F} < 00. If we filter M} as above then we have a filtration g-k M} of M} so that M} is a filtered module. The argument above implies that dimg-kM1

::;

Ceo

Thus M} is of finite length (see Appendix 1, Theorem 2) as a D(V)-module. We will now show that this implies that M is of finite length as a D( V) G -module. Indeed, let M ::J MI ::J M2 ::J ..• ::J Mk ::J ..•

be a decreasing family of submodules of M. Set N k equal to the canonical image of D(V) ®D(V)G Mk in MI. Then N k ::J N k+ l • Since MI is of finite length, there exists k such that N k = N P for all p 2': k. This implies that if v E Mk and if p 2': k then there exist Di' d i E D(V), ei E D(v)G, Vi E M P , Wi EM such that (the following tensor products are over C) 1 ®V = LDi®Vi+ L(diej®wl-di®ejWI)' ijl

Thus, in the notation of Lemma 1.1, we have 1 ® v = L P(D) ® Vi

+ L(P(di)ej ijl

® WI - P(d) ® ejwl )·

Hence v = EP(Di)V i E M P • Thus M P ::J Mk. Hence M P = Mk for p 2': k. The proof in this case is now complete. The proof of the parenthetic statements is the same after we have reversed the roles of 9'(V). and S(V). Let lfG (resp. lf~) denote the category of all finitely generated D(V)Gmodules such that if mE M then there exists k such that if p E Sk(V)G (resp. p E 9'k(V)) then pm = O. If G leaves invariant a symmetric nondegenerate form and if ¢ is as above then the functor M --+ M d~fines an equivalence of categories between lfG and lf~. The following result is a direct consequence of Proposition 1.3. Corollary 1.4. Assume then G is finite. If M E lfG (resp. C~) andif m E M, p E 9'(V)G - {O} (resp. S(V)G - {O}) is such that pm = 0 then m = O.

We define a D(V) module structure on S(V) as follows. Let Co be the 9'(V)-module, C, with f·l = f(O) . Then as an S(V)-module, D(V) ®9'(V) Co is S(V) ® 1. We look upon S(V) as a G-module in the usual way. Thus with this D(V)-module structure S(V) is a (D(V)G, G)-bimodule (as is 9'(V)). This is the desired structure on S(V). 9'(V) E lfG and S(V) E lf~.


N. R. WALLACH

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~position 1.5. Assume that G acts completely reducibly on V. Let .7 (resp . .7) denote the set of all isomorphism classes of irreducible finite-dimensional Gmodules, M~ such that HomG(M, .9'(V)) :f 0 (resp. HomG(M, S(V)) :f 0) . If A E .7u.7 thenfix J). EA. To each A E.7 (resp. A E.7) there corresponds an irreducible D(V)G-module V;' (resp. V;') such that:

(i) If V;' (resp. V;') is equivalent with Vil (resp. VIl) as a D(V)G-module then A = Jl; and (ii) As a (D(V)G, G)-bimodule .9'(V) (resp. S(V)) is equivalent with

EB V;' ® J).

;'EsP

(res p .

EB V;' ® J). ).

;'EJP

Proof. If g E GL(V) , let gt denote the element of GL(V*) given by gt Jl = Jlog- 1 , Jl E V*. Set Gt = {gtlg E G} c GL(V*). Let xl' ... , xn be linear coordinates on V. Let Ij/ be the automorphism of D( V) defined by

Ij/(x) = 8 j and 1j/(8j ) = -xj • Then Ij/(D(V) ) = D(V) , 1j/(.9'(V)) = S(V), and Ij/(gf) = gt Ij/(f) for f E .9'(V) , g E G. Thus if we prove the result for .9'(V) then the result will follow for S(V). We will be using the following construction throughout the proof of this result. Let A E .7 , and let A* be the class of the contragredient representation of J)., J).* . Then A* E .7. Let e 1 , ••• , ed be basis of J)., and let e~ , ... , e; be the dual basis in J).* . If T E HomG(J)., .9'(V)) and S E HomG(J).* , S(V)) then set DT,s = Lj T(e)S(e7). Then DT,s E D(V)G . If A E.7 then let .9'(V)[A] denote the A-isotypic component of .9'(V). It is clear that as a (D(V)G, G)-bimodule .9'(V) splits into a direct sum of the invariant subspaces .9'(V)[A]. Fix Z;. a nonzero G-invariant irreducible subspace of .9'(V)[A]. Let Me .9'(V)[A] be a nonzero D(V)G-invariant subspace. Let E = Lj x j 8j • Then E E D(V)G . Thus EM eM. Hence if f E M then every homogeneous component of f is in M. (1) MnZ;.:f{O}. Indeed, let f EM - {O} be homogeneous of degree r. Then spandgf} is a direct sum of irreducible G-submodules in the class A. Let T be a nonzero element of HomG(J)., Z;.). Let S E HomG(J).* ,Sr(V)) be such that there exists Jl E J).* with S(Jl)f = 1. Sand Jl exist since f is homogeneous and the pairing between Sr(V) and .9'r(V) given by (plf) = pf E C is perfect. Fix a basis e 1 , ••• ,ed of J). such that e~ = Jl and S(e;)f = 0 for i > 1. Then DT ,sf = T(e 1 )· This implies (1). Let fEZ;.. Put M = D(V)G f. We use the notation C[G] for the group algebra of G thought of as an abstract group. (2) M n Z;. = Cf. Indeed, if gEM n Z;. then there exists D E D(V)G such that Df = g. Thus C[G]g = DC[G]f. So D1Z.l E HomG(Z;., Z;.). Schur's lemma implies that D1Z.l = CI . Thus g E Cf. G


G'


785

(3) M is irreducible as a D(V)G-module. In fact, if MI is a nonzero D(V)G -invariant subspace of V then MI nz;. =I- 0 by (1). Hence f E MI by (2). Thus MI = M. Let f l , ••• , fd be a basis of Z;.' Set Y; = D(V)G1;. We assert that the sum V; + ... + JId is direct. Let u ik E qG] be such that Uikfi = tJ1k 1; (this is possible since qG]lz. = End(Z;.)). Let Vi E Y;, and assume that Li Vi = O.

Then Vi = Di1; with Di E D(V)G . Thus 0= uij(v i

+ ... + v d ) = 'LDkuijfk = Di1; = Vi' k

Similarly, u ij defines a D(V)G-module isomorphism of ~ onto Y;. Let V;' = D(V)G f for some fEZ;. - {O}. Then since there exists U E qG] with uJ; = f, V; is isomorphic with V;' as a D( V) G-module. This implies that 9"(V)[A] is isomorphic with V;' ~ V;. as a (D(V)G, G)-bimodule. This proves (ii). If A E!7' then let j(A) be the minimum of j such that 9"/V) n9"(V)[A] =I-

{O}. Suppose that V;' is equivalent with Vil with A, f..t E !7'. We take these modules to be realized as above. Let A implement the equivalence. Let j = j(A). Since AE = EA, A(V;' n 9"/V)) C Vil n 9"j(V), We may assume that Z;. C 9"j(V), Let T E HomG(V;., Z;.) be nonzero, and let !7' E HomG(V;.* ,Sj(V)) be such that S(V;.*)Z;. =I- O. Set D = DT ,s' Then D(V;' 9"j(V)) =I- O. Thus 0=1- A(D(V;' n9"j(V))) = DA(V;' n9"/V))

C

n

Vil n9"j(V),

But the fonnula for D implies that D9"/ V) C Z;.' Thus Vil n Z;. =I- O. This implies that A = f..t. This completes the proof of (i) and, hence, of the proposition.

Theorem 1.6. Assume that G is finite. Then (resp. V;' E ~~) be as in Proposition 1.5.

!7' =

G.

If A E

G,

let V;' E ~G

(i) If M E ~G (resp. ~~) and V is irreducible then M is equivalent with -;. (resp. V) for some A E G. (ii) If M E ~G (resp. M E ~~) then for each A E G there exists m;. E N such that M is equivalent with V

;.

~

Eem;.V;' ;'EG

(resp .

~m;.v;,). ;'EG

That is, every object in ~G (resp. ~~) splits into irreducible components with finite multiplicities. Proof. As above it is enough to prove the result in the case of ~G' For the sake of completeness, we give the standard proof that !7' = G. Ifg E G and g =I- I then set Ug = {v E Vlgv =I- v}. Then V - Ug = ker(g - I)


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which is a proper subspace of V if g i- I. Since G is finite, this implies that V' = ngEG-{I} Ug i- 0. Fix v E V'. Set J = {f E 9'(V)lf(Gv) = O}. Then J is G-invariant and 9'(V)/J is equivalent with qG)* ~ qG] as a G-module. (1) If M E ~G then D(V) ~D(V)G M E ~ (= ~I})' Indeed, we set U = D(V) ~D(V)G M. We note that as a D(V)G-module U E ~G' We must show that if p E S+(V) and m E U then there exists k such that pk m = O. Since S(V) is integral over S(V)G , there exist j > 0 and ao ' ... , aj _ 1 E S(V) G such that Pj

+

j-I '~p " ' i ai

= O.

i=O

Clearly, we may assume that ai E S+(V)G. Let m E U. Set Z = S(V)G m. Then ZI = Z + pZ + ... + pj-I Z is p-invariant. Since S( V) is commutative, ZI is S(V)G-invariant. Let Z? = {z E Zllaiz = 0, i = 1, ... , j - I} ZII = {z E Zllarasz = 0, r, s ::::; j - I}, .... Then since ZI is finite dimensional, we have Z? c ZII c .. , c Zi = ZI' Clearly, pZ; c Z;. (*) implies that . 0 . . I ." I pi ZI = O. We note that aiZ; C Z;- . Thus pi Z; c Z;- (by (*». Hence pj(q+I)Zi = O. Hence pj(q+I)m = O. Lemma 1.3 now implies that D(V) ~D(V)G M is a finite multiple of 9'(V) as a D(V)-module. Lemma 1.1 implies that M injects in D(V) ~D(V)G M as a D(V)G-submodule. Thus Proposition 1.6(ii) implies that M has a decomposition as in (ii). Since (i) is a consequence of (ii), the theorem follows. 2.

DIFFERENTIAL OPERATORS INVARIANT UNDER A WEYL GROUP

Let ~ be a real n-dimensional vector space over R with inner product (... , ... ). Let V = (~)c = ~ ~R C. We will denote the Hermitian extension of (... , ... ) by the same symbol. Let be a (reduced) root system contained in ~*, and let W c O(~) be the (finite) group generated by the reflections about the hyperplanes a = 0, a E . Let U I " ' " un be a set of basic homogeneous invariants which we take to be real valued on ~. Fix +, a system of positive roots for . Let XI' ••• ,xn be a set of linear coordinates on ~ corresponding to an orthonormal basis. Let c;l"'" c;n be the dual coordinates on ~* . If f, g E 9' (~ x ~*) then let {f, g} (the Poisson bracket of f and g) be as in Appendix 1. We say that the pair (W, V) is "good" if the smallest subalgebra of 9'(V x V*) containing 9'(V)w and 9'(V*)w and closed under {... , ... } is 9'(V x V*)w (here W acts under the diagonal action s(x, A.) = (sx, A.OS-I». Note that if ~ = ~I EB ... EB with acting irreducibly on then d W is equal to WI x ... x W with Wi a finite subgroup of V: generated by reflections. Set Vi equal to the complexification of V: . Then D( V) W is equal

V:

W


V:

787


to D(VI)Wl Q9 ••• Q9 D(Vd)Wd and 9'(V)w, S(V)w split into corresponding tensor products. Also, the Poisson bracket "splits" in a manner consistent with the above direct sum decomposition. We therefore see that if (Wi, Vi) is good for i = 1 , .,. , d then (W, V) is good. Proposition 2.1. If (W, V) is good then D( V) W is the algebra generated by 9'(V)w and S(V)w in D(V). Proof. We denote by .!B the algebra generated by 9'(V)w and S(V)w in D(V). We set .!B k =.!B nDk(V) and Dk(V)W = Dk(V) nD(V)w (here we are using the notation of Appendix 1). (J defines an isomorphism of GrD(V) onto 9'(V x V*). W x W acts on ~ x ~* as a finite group generated by reflections under (s, t)(v, A) = (sv, AO t- I ), and W acts on ~ x ~* by the diagonal action (as above). We identify W with the subgroup {(s, s)ls E W} c W x W. Under these identifications GrD(V)w = 9'(V x V*)w. We set B = Gr.!B. Then B is a subalgebra of 9'(V x V*)w containing 9'(V)w, 9'(V*)w and closed under {... , ... } (see the formula for the "top order symbol" of [D I ' D 2 ] in Appendix 1). Since (W, V) is good, B :J 9'(V x V*)w. Hence B = 9'(V x V*)W. So .!B = D(V)w.

Theorem 2.2. If ~ has no irreducible component isomorphic with the Weyl group of E 6 , E 7 , or E8 then (W, V) is good. In particular, the conclusion of Proposition 2.1 holds.

Proof. We need only show that if (W, V) is irreducible and not of type E then

(W, V) is nice. If (W, V) is not of type F4 this follows from Propositions 2 of Appendix 2 since (in the notation of that appendix) P f = - {u, f} with u =

Lic;~,

i-

Pjf = -{qJi(c;) , f}. In the case of F4 the proof of ;:goodness" is

somewhat complicated and will be given elsewhere.

For the rest of this section we will study the module theory of the algebra, .!B, generated by 9'(V)w and S(V)w. Let XI' ... ,xn be linear coordinates on ~ corresponding to an orthonormal basis. We define a conjugate linear anti-automorphism D I-+ D# of D(V) by Xi I-+ 0i' 0i I-+ Xi' If f, g E 9'(V) then we set (f, g) = g# f(O). (... , ... ) is an inner product on 9'(V). If DE D(V) then (Df, g) = (j, D#g). Let n be the product of a system of positive roots for . We observe that n 2 E 9'(V)w .

Proposition 2.3. There exists ko such that n 2koD(V)w c.!B . To prove this result we will introduce some notation and results that will be used throughout this paper. Chevalley's theorem implies that we can choose u l ' ... , un with u i homogeneous of degree d i , d i :S d i +1 such that det[oiu) = n . Let S+ (V) denote the ideal of elements of S( V) , p , such that pi = O. Put S+(V)w = S+(V) n S(V)w. We set JIt? = JIt?(V) = {f E 9'(V)lpf = 0, P E


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N. R. WALLACH

S+(V)W}. Then the map 9'(V)W ®Jr'

-+

9'(V)

given by f ® h 1-+ fh is a linear bijection. We note that the Uj can be chosen such that Uj are real valued on ~ and such that if pES+ (V) Wand degp < d j then pU j = O. Let W denote the set of equivalence classes of irreducible representation of W. If A E ---W, fix V;. EA. We look upon Homw(V;.' 9'(V» as a 9'(V) W module under the action fT(v)(x) = f(x)Tv(x). Then Homw(V;.' 9'(V» is a free 9'(V)w-module on generators Homw(V;.' Jr'). Furthermore, dim Homw(V;. , Jr') = dim V;.. It is easy to see that it is enough to prove Proposition 2.1 under the assumption that the action of W on V is irreducible. We define Tix;) = 0iUj. Then j

(1) If D

DE!I.

E

D(V)w is of order (as a differential operator) at most 1 then

Indeed, Dl E 9'(V)w c!l. Thus replacing D with D-Dl we may assume that Dl = O. This implies that D is completely determined by its restriction to V·. We therefore see that DW. = E j 'PjTj with 'Pj E 9'(V)w. Thus if we define D j

= Ej(ojuj)Oj

= Ei 'PPi. We set [d, u j ] = 2D + dU i .

then D

d

= EO 2 j

•

Then

j

So D j E !I . Hence D E!I as was to be proved. (2) If DE D(V)w then there exists k such that 7C 2k DE!I . The proof of this assertion will take some preparation. If f is a polynomial in indeterminates Y1 ' ••• , Yn then

a

af(u 1 , Xi

••• ,

un) =

of E a(u Y j

j

1 , ••• ,

o~

Un)a· Xi

Set [aij] equal to the inverse to the matrix [Oju). Then 7Ca jj E 9'(V). Set 2W • OJ = E j aijoi' Then OJ = 7C OJ E D(V) IS of order 1. Hence OJ E !I. We note that 0jf(u 1 ,

••• ,

un)

= 7C

20f a(U 1 '

Yj

••• ,

un)'

Let X E ~ be such that 7C(X) i:- O. Let Ube an open connected neighborhood of X in ~R such that U 1 " ' " un define local coordinates on U. If D is a differential operator on U with COO coefficients then we can write


789


If DE D(V)w then DC[u l

, ••• ,

un] C

qUI' '" , Un]' Thus

a i ' ... 1

'i

n

is a

polynomial. Thus if D E D( V) W has order k then there exist polynomials b.'t;···,11I. in n indeterminates such that ~b ( n 2kD = L...J i ... i U I " ' " 1 •

~in un )~il U I ... Un

on U and hence, on all of ~. This implies (2). If we use the Bernstein filtration on D(V) (see Appendix 1) then the corresponding graded ring is .9 (V x V*). The subring corresponding to the induced grade of D(V)w is .9(V x V*)w relative to the diagonal action of W on V x V* . The Chevalley theorems applied to W x W imply that .9(V x V*)w is free as a .9(V x V*)wxw_module on generators (2(V) ® 2(V*))w. Let E be a filtered subspace of D( V) W such that the corresponding graded subspace of .9(V x V*)w is (2(V) ® 2(V*))w. Then it is easily seen that as a (.9(V)w, S(V)w)-bimodule (.9(V)w acts by multiplication on the left and S(V)w acts by multiplication on the right), D(V)w is isomorphic under the obvious map (multiplication) with .9(V)w ® E ® S(V)w. Since dimE < 00, (2) implies that there exists ko such that n 2ko E c.£O . Hence n 2koD(V)w c.£O . This completes the proof of Proposition 2.3. If A E W, let .9(V)[A] denote the A-isotypic component of .9(V) with respect to the action of W. In Proposition 1.5 we have seen that for each A E W there exists an irreducible D( V) W -module, VA, such that as a (D( V) W, W)bimodule, .9(V)

==

EB VA ® V;. ..

AEW

Furthermore, .9(V)[A] == VA ® V;.. The moduJe VA can be realized as follows. Let Z c .9(V) be a W-invariant irreducible subspace in the class of A. Let h E Z be nonzero. Then D( V) W h is an irreducible D( V) W module isomorphic with VA. Furthermore, (3) D(V)wh

nZ

= Ch.

W then

.£0 acts irreducibly on VA. Proof. Fix Z as above and realize VA as D(V)w h = Iv. Let .9(V)[n;2) denote the subalgebra of the algebra of rational functions on V with denominator powers of n 2k . That is, .9(V)[n;2] is the algebra of functions generated by XI' ••• , x n ' n- 2 . Let N[n;2] be the subspace of .9(V)[n;2] given by qn- 2]N. W W Let D(V)[n;2] (resp. ~n;2)) be the algebra of operators generated by D(V) (resp. .£0) and n- 2 . Then it is easy to see that N[n;2) is a D(V)~2fsubmodule of .9(V)[n;2). Also one can see that if D E D(V)~2) then there exists k such that n 2k D E D( V) W • Proposition 2.3 implies that Theorem 2.4. If A E

(i) D(V)~2) = ~n;2] , and


790

N. R. WALLACH

(ii) N[n2) is irreducible as a D(V)~2rmodule. W

Indeed, let M c N[n2) be a nonzero D(V)[n2rsubmodule. Let m E M be

nonzero. Then there exists k such that n 2k mEN. Thus D( V) W n 2k m = N. Hence M = N[n2) .

Let M be a nonzero ~-submodule of N. If m/n 2k E M[n2) and if D E ~ then D(m/n2k) = n- 2r D'm with D' E D(V)w and some r. There exists p such that n 2p D' E ~. Hence D(m/n2k) E Min2). Thus M[n2) is a ~n2) = D( V)~2rsubmodule of N[n2). So M[n2) = N[n2) . This implies (iii) If MeN is a nonzero ~-submodule and if v E N then there exists k such that n 2k v EM. Let M be a nonzero ~ -submodule of N. (iv) As a 9'(V)w-module N (resp. M) is free on generators any basis of N n:Jt' (resp. M n:Jt') . In light of the Chevalley theorems (described above) it is enough to show that N = 9'(V)w (V n:Jt') (resp. M = 9'(V)w (M n :Jt')). Let B denote either N or M. Let 9'i (V) denote the space of homogeneous polynomials of degree j . Set Bi = B n 9'i (V). Since the Euler operator E x j8 j E ~ (see (1) above), B is the direct sum of the Bi. Let jo be minimal such that Bio i- o. W . . . If p E S+(V) then pBJo c EiO


N. R. WALLACH

796

with bp,Q a regular function on V. The ring of regular functions on V is

d; b

.9'(g)[d[-I]. Let kEN be such that p ,Q E .9'(g). Then Hence k G d[ D E D(g)r(g) n D(g) .

d; DE D(g)r(g).

This completes the proof of the lemma. 5. A

THEOREM OF HARISH-CHANDRA

Let go be a reductive Lie algebra over R. Let 9 denote the complexification of go. We assume throughout this section that 9 is nice (see the end of §3). We choose B such that there is a real form gu of 9 such that Big. is negative definite and such that B(go' go) cR. This clearly can be done. We denote by Gu the subgroup of G generated by {e ad x IX E gu}. Let .9' (go) be the usual Schwartz space of go with the usual Frechet space topology. If f E .9'(go) then we use B to define ~(f) = !T(f) == E .9'(go) as in Appendix 1. Let 1> be defined as in Appendix 1 for an orthonormal basis of 9 with respect to B. We will also write 1>(D) = D. We note (as in Appendix 1) that !T(Df) = D!T(f) for DE D(g) and f E .9'(go). Let J be as in the previous sections.

J

Lemma 5.1. If D E J then D E J . Proof. We first observe that if .9'(gu)G. = {f E .9'(gu)lf(Ad(u)x) = f(x) , u E

Gu' x E gu} then (1) If DE D(g)G then DE J if and only if D.9'(gu)Gu = o. Indeed, Lemma 4.1 implies that D.9'(gJ G• = 0 if D E J. If D.9'(gu)G. = 0 and if f E .9'(g)G, x E gu let rp E C;'(gu) n .9'(gu)G. be such that rp is identically equal to 1 in a neighborhood of x in gu. Then 0 = (D(rpf))(x) = Df(x). Thus DE J «3) in §3). G , Now if DE J then !T(Df) = 0 for all f E .9'(gu) u. Thus Df = 0 for all f E .9'(gu)G• . Since !T is a linear bijection of .9'(gu)G. onto itself, we see G that D.9'(gu) • = o. Thus (1) implies that DE J . ~

~

~

Lemma 5.2. D E J

if and only if there exists

kEN such that

1>(d;)D E (D(g)r(g)) n D(g)G. Proof. We note that if X

E

9 then (r(X)) = -r(X). Thus G

G~

«D(g)r(g)) n D(g) ) = (D(g)r(g)) n D(g) .

Let D E J , and let DI E J

be such that DI = D. Let kEN be such that G

k

d[ DI E (D(g)r(g)) n D(g) . Then

'k

'k

k

G

d[ D = d[ DI = (d[ D 1 ) E (D(g)r(g)) n D(g) . The converse is proved in the same way. ~

~



797

Let GO be the subgroup of G generated by {eadXlx Ego}. If 0 is a subset of go' it will be called invariant if gO c 0 for all g E Go. If 0 is an open invariant subset of go then 0 will be called completely invariant if, for each x EO, Xs EO. Here if X E go then X s ' XnE go are uniquely determined by the conditions that [Xs ' Xn] = 0, ad Xs is diagonalizable on g, Xn E [go' go]' and ad(Xn) is nilpotent. Set .IY = {X E golX = X n }. Then .IY = {X E golf(X) = f(O) , f E 9'(g)G} (cf. [RRGI,8.A.4.2]). If Q is an invariant open subset of go and if T E 9' (Q) (distributions on Q) then we define gT(f) = T(g -I f) (gf(x) = f(g -I x)). We set 9' (Q)Go = {T E 9'(Q)lgT = T, g E Go} We now recall a basic theorem of HarishChandra ([H2], cf. [RRG1, Theorem 8.3.5]. Theorem 5.3. Let Q be an open completely invariant subset of go' and assume that T E 9' (O)Go is such that dimS(g)G T < 00. If 1Inngl = 0 then T = O.

We will now give a new proof of the following theorem of Harish-Chandra. We note that an affirmative answer to the question in the last section would directly imply the theorem. If 9 is nice our proof is independent of Theorem 5.3 and for such go we show how the next result (combined with our theory) implies Theorem 5.3. Theorem 5.4. If 0 is an open, nonempty, completely invariant subset of go then

J = {D

E D(g)G 1D9' (Q)Go =

O}.

The first part of our proof follows the same line of the original argument of Harish-Chandra (we will refer to the exposition in Varadarajan [Var]). The proof is by induction on dimg. If dimg < 3 then J = {O}, so the result is clear. Assume the result for 2 :::; dim 9 < r. We look at the case dim 9 = r. We refer to [Var, pp. 149-150] for the reduction of the inductive step to the case when 9 is semisimple and if DE J, suppDT c.IY. Thus Lemma 2 in Appendix 3 implies that DT extends to a tempered distribution on go. Since suppDT c.IY, if f E 9'+(g)G then there exists kEN such that /' DT = O. Applying the Fourier transform, this implies that if p E S+(g)G then there exists k

~

kEN such that p (DT) = 0 . We now assume that 9 is nice. Let DI E J; then dimS(g)G DI (DT)
q; we now prove it for p = q. Let X = Xq . Let {X, Y, H} be an s-triple containing X . Set V = g~ . As in [RRGI, 8.3.6, p. 299] we choose U an open neighborhood of 0 in V so that if cl>(g, Z) = g(X + Z), g E Go' Z E U, then cl> is a submersion of G x U onto an open subset Q of go such that: (i) Q n ~ = &p , and (ii) (X + U) n ~ = {X} . Since cl> is a submersion, we may define, for T E 9J~(go)Go, cl>°(T) E p

9J'(U) with suppcl>°(T) = {O} (see [RRGI, p. 301]). Let -J.ll' ... , -J.ld be the eigenvalues of ad H on V counting multiplicities. Then we can choose linear coordinates on V, Y 1 ' ••• , Yd such that cl>°(EgT) =

(L (~J.lj + 1) Yj8~J cl>°(T).

We note that 'E(J.lj + 1) = dimg and that dim V = dimgX by TDS theory. We also note that cl>°(T) E S(Vdt5v,o = 9J (t5 v ,o the Dirac delta for V supported at 0). The eigenvalues of 'E(!J.lj + l)y j 8/8Yj on 9J are of the form - 'E(!J.lj + l)a j with a j > 0, each has finite multiplicity and the eigenspace for - 'E(!J.lj + 1) is Ct5 v ,o' Now the observations using TDS theory imply that L(!J.lj+ 1) = !(dimg+dimgx ).

If cl>°(T) = 0 then T E 9J:/ (g )Go . The first assertion of the lemma now o/'p+1 ° follows. As for the second, the proof of the first part implies that if T E 9J '.,. (g )Go is an eigenvector for E with eigenvalue - -21 (dim 9 + dim gX) then ~x

°

9



~o (T) E Cc5 v . 0 and if ~o (T) of the lemma.

= 0 then

T

= O.

801

This proves the second assertion

If X E go then we recall that there is a canonical Go invariant measure (the Kostant-Kirillov measure) on &x given as follows. If y E &x and if U, v E T(&x)y with U = [y, U], v = [y, V], U, V E go' then set wy(u, v) = B(y, [U, V]). Then w defines a symplectic structure on &x and IIx = wm(X) defines a volume form on &x. In [RR] it has been shown that if X E,AI" and f E Cc(go) then

r

l~x

fllX = Tx(f)

is defined by an absolutely convergent integral and that Tx defines a Go invariant Radon measure on go. Thus Tx E 9~(go)Go. Clearly, supp Tx c &x. Theorem 6.1 implies that as a D(~)w-module we have D(g)G Tx =

EB mx(A)V)..

).EW

We have therefore assigned to each X E,AI" a function m x: W -+ N. If A E W then set j(A) = min{jl Homw(J';', 2j) =I O}. Put j(X) = min{j(A)lmx(A) =I

OJ.

Theorem 6.3. If X E ,AI" then j (X) = ! (dim gX - l). Furtherm~re, {A. Wlmx(A) > O} consists of one element, AX with mx(Ax) = 1, and

E

dim HomwU;'x ,2j(X)) = 1. This result will take some preparation. We note that there are choices of invariant measures on Go and G: = {g E GlgX = X} such that if f E ~(go) then Tx(f) =

1

Go/G;

f(gX)d(gG:).

If hE Go and if hX = cX for some c E R then

r

lGo /G0x

f(ghX)d(gG:)=det(h l x)-I go

r

lG0 /G0x

f(gX)d(gG:)

for f E Cc(go) . As above, we will look upon the N-module NTx as a g-module via c5.

Lemma 6.4. Let XE,AI". Then E"Tx=-!(dimg: +1)Tx. Proof. We note that EgTx(f) = Tx(e~f) = -mTx(f) - Tx(Egf). If X = 0 then To is the Dirac delta function at O. Thus To(Egf) = O. Thus in this case EgTO = -mTo . Otherwise, let {X, Y, H} be an s-triple containing X. Then f(ge(logt)ad H/2 X) = f(tgX)


802

N. R. WALLACH

for g E Go' t > O. Thus

E f(gX)

=!!:.-

f(ge(IOgt)ad H/2 X ). dtlt=l Hence, if we apply the observation above, we have g

EgTX =

(~tr ad H1gx -

m) Tx-

Let gX = EB g; with g; the A eigenspace for ad H in gX. Then standard TDS theory implies that

L dim(g;)(A + I) = dim g. J.

Hence

tr ad H1gx = dimg - dimgX

We have thus shown that

EgTX =

(*)

-~(dimg+dimgX)Tx-

This formula is valid if X = 0 . We note as above that f5(Eg + !f) follows.

= E~ + i.

Since m

= dimg,

the lemma

We will now give another interpretation of the D(~)-module, S(~). Let f50 E g;(~)* be defined by f50 (f) = f(O). If D E D(~) and if A E g;(~)* then set DA = A0 DT. Then our module structure on S(g) is just the module D(~)f5o

.

Lemma 6.5. If A E

W

then the highest eigenvalue of E~ on VJ. is -I - j(A) and it has multiplicity dim Homw(J-i '~(J.))'

Proof. We realize VJ. as tBpf50 with p E ¢(~(J.)[A]). We note that E~pf5o =

(-I - j(A))pf5o'

The lemma now follows from Theorems 1.6 and 2.7. Lemma 6.6. If X

then the highest eigenvalue of Eg on Sf'Tx is (dim 9 + dim gX) and it occurs with multiplicity I. Proof. In light of (*) in the proof of Lemma 6.4, the eigenspace for Eg with eigenvalue - t (dim 9 + dim gX) has positive dimension. The result now follows from the second assertion of Lemma 6.2.

-t

E ,AI'

Proof of Theorem 6.3. As a tB-module, M = Sf'Tx is isomorphic with EBJ.EWmX(A)VJ.. Thus the highest eigenvalue of E~ on M is -1- j(X) with multiplicity equal to a=

L

J.EW j(J.)=j(X)

dim Homw(J-i '~(x))mx(A).


DIFFERENTIAL OPERATORS ON A REDUCTIVE LIE AWEBRA

803

Lemma 6.3 in combination with Lemma 6.4 implies that -/ - j(X) = _!(dimg X + /) and that a = 1. Thus j(X) = !(dimgX - /) and, since a = 1, {A. E Wlmx(A.) > 0, j(A.) = j(X)} consists of one element AX with mx(A x ) = 1 and dim Homw(J'J.x ,~(X)) = 1. On the other hand, M is generated by one element in the -/ - j(X) eigenspace of M. Since M splits into a direct sum of modules isomorphic of the form VA and mx(A) = 0 for j(A.) < j(X) , mx(A) = 0 if j(A) = j(X) and A =f. AX' it follows that M ~ VAx. This completes the proof of the theorem. Since &x is a cone for X E ,AI, Tx is a homogeneous distribution on go' Hence Tx is tempered. We will now analyze Tx. Choose £) such that £)0 = £) n go is a real form of £). We note that 8+(g) Tx = O. Thus Tx is Isons a real analytic function which completely determines Tx (cf. Theorem 5.3). G~

~

I

Theorem 6.7. Let X E ,AI . If C is a connected component of £)0 n £)' then there exists he E .:fj(X) [Axl such that Tx,c = he/n. Proof. If f E .9'(£)0) then we define c;zJ

1

A

f(x) = f(x) = (2n)I/2

(

1f)0 f(y)e

-iB(y x)

'

dy.

Here dy is the Lebesgue measure on £)0 corresponding to a pseudo-orthonormal basis of £)0' If T

E .9"(£)0)

and if f

~/£)) (resp. p

E

E 8 j (£)))

then (fT) =

¢>(f)i (resp. (pt) = ¢>(p)i, where ¢> is defined as in Appendix 1 corresponding to an orthonormal basis of £). Thus we have (i) o(D) = ¢>(o(D)) . We also note that if M is the ~-module VA with action given by D· m = Dm then M ~ VA . With these observations in hand we can prove the theorem. Set T = Tx' As a ~-module NT is isomorphic with vAx. Let C be as in the statement. If DEN then r(D)~e = n-IO(D)n~e' Since ~+(g)GT = G~

0, 8+ (g) T = O. Thus

w

~

nr(8+(g) )1Ie

This implies that n~e

= he

= 8+(£))

with he

E

w

~

n1le

Jt'. Since

= O.

Ef) = -Ef) + / , we see that

Ef)He = j(X)he' The above observations imply that D(£))w he ~ VAx as a

D(£))w-module. Hence he

E .:fj(x)[A.xl.

This completes the proof.

We now look at a special case of these results. Let gl be a semisimple Lie algebra over C. Let go denote gl as a Lie algebra over R. Let u l be a compact form of gl . Let X denote complex conjugation of X E gl with respect to u l Then we identify go with the subalgebra {(X, X) E gl x gliX E gl} of gl x gl . If £)1 is a Cartan subalgebra of gl then £)0 = {(X, X)IX E £)I} is a Cartan subalgebra of go' With these identifications 9 = gl X gl and £) = £)1 X £)1 .


804

N. R. WALLACH

W(9, ~) = W(91' ~I) X W(91' ~I)' We set W = W(9, ~), »'t = W(91' ~I)· WI there exists g E Go such that gl~ = s}. If

Then {(s, s)ls E WI} = {s E

A, J.l

~ then we denote by A®J.l their exterior tensor product as an element of W. We fix a system of positive roots, PI' for the roots of 91 with respect to ~I • Then P = {(o, 0)10 E PI}U {(O, 0)10 E PI} is a system of positive roots for 9 with respect to ~. This implies that if 1t1 = TIaEplo and 1t = IIaEpo then E

1t«X, X» =

1t1 (X)1t1 (X).

We note that .w'(9) = .w'(91) ®.w'(91) and g(9, h) = g(91' ~I) ®g(91' ~I)· Thus the irreducible g (9, ~ )-module corresponding to A ® J.l in the category -;'-p. ' CwisV®V.

Theorem 6.S. If X E 91 is nilpotent then there exists axE WI such that A(X,X)

=

!(dim 9; -

if axI

ax®ax

dim ~I). Furthermore,

= ax then GIXI = G I X 2 · 2

if

=

and j(ax ) = XI' X 2 are nilpotent elements of 91 and

(here o®P(SI' S2)

o(SI) ® P(S2»

Note. The above result implies that for a semisimple Lie algebra over C, we

have constructed an injective map from the set of nilpotent orbits of the adjoint group into the set of equivalence classes of irreducible representations of its Weyl group. At the end of this section we will use the results of [BV2, BV3, HK] to show that this correspondence is the Springer correspondence [S].

Proof· Since ~o - (~o n ~') is of (real) codimension 2 in ~o' we see that ~o n ~' is connected. Set T = T(X,X) and h~on~' = hx . If we had chosen a different Cartan subalgebra then it would be of the form g~o with g E Go. Since T (hence T) is Go-invariant, the corresponding" h" would be given by h(gH) = hx(H) for H E ~o. Also by the Go-invariance of T and the ~-invariance of

we see that w hx = hx for w E ~. This implies that (II). ) wo =f. O. Schur's x Lemma, the fact that every irreducible representation of »'t is defined over R, and Theorem 6.3 imply that: 1t ,

(i) AX = ax ® ax' and (ii) dim(II).)w" = dim(~(x,X)[Ax])wo

= 1.

This implies that if XI' X 2 are nilpotent elements of 91 and if A(XI ,XI) = A(X2 ,X2 ) then there exists c E C such that 1( XI'X I ) = C1(X2 ,X2 ) on 9~ and, hence, on 90 by Theorem 5.5. This implies that T(XI'X I ) = c1(X2'X2 ). Since 19(XI ,XI ) is the unique open orbit in its closure, this completes the proof.

We now return to the general case. The main result in this section is Theorem 6.9. Let X E./Y. Ifwe look upon X as a nilpotent element of 9 then

Ax=ax · Proof. We note that GoX is open in GXn90. We look upon GX as a complex submanifold of 9. By its very definition, lIx extends to a holomorphic 2m(X) form on GX. We will use the same symbol for this extension. Then up to



805

a scalar multiple, lI(X,X) = lIx 1\ v x · Let Yx = {f E 9'(g)lf(&x) O}. If x E &x then there exists an open neighborhood, no of x in go' and It ' ... , f m - 2m (x) E Yx that are real valued on no and such that no n &x = {y E nolf;(y) = O}. We may also assume that there is an open neighborhood, n, of GX in 9 such that n n go = no and n n GX = {y E'nlf;(y) = O}. If we shrink 0, we may assume that 0 n GX is connected and there are local holomorphic coordinates Xl' ••• ' Xm on n such that Xj(x) = 0 and x j+2m (X) = f; for i ~ 1 and that Xl' ••• ' Xm restricted to no are (real) local coordinates on no. Furthermore, may assume that these coordinates on no satisfy the condition of Lemma 3 in Appendix 3 for M = &x and that after reordering {RexI' ... , Rexm , ImxI ' ... ,Imxm } satisfy the conditions of Lemma 3 in Appendix 3 for M = GX. In the notation of Lemma 3 in Appendix 3, if rto is the" rt" for (J) = lIx then rto extends to a holomorphic function on GX n n and the" rt" for lI(X ,X) is rtotl o . If D E D(g)G then we can think of D as a holomorphic differential operator on 9 and then we denote it by D® 1 . We can think of D as an antiholomorphic differential operator on 9 and then we denote it by 1 ® D. Then D(g x g)GXG is just D(g)G ® (D(g)G)- . Also the corresponding".9f " is .9f ®.9f . Set .fx = {D E D(g)G 1DTx = O}; then Theorem 5.4 implies that .fx ::> J. The local criterion in Lemma 3 in Appendix 3 implies that if D E .fx and if (in the notation of Appendix 3) D

then for each J we have (1)

T

I

~

= L..tal,iJ a

J

I,J

~ III I L..t(-I) a (rtoal,J) =

0

I

on 0 0 n &X. Since all the terms extend to holomorphic functions on 0 n GX , (1) is true on 0 n GX. Now the local condition that (D ® J)T = 0 is that for each J L(-I)l/lal(rtol1oal,J) = 0 I

.

on n n GX with al looked upon as partial derivatives from the holomorphic tangent space. Thus L(-I)l/lal(rtotloal,J) = tloL(-1)l/lal(rtoal,J) = O. I

I

Lemma 3 in Appendix 3 implies that (D ® 1) 1(x ,X) = O. Similarly (1 ® D) T(X ,X) = O. Thus the cyclic .9f ®.9f -module generated by 1(x, X) is a quotient of (~/.fx) ® (~ /.fx). But as a g-module, .9f /.fx is isomorphic with VAX (Theorem 3). This implies that Vax ® Vax is isomorphic as a -g®g-module with a quotient of V x ® V x . Thus Ax®Ax = (Jx®(Jx. Hence ~A

~A

Ax = (Jx·

We now close the circle of ideas that we have developed in this section by quoting an important theorem of [BV2; BV3; HK, Theorem 8.2, p. 357].


806

N. R. WALLACH

Theorem 6.10. If 9 is a semisimple Lie algebra over C then the correspondence &'X f-t a x is the Springer correspondence. Proof. In [HK] it is shown that (1(x ,X))I~I = h/(1C 0 it) and that h is a (W x W)-harmonic polynomial whose cyclic space under W x W is in the class of A 0 A with A the Springer representation associated with X. Theorem 6.7 now implies the result.

We conclude this section with a complete description of g-.:r(g)Ga as a s(module in the special case when go is a semisimple Lie algebra over C looked upon as a Lie algebra over R. We use the notation and conventions established above for this special case. Theorem 6.11. As a !B(g, ~)-module g-.:r(go)Go is isomorphic with

E9 V"0 V".

"EW, Proof. Theorem 6.1 implies that as a !B -module ,

g-.#'(go)

G

0

~

ffi

-"

~11

W m",11 V 0 V . ",I1EW,

Let M be an irreducible nonzero s( -submodule of g-.:r(go)Ga • Then since 9'+(g)G acts locally nilpotently on M, there exists T =1= 0, T E M such that 9'+(g)w T = O. Thus S+(g)G = O. This implies that there exists h E ,;r'(~I) 0,;r'(~1) such that

r

r(X, X) = heX , X)/1C I (X)1C I (X)

for X E ~o n g'. Furthermore, !B h is irreducible and in ~w. Thus, !B h ~ V 0 VI1 for some A, f.l E ~ WI' Hence, h E 9'(~)[A 0 f.l]. But, if s E WI then r(sX, sX) = r(X, X) and 1C I (SX)1C I (sX) = 1C(X)1C(X) for all x E ~I' Thus relative to the diagonal action of ~ on TJ. 0 VI1 there is a nonzero fixed vector. This implies that A = f.l. Let Wo be the diagonal subgroup of WI x WI . Then we have shown that 1C.M;~angl is an irreducible !B-submodule of 9'(~)[A0A]wa .

"

The results of §1 easily imply that 9'(~)[A 0A]wa is irreducible as a !B-module. We therefore see that m",11 :::; 1 and m",11 = 0 if A =1= f.l. To prove that m" " = 1, we will make use of Harish-Chandra's theory of orbital integrals. Let 'n l denote the sum of the root spaces in gl corresponding h to a E PI' Set n = {(X, X)IX E n l }. Let, for f E 5"'(go) , ; be as in [RRG1, 7.3.6]. There is a choice of Lebesgue measure on n such that (see [RRG1, 7.3.8(4)]) if H E ~o n g' then ja(H) =

1

;(H +X)dX

where if K is the maximal compact subgroup of Go corresponding to u and if dk is normalized invariant measure on K then leX) = tf(kX)dk.



This implies that if then we set

1 E .9'(go) then ~

1

Sh (I) =

h(X,

~o

E

.9'(~o) . If h

E

807

K(~)w" n9'(~)[A.®A.]

X)~ (X) dX.

Here we choose the Lebesgue measure on ~o corresponding to a pseudo-orthonormal basis relative to the form B. If p E .9'(g)G then !j =

c5(p)~.

D(~o) then Sh(Pf) = SJ(p)Th(/) for P E S(g)G. This implies that if we set Th(f) = Sh(j) then 9'+(g)G Th = o. Hence supp Th c./Y n go . Since h(sX, sX) = h(X, X) for X E ~I ' we can define a COO function 'Ph on gong' by 'Ph(gH) = h(H) for H E ~ong'. Then noting that if II = dim~1 then 1=211 and d/(X) ~ 0 for X E go' we have Thus if DT is the formal adjoint of D

Th(/) =

1

E

'Ph (X)

gong'

d 2 /(X)

1/2 /

A

(X) dX.

~;"'(go)Go. Now Th(f) = Th(.9T- I I) = Sh(.9T(.9T- I f)) = Sh(f). So Th = Sh and S (X) _ h(X)

Thus Th

E

h

-

n(X)n(X)

for X E ~o n g'. In light of the material at the b~ginning of the proof of this theorem, it follows that .SiI Th is isomorphic with VA. ® VA. when looked upon as a B-module. ApPENDIX

1.

POLYNOMIAL DIFFERENTIAL OPERATORS

The purpose of this appendix is to compile basic theory of the Weyl algebra that will be used in the body of the paper. Let V be an n-dimensional vector space over C, and let V* denote the dual space. Let 9'(V) denote the algebra of all polynomials on V, and let S(V) the symmetric,algebra on V. 9'k(V) (resp. Sk(V)) will denote the space of elements in 9'(V) (resp. S(V)) homok k geneous of degree k. We set 9'+(V) = Ek>o9' (V), S+(V) = Ek>oS (V). We also use the notation D( V) for the algebra of differential operators on V with polynomial coefficients. If v E V then we look upon v as a differential operator on V using the operation vl(x) = 8vl(x) = ftl(x + tV)lt=o for 1 E 9'(V). The corresponding map v 1--+ 8v induces an injective algebra homomorphism of S(V) into D(V) with image the algebra of constant coefficient differential operators on V. Thus if U E S(V), we will look upon U as a constant coefficient differential operator on V. If 1 E 9'(V) then we look upon 1 as a differential operator on V relative to polynomial multiplication. With these identifications the map 9'(V) ® S(V) to D(V) given by 1 ® U 1--+ Iu


N. R. WALLACH

808

defines a linear bijection. If VI' ••• 'V n is a basis of V and if Xl' ... Xn is the corresponding dual basis then we have identified Vj with a~ . We use standard multi-index notation. If I = (iI' ... ,'in), i j E N (the nonnegative integers), then we set III = i l + ... + in' We write Xl = X;I ... x!n and I

a

=.

alII

. .

axil1 ... ax'n n

We will also write aj = a~;' If D E D(V) then D = EIII+IJI9 aI ,JXI a J for some kEN and aI, J E C. The minimum of such k is called the Bernstein degree of D. We set Dk (V) equal to space of all D with Bernstein degree less than or equal to k. With this filtration D( V) is a filtered algebra with corresponding graded algebra, Gr D( V) , isomorphic with .9 (V x V*) = .9 (V) ® .9(V*). If el , ... ,en are the dual linear coordinates on V* to Xl"" , Xn on V (i.e., ej(A.) = A.(v)) and if we set (Jk(D) = EIII+IJI=kaI,JxleJ then the map (Jk: Dk(V)/Dk-\V)

-+

.9 k(V x V*)

gives the isomorphism. We also note that if DiE Dk; (V), i = 1, 2, then [D l , D 2 ] E D k l+ k 2- 2 (V). Furthermore, (Jk +L- _2([D l , D 2]) I

"'2

= {(Jk (D l ), (Jk (D2)) I

2

with {... , ... } the usual Poisson bracket given by {f, g} =

",(afa g afa g ) ~ ax. ae . ..,.. ae· ax. j

I

I

I

I

for f, g E .9(V x V*). We now record some (standard) results. Proposition 1. D( V) is a simple algebra over C. For a proof see, for example, [Eh, Proposition 1.1]. If M is a D( V)-module with a filtration, Mo c Ml C M2 C ... such that Di(V)Mj c Mj+j' then we say that M is a filtered D(V)-module. We note with the obvious action of D(V) on .9(V) and the degree filtration that .9(V) is an irreducible D( V)-module. Theorem 2. Suppose that M is a filtered D( V)-module with dim Mq < 00 and that dimMq :::; cqP /p! for some 0 < c < 00 and all q sufficiently large. If p < n then M

=0

and if p

=n

then M has length at most c as a D( V)-module.

A proof of this result of J. Bernstein can be found in [Eh, Propositions 1.12 and 1.15]. The next theorem (an algebraic version of the Stone·-Von Neumann theorem) is a special case of the Kashiwara Lemma of 9 -module theory. Since it plays a critical role in this paper, we will record a relatively simple proof.



809

Lemma 3. Let M be a finitely generated D( V)-module such that if m E M and PES+ (V) then there exists k such that pk m = O. Then M is D( V)isomorphic with a finite multiple of .9(V). In particular, M splits into a finite direct sum of irreducible D( V)-modules. Proof. Let !?f be the category of all D( V)-modules M satisfying the S + (V)nilpotence assumption of the lemma. We look upon V as an abelian Lie algebra and M as a V-module under the action V· m = 8v • m. We show that if M E !?f then HI(V, M) = O. Let Mk = {m E MISk(V)m = O}. Then ~ = {OJ and Uk>o Mk = M. Let Xl' ... ,xn be linear coordinates on V. Put E = EXj 8 j • Then EMk c Mk for all k. Let WE ZI(V, M); then there exists k such that w(V) c Mk. Since W E ZI(V, M), 8 j w(8) = 8 j w(8J. Define T(w) = Exj w(8J E Mk+l . Then dT(w)(8) = L8j x j w(8j) = w(8) j

+ LXj8 j w(8j) =

(E

+ 1)w(8).

j

If mE Mk then we assert that (E -k+ l)m E M k- l . Indeed, if mE Mk and v E Sk_l (V) then vEm = [v, E]m + Evm. The formula for E implies that Evm = O. Now [v, E] = (k - l)v. Thus veE - k + l)m = O. We therefore see that (dT(w) - kw)(V) c M k- l . Since ~ = {O}, this implies our vanishing assertion. We note that Ml = ~(V, M). If mE Ml - {OJ then D(V)m = .9(V)m. Thus D(V)m is a nonzero quotient of the irreducible D(V)-module, .9(V). So it is equivalent with .9(V). The above vanishing of first cohomology easily implies that, as a D(V)-module, M is isomorphic with .9(V)®M I with D(V) acting on the first factor. This completes the proof. We define an isomorphism, ¢>, of D(V) by ¢>(x) = iaj8xj and ¢>(8j8xj) = iXj • We note that ¢> depends on the choice of basis {xJ of V*. If ~ is a subalgebra of D(V) such that ¢>(~) = ~ and if M is a ~-module then we denote by Xi the D( V)-module with total space M and with action D . m = ¢>(D)m . Let ~ be a real form of V, and let B be a nondegenerate, symmetric, bilinear form on ~. We will also denote by B the complex bilinear extension of B to V. Let dx denote the Lebesgue measure on ~ corresponding to a basis {Vj} of ~ such that B(vp v) = ±c5 jj . Let 9(~) denote the Schwartz space of ~. We use B to define a Fourier transform !Fn(f) = J by fA(x) =

1 ( f( )e-jB(y,x) d . (2nt/21v" y y

If U is an open subset of ~ then we look upon D( V) as a subalgebra of the algebra of differential operators on U with COO coefficients. If {xJ is defined as above with {v J an orthonormal basis of V then !Fn(Df) = ¢>(D)!Fn(f).


810

N. R. WALLACH

ApPENDIX

2.

SOME INVARIANT THEORY

Let ~ be a vector space over R with inner product (... , ... ). Let be a root system in ~ (cf. [Bo]), and let W denote the corresponding Weyl group. Let V denote the complexification of ~. We extend (... , ... ) to a symmetric nondegenerate C-bilinear form on V. We choose u l ' ••• , un to be basic (homogeneous) W-invariant polynomials in 9'(V) such that u j is real valued on ~. Fix + a system of positive roots in . Let e l , ••• ,en be an orthonormal basis of ~, and let Xl' '" ,xn be the corresponding linear coordinates on ~ and V. Let l , ... ,en be the linear coordinates on ~* corresponding to the dual basis to e l , .•. , en' We look upon V* as a Wmodule under the contragredient action and write sA. = A. 0 S -1 for A. E V*, S E W. In this section we will be studying the action of W on V x V* given by s(v, A.) = (sv , sA.) . We set P = Eje/J/8xj. If f E 9'(V) then pkf is called a polarization of f. We identify 9'(V) (resp. 9'(V*)) with the polynomials on 9'(V x V*) that depend only on Xl' ••. 'Xn (resp. l , •.• , en)' The basic question of this section is to find a method of generating 9' (V x v*) W using 9' (V) W • Based on a result in [W] for the symmetric group (and its easy extension to Weyl groups of type Bn = en) it has been suggested (sometimes conjectured) that 9'(V x V*)w is generated as an algebra by the polarizations of the elements of 9'(V)w. We will call this suggestion the polarization hypothesis. Unfortunately, this hypothesis is false for Weyl groups of type Dn for n ~ 4. Before we go on to positive results we give an example for D 4 of an invariant that is not in the algebra generated by polarizations of elements of 9'(V)w . We will use the notation of [Bo]. Let W be the Weyl group of type D4 , and use the coordinates Xj = ej . We take u j = E j xJj for i = 1, 2, 3 and u4 = X l x 2 x 3 X 4 • We note that, if 1 ~ i ~ 3, u j is invariant under the bigger Weyl group, W', of B 4 • Let (() E Wi be the element such that wei = ei , i = 1 , 2, 3, and we 4 = -e4 • We set

e

e

U = u(x,

3 3 3 e) = e3 l X2X3X4 + Xl e2X3X4 + Xl X2e3X4 + Xl x2x3e4·

Then U E 9'(V x V*)w. We assert that U is not contained in the algebra generated by the polarizations of the elements of 9' (V) W • Since P is a vector field it is enough to show that U is not in the algebra generated by 1 and pk Uj for k = 1, 2, 3, 4, 5, 6 and j = 1, 2, 3, 4. So assume that U is in this algebra. Since wu = -U, this implies (assuming the polarization hypothesis for B 4 , which is correct) that U

1;

= fau 4 + ftPu 4 + J;p

2

U4

+ J;p 3 U4 + ~P 4 u4

in the algebra generated by the polarizations of u l ' u 2 ' u 3 and v = Ej We may assume that the 1; are homogeneous in Bidegree considerations imply that fa and ~ are O. Also, we may assume that 1; has bidegree (i - 1, 3 - i) (the first coordinate is the degree in x, the second is the degree with

xJ .


x, e.

gil


in ~). Degree considerations imply that J; = Ci P 3- i U I • If we expand u l p 3 u4 into monomials, we see that the term ~2~3~4 occurs with a positive coefficient but cannot occur in the expansion of u or p3-iUIpiU4 for i = 1,2. Thus h = o. If we expand PU IP 2U4 then the monomial X~X2~1~3~4 has a positive coefficient but cannot occur in U or P 2UI Pu 4 . Hence fz = o. But now we have the contradiction that U = ci P 2UI Pu 4 . Thus the polarization hypothesis is false in general. We now introduce a new hypothesis that we will prove all irreducible Weyl groups but F4 , E 6 , E 7 , E g • We return to the general notation of this appendix.

xi

Revised polarization hypothesis. Set ~

au.

a

Pi = ~ 0:/ (~) ax j'oJ

I

;

then g (V x V*) W is generated as an algebra by 1 and the elements p.II . .. P.I, UJ. , r = 0, 1, ... , i k = 1, ... , n, j = 1 , ... , n. We state a simple lemma whose proof will be left to the reader. Lemma 1. If the revised polarization hypothesis (resp. the polarization hypoth-

esis) is true for every irreducible factor of W then it is true for W. If after adjoining a variable xn+1 and dual variable ~n+1 with sXn+1 = xn+I' SEW, the revised polarization hypothesis (resp. the polarization hypothesis) is true then it is true for W on ~.

Proposition 2. The polarization hypothesis is true for Weyl groups of type An' En = C n ' G2 . The revised polarization hypothesis is true for D n , n ~ 4. Proof. We first consider the case of G 2 . We note that the argument that we

use would apply to any dihedral group. Let ai' a 2 be simple roots for + . Set XI = al/llaill , and take e2 such that a l (e 2) = o. This defines XI' x 2 . Set Sl = sal· We take as basis generators for the invariants of Sl' = X~ + and x 2 • The polarization hypothesis is trivial for the group generated by Sl • Thus if f E g(V x V*)w then f is a polynomial in u, Pu, x 2 ' and ~2. We write f = L.p,q'Ppq(u, Pu)~~i. Let Av(g) = I~I L.sEwsg. Then since su = u, SPu = Pu for SEW,

u

xi

f = Av(f) = L 'Pp,q(U, PU) Av(xf~i)· p,q

If q = 0 then since Av(xf) is W -invariant it is a polynomial in u I ' u 2 . If q > 0 then g = Av(~+q) is invariant, so it is a polynomial in u I ' u 2 and f= CL'Pp,q(u, Pu)Pq(g) p,q

with C-I=(p+q)(p+q-l) ... (p+l). We note that this argument is a variant of Weyl's original argument of An' We now consider the cases An_I' En' Dn. For all of these cases we take Xi = ei


N. R. WALLACH

812

as in [Bo]. We will use the symbol Tk .l'Sk) _ 1 '"' (XITI.l'SI ~I ... x k ~k - , L..J n.

TI .l'SI

xwl~wl

Tk .l'Sk ... xwk~wk'

1 $ k $ n.

wESn

Then in the An_I case every j E .9'(V x V*)w is a linear combination of terms as above. For Bn the same is true as long as we restrict our symbols to rj + Sj even. For Dn this is true if for k < n the rj + Sj are even and for k = n the rj + Sj are either all even or all odd. Let A be the algebra generated by 1 and the polarizations of the basic invariants. We show that the symbols above for k < n are in A by induction on k. If k = 1 then (rl +sl)···(r l + l)(x;I
T S

T S

n-k+1

T S

T S

.

(XII
by induction on the number, q, of sJ =1= o. If q = 0, then U E .9'(V)w CAl. Assume 0 $ q $ k - 1 . We now consider the case when q = k. Set p = n - k. Then we may (after relabeling) assume that

_ ... (T1-I ... Tp-I Tp+I-I.l'Sp+I ... Tn-I.l'Sn) U - XI xn XI xp Xp+I ~p+1 xn ~n· The result for Bn now implies that U EA. Thus we may assume that rp+ 1 =

Consider

L

(TI Tp Tp+2.l'Sp+2 Tn .l'Sn) Sp+1 XI· .. Xp Xp+I Xp+2~P+2 ... Xn ~n p T T.-I S T T S l 1 XPX ='"'r.(x

o.


813

where v is a sum of symbols with at most q - 1 nonzero exponents for the ~j • We note that all of the terms in the first sum have at least one pair (rj' s) = (1 , 0) . Thus we may assume that If we now repeat the above argument then 'p 'p+2):Sp+2 .rn):sn) L ( S

p+l

XI··

·xp x p + I X p+2"'P+2·· ·xn "'n ,.-1

p

= Lrj(x l •· ·x/

S

~rl

j=2

..

,

·X;Xp+I •.

,

S

·xnn~nn)

+ 2u + V

with v in Al (smaller number of nonzero ~j exponents) and each term having two exponents (1, 0). If we continue in this way, we may assume that (x x ... X

U -

-

Now

Ls

p+l

with v

E

I 2

):Sp+l •••

p"'p+1

(X I X 2 •• ·XpXp+I •• • x~n~~n)

x'n):sn) n "'n

= (p

.

+ l)u + V

A I . This completes the proof. ApPENDIX

3.

SOME OBSERVATIONS ABOUT DISTRIBUTIONS

In this appendix we first give a slight generalization of the well-known result that a homogeneous distribution on Rn is tempered. We use it to show that certain distributions, locally defined, on a real semisimple Lie algebra extend to tempered distributions. If n is a compact subset of R n then we set C; (Rn) equal to the space of all smooth functions on R n such that supp fen with the topology given by the seminorms ql(f) = sUPxEn 18 1f(x)1 for I = (iI' ... , in)' i j E N (= {O, 1, 2, ... }). Here we use standard multi-index notation as in Appendix 1. As usual, a distribution on R n is a linear function, T, on C;"(Rn) such that T is continuous on each space C;(Rn ). As a customary, we set 9'(Rn ) equal to the space of all distributions on Rn . If f E C,oo(Rn) then we set Pk

,

I(f)

= xER sup IIxll k l8 1f(x)l· n

Here IIxII 2 = x~ + ... + x; (as usual) and k ~ O. As is usual, we denote by .9"(Rn) the space of all f E Coo(Rn) such that Pk ,/(f) < 00 for all k ~ 0, I endowed with the topology induced by the seminorms Pk, I. If T E 9' (Rn) then (as usual) we say that T is tempered if T extends to a continuous functional on .9"(Rn). We set E = Ei x j 8 /8x j • The following is a mild extension of a well-known result and is no doubt also well known. The proof involves standard methods of distribution theory.


814

Lemma 1. If T

N. R. WALLACH

E 9' (Rn)

is such that dim C[E]T < 00 then T is tempered.

We will now apply this lemma to the case of interest in this paper. Let G be a semisimple group of inner type (cf. [RRGI, 2.2.8]). Let g be the Lie algebra of G, and let 0 be a Cartan involution of G. As usual, B will denote the Killing fomi of g. We note that (X, Y) = -B(OX, Y) defines an inner product on g. Using an orthonormal basis of 9 with respect to (... , ... ) we may identify g with R n with n = dim g. If X E 9 then X is said to be nilpotent if Ad X is nilpotent. Let ,Af denote the variety of nilpotent elements of g. If f E Cm, IXj(Y)I

INVARIANT DIFFERENTIAL OPERATORS ON A REDUCTIVE LIE

INVARIANT DIFFERENTIAL OPERATORS ON A REDUCTIVE LIE

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