Minimal representations, geometric quantization ... - Semantic Scholar

Proc. Nadl. Acad. Sci. USA Vol. 91, pp. 6026-6029, June 1994 Mathematics

Minimal representations, geometric quantization, and unitarity (Joseph ideal/nilpotent orbit/syMplec

manfd/half-form/hypergeometrlc fuctio)

RANEE BRYLINSKItI AND BERTRAM KOSTANT§ tPennsylvania State University, University Park, PA 16802; Institute of Technology, Cambridge, MA 02139

*Harvard University, Cambridge, MA 02138; and §Department of Mathematics, Massachusetts

Contributed by Bertram Kostant, March 14, 1994

in H from the associated nilpotent orbit in g appeared first in ref. 8, and there Vogan worked out the existence, unitarity, and K-type decomposition for several cases including some of the cases in the table. In. ref. 9 Vogan proved that, as a virtual K-module, H must equal a multiple of the space of sections of a K-homogeneous half-form line bundle on Y minus some finite-dimensional K-module. Let 0 C o* be the minimal (nonzero) coadjoint orbit so that o corresponds under the Killing form to Ornin. Then 0 is a complex symplectic manifold with respect to the KirillovKostant-Souriau symplectic form. We have g* = f* e p* in the obvious way. A necessary condition for Go to admit a minimal representation is that the set Y = o n p* is nonempty-we assume this condition from now on. The set Yis then a smooth K-homogeneous Langian submanifold of 0; in fact, Y is the cone of highest weight vectors in the simple K-module p*. In particular dim Y = Y2dim 0 and Y = K*C where E p*. The K-action on Y defines a Hamiltonian K-action on the cotangent bundle T* Y (equipped with its canonical symplectic form) with moment map IK: T* Ye f*. We find that although 0 admits no K-invariant polarization, in fact T* Y and a (ramified) 2-fold covering of 0 share a common smooth symplectic K-invariant open dense set M. Consequently the Hamiltonian G-action on 0 defines an infinitesimal Hamiltonian g-action on M-i.e., an infinitesimal action by vector fields which admits a moment map pu: M-- g*. In this way, the infinitesimal f-action on T* Yis being enlarged, over M (but not over all of T* Y), to an infinitesimal g-action. We may thus regard M as "Hamiltonian (g, K)space." Our construction of M is intrinsic to the geometry of T* Y-one characterization is that M is equal to the union of all K-orbits on T* Y of principal orbit type. To state our results in more detail we introduce the following. Let R(X) be the algebra of regular functions on an algebraic variety X. Then we have an inclusion R(T*Y) C R(M) of Poisson algebras where the symplectic form on T* Y defines a symplectic form on M. Each 4 E R(M) defines a Hamiltonian vector field f# on M. As Y is K-homogeneous, each x E f defines a vector field

In the framework of geometric quantization ABSTRACT we exlicitly construct, in a uniform fashion, a unitary minimal representation irT of every simply-connected real Lie group G. such that the maximal compact subgroup of G. has finite center and G. admits some minimal representation. We obtain algebraic and analytic results about frf. We give several results on the algebraic and symplectic geometry of the minimal nipotent orbits and then "quantize" these results to obtain the corresponding representations. We assume (Lie Go)C is simple.

Let Go be a simply-connected simple real Lie group and let K0 be a maximal compact group. Let g be the complexification of %o = Lie Go with g = f + p a complexified Cartan decomposition corresponding to go. Let G be a simplyconnected complex Lie group with Lie algebra g and let K be the subgroup corresponding to I. We assume throughout this note that the pair (G0, K0) satisfies the following equivalent conditions: (i) the associated symmetric space GO/Ko is non-Hermitian, (ii) K0 has finite center, and (iii) p is irreducible as a f-module. We will call an infinitesimally irreducible representation ire, of Go on a complex Hilbert space minimal if its annihilator Ann ir, in the enveloping algebra U(g) is equal to the Joseph ideal J (1)-i.e., 7rO, is "attached" to the minimal nilpotent orbit and the primitive ideal Ann ir, is completely prime. We assume throughout g is simple. In this note we extend the methods of ref. 2 to obtain, in a uniform way, analogous results on the construction, unitary structure, and harmonic analysis of a minimal representation 'rr of Go for each case where 'rw exists. In addition, we explain some of our results on the symplectic geometry of the minimal nilpotent orbit Oiam C g and show how these results underlie our explicit construction of the corresponding minimal representation. In our models the Harish-Chandra module H for 1r, is the space IMY, N1/2) of algebraic sections of the (K-homogeneous) half-form line bundle N112 on a (Khomogeneous) Lagrangian Y of Ongn. (H is spherical if and only if N1/2 is K-equivariantly trivial-this occurred for the three cases treated in ref. 2.) Furthermore, the elements of g act by explicit pseudodifferential operators on r(Y, N1/2). This places our results clearly in the scheme of geometric

nx on Y.-- Let #K(X) E R(T* Y) be the symbol of 'f and let OK

quantization. Finally, we give explicit formulas in terms of hypergeometric functions for the unitary structure of iro and also for a matrix coefficient of 7rT, restricted to an SL(2, R) root subgroup. The pairs (g, f) occurring here fall naturally into 10 casessee Table 1 below. The existence of Hilbert spaces carrying these unitary minimal representations, along with the K-type decomposition of H, had already been established by methods adapted to the individual cases, with the exception of case x if q 2 6: for cases i-iii see ref. 3; for iv-vii see ref. 4; for viii see ref. 5; for ix see refs. 6, 3 (ifp = q), and 7; and for x see ref. 5 if q = 4. The notion of determining the K-types

S(f) R(T* Y) be the corresponding algebra homomorphism, where S(f) = @,i:oS'(f) is the graded symmetric algebra of f. Then qK is the comorphism to AXK. Let A E R(T* Y) be the symbol of the Euler vector field E :

on Y. (The Euler vector field on a vector space V, or on any cone inside V, maps each linear function to itself.) Let a be the unique vector field on T* Ywhich coincides with the Euler vector field on every fiber of the cotangent fibration ty: T* Y -+ Y. Then we have a K-invariant algebra bigrading R(M) = EJpqezRpsq(M), where Rp,q(M) is the intersection of the p-eigenspace of {A with the q-eigenspace of a. Then OK(Sq(f)) C R(oq)(M), while Rp(Y) C R(p,o)(M), where Rp(Y) is the p-eigenspace of E and R(Y) C R(T* Y) is the inclusion defined by ty. The Poisson bracket satisfies {R(p,q)(M), R(p',q')(M)} C

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R(p+p',q+q'-1)(M). Consequently, R,(M)

6026

=

@p+q=1Rp,q(M),

Proc. NatI. Acad. Sci. USA 91 (1994)

Mathematics: Brylinski and Kostant

6027

the 1-eigenspace of the vector field CA + a, is an (infinitedimensional) Lie subalgebra of R(M) containing MK-. Let A' be the group of symplectic diffeomorphisms of T* Y defined by the group sa of K-invariant automorphisms of Y. Then al' A C*, as .s is the one parameter group giving the flow of E. THEOREM 1. Let M C T*Y be the complement ofthe divisor (A = 0) so that M is a Zariski open dense K-invariant subset. Then, up to the action of d', there is a unique finitedimensional subspace r C R,(M) such that r contains 4'0dt), r is closed under Poisson bracket, and r is isomorphic as a Lie algebra to g. The image of the moment map tt: M -g* corresponding to r lies in 0, and furthermore, p(M) C 0 is a Zariski open dense K-invariant subset. Finally, pu defines a 2-to-1 cover M -* A(M) and the corresponding lifting of the Euler vector field on p(M) is equal to kA+,. In fact, A2 = Fo°., where F E R(O), and then p(M) C 0 is the complement of the divisor (F = 0). Moreover, A and Fare primitive K-invariants in that R(T* Y)K = C[A] and R(O) =

From now on we exclude two cases: (i) g = ?t(n, C), n 3, since there the Joseph ideal is not defined, and (ii) g = ?w(p + q, C) with p + q odd and p, q .4 since there (Howe and) Vogan (8) has shown that no minimal representation exists and, indeed, no K-homogeneous half-form bundle exists. We note that the requirement that 0 meets p* has already excluded the following five cases of non-Hermitian symmetric pairs (g, f): (KI(2n, C), ?op(2n, C)) where n 2 2, (Bo(p + 1, C), Bo(p, C) + 5o(l, C)) where p 2 3, (4p(2p + 2q, C), ?p(2p, C) + 4p(2q, C)) wherep, q . 1, (F4, !5p(9, C)), and (E6, F4). This list follows from the classification in ref. 10 (see Table 1 in ref. 11), as we show that 0 fails to meet p* if and only if K has a Zariski dense orbit on 0. Now Y admits a unique (up to isomorphism) line bundle N1/2 such that N112 ® N112 = N, where N is the top exterior power of the cotangent bundle on Y and, furthermore, N112 has a unique K-homogeneous structure. N112 is then the "half-form" line bundle on Y. The K-action on N1/2 defines a K-module structure on the space

Next we explicitly describe (a choice of) the Lie algebra r in terms of the 4K(x) (x E f), A and the functions on Y. Let Z(V) be the K-orbit of highest weight vectors in an irreducible K-module V. Let v E Z(P). Then the Weyl group translates of v form a basis by weight vectors of p. Associated to v we have the isotropy subgroup Kv, the isotropy subalgebra fv, the nilradical ftv of fv, and also the subset 'iv C p defined by w E ',v if and only if w E p and ([v, w], v, w) is an S-triple (i.e., we have (h, v] = 2v and [h, w] = -2w, where h = [v, w]). The subalgebra f is abelian and so the symmetric algebra S(O ) identifies with the universal enveloping algebra Uffv). Furthermore, [fv, If, v]] = Cv, and hence we have a bilinear pairingj: S4(fV,) X S4(f) C determined uniquely by j(X4, b4) = 24c4, where x E f v, b E I, and [x, [b, v]] = cv. The action of f on p defines a homomorphism T: U(f) End p. PROPOSITION 1. Let v E Z(p). Then there exists a unique Kv-invariant polynomial Pv E S4(f V) such that (i) (RPv)(Cw) = Cv for all w E %, and (ii) j(P, b4) = 1 if b E f and (7b4)(v)

H = F(Y, N1/2)

C[F].

E

%V.

In the proposition, condition i determines Pv up to a nonzero scalar and then ii is a normalization condition. There is a natural isomorphism f: p R,(Y), v fv, defined by the inclusion Y C p*. THEOREM 2. The rational functions Oc(Pv)/fv on T*Y, where v E Z(p), are in fact regular (i.e., have no poles), and furthermore these functions span a simple K-submodule of R(T*Y) equivalent to p. Let r C R,(M) be as in Theorem 1 and let : g -->t be a Lie algebra isomorphism extending OK. Then, afterpossibly modifying 4 by the action ofA', we have

+(v) = fv - gv, where gv =

1

K

[1]

if v E Z(p). The functions fv, v E p, generate the (Poisson commutative) K-stable subalgebra R(Y) of R(T*Y). But also the functions gv, v E Z(P), Poisson commute and generate a K-stable subalgebra of R(M) isomorphic to R(Y). Notice that 4K(I) C R(o0,)(M), while fv E R(l,o)(M) and gv E R(-1,2)(M). The action of d fixes EK(E) and transformsfv - gv into cf, - c-lgv, where c E C* s '. The formulas in 1 give the unique expression for 4+(v) in terms of the (local) coordinate system on M formed by A together with functions on M corresponding to a basis of the Heisenberg Lie algebra given by the nilradical of gv. We next "quantize" the functionsfv and gv in 1-i.e., we convert these functions into pseudodifferential operators on the space of global sections of a (K-homogeneous) "halfform" line bundle N1/2 on Y.

of algebraic sections of N1/2 and also a corresponding algebra homomorphism irK: U(f) -- End H. We have some natural differential operators on H whose symbols are the functions occurring in 1. Indeed, each function fv, v E p, defines a multiplication operator. Corresponding to a vector field r on Y we have the order 1 differential operator X, where T denotes the Lie derivative. If x E I, then OK(x) is the symbol of T,,, and also TKW(x) = Finally, A is the symbol of E' = 5E. If L is a K-module then let (End L)K be the algebra of K-finite endomorphisms of L. The "quantization" of Theorem 2 is THEOREM 3. If v E Z(p), then the operators mx(P,) and f, on H have the same image and, furthermore, they commute. Consequently the quotient wrk(P,)/f, defines a global algebraic differential operator on N1/2. The linear span of these operators wK(Pv)/fv, v E Z(P), is a simple K-submodule of (End H)K equivalent to p. The operator E' on H is diagonalizable with positive spectrum. Let ir: gD--End H [2] be the linear map such that lr(x) = ITK(X) if x E f and

=wK(P Ev) W~)= f, - T,,, where Tv =E(E 1)f [3] 1

if v E Z(p). Then r is a Lie algebra homomorphism. Hence ir is a representation of o by global algebraic pseudodffer-

ential operators on N"/2. The operators Tv, v E Z(p), (respectively, fv, v E p) commute and generate a K-stable subalgebra of (End H)K isomorphic to R(Y). The proof of Theorem 3 again (cf. ref. 2) relies on an application of the generalized Capelli identity established in ref. 12. Let Hs, s E Z, be the s-eigenspace of E' on H. Concerning the representation theory of ir we prove THEOREM 4. The representation 2 constructed in Theorem 3 is irreducible. The annihilator of the algebra homomorphism ;i: OIL(g) -- End H corresponding to iris the Joseph ideal J. Furthermore, H admits a unique (up to scaling) &0-invariant positive definite Hermitian inner product B. The (g,K)-module H is then the associated Harish-Chandra module of a unitary minimal representation ir. of Go on a (complex) Hilbert space We. The spectrum of E' on H is equal to {r, r 1 +. ..}, where r is the positive integer or half-integer given in Table I below. The eigenspaces of E' on H are simple inequivalent K-mod-

6028

Mathematics: Brylinski and Kostant

Proc. Nad. Acad. Sci. USA 91

Table 1. Cases of minimal representations Case g i E6 !90(8, C) ii E7 K(8, C) iii E8 4o(16, C) iv F4 ip(6, C) + Q[(2, C) v E6 ?R(6, C) + ?f(2, C) vi E7 ?o(12, C) + QR2, C) vii E8 E7 + BR(2, C) viii G2 QR2, C) + Q(2, C) ix Bo(p + q, C) 0o(p, C) + ?o(q, C) 3 x

p

I

8 < p + q is even 4o(3 + q, C) 4 s q, q is even

Kf(2, C) + 4o(q, C)

=

fn

nO n!

an(b)" -\xn = 2F((a) b; r + 1; x) =

n!/ft

r

V C C C C @ SKC2) C S2(C2)

5/2 4 7 2 3 5 9

2

S4(C2) S(C2) S2(C2) @ C

p

S(q-P)/2(Cp) S C

1 Y2(q

3

Sq-3(C2) @ C

½(q - 2)

C C

-

2)

q,

ules and so in particular H is a multiplicity-free K-module. V We have Hp+r Vp*+,,(p 2 0), so that H Ep2oVp#+,v where is the minimal and K-type. V, V# As noted above, the K-type decompositions were already known. The operator E' and its minimal eigenvalue r are completely new, however. In Table 1, 1 is the rank of the symmetric pair (g, I). In case viii, p = S3(C2) 8) C2. We may group cases ix and x together by setting p = 3 in x. Next, we explicitly describe the Hilbert space inner product. First we choose (as we can) nonzero vectors z E Z(p) and so E Z(Hr) such that ((z, zf, z, Z) is an S-triple and so is Kz-invariant. Here z z-P Z denotes the complex conjugation map on g defined by go. In particular, then so and z are highest weight vectors for the K-action with respect to a common choice of positive system. If H is spherical then so is, up to scaling, the constant function 1. We normalize B by requiring that B(so, so) = 1. Since B is K-invariant it follows that Hp and Hq are B-orthogonal if p # q and, furthermore, that each restriction BIHp is determined up to a scalar. Hence it suffices to know B on the highest weight vectors 'r(z))(so) E Hn+r, n - 0. It turns out that these values are given by the coefficients of a Gaussian hypergeometric function. Define m to be the eigenvalue of 'n([z, z]) on so. THEOREM 5. We have lr(z)n(so) = fnso where n - 0. The normalized G.-invariant Hilbert space structure on XC is such that

B(so B

I 6 7 8 4 4 4 4

(1994)

n2-o n!(r +1)n

xn'

where a and b are given asfollows: (1) in cases i-viii in Table

1-i.e., when g is of exceptional type-we have a = 1 + d/2 and b = 1 + d, where d is defined by 1 + 3d/2 = r + m; and (2) in cases ix-x, we have a = (q - 2)/2 and b = (q - p + 2)/2. In every case we have a + b = r + 1 + m.

The values of d in (1) turn out to be i, 1; ii, 2; iii, 4; iv, 1; v, 2; vi, 4; vii, 8; and viii, Y3. Generalizing the notion of spherical function, we have the matrix coefficients corresponding to the minimal K-type in H. We cannot as yet explicitly determine these matrix coefficients. But we can determine their restriction to the root subgroup L, = SL(2, R) of G, corresponding to the span of h = z + z, z, and z. As an example we consider the matrix coefficient co given by co(g) = B(g-so, so), g E Go. (Note that co is just the spherical function in case H is spherical.) The function 4. given by q6(t) = co(exp th'), t E R, determines the value of c0 on L0. THEOREM 6. The function 4,: R -- R has values

0(t) = 2F,(a, b; r + 1; - sinh2 t)

where a and b are as in Theorem 5. A feature of our results is that we can construct the representations iT in any model of H so long as we are given both the K-module structure and the R(Y)-module structure on H. In particular, the half-forms can be completely suppressed in the model. We illustrate this by two examples. These cases are particularly simple since (unlike the cases i-iii explained in ref. 2) here the polynomials P, (see Proposition 1) factor into a product of 4 linear terms. Example 1: Let g& = 4o(4, 4). Then f = Qf(2, C)4. As K-modules we have H R(Y) @Ena0*S(C2)04. A model of H is given in the following way. Let S be the polynomial ring in 8 variables xp,, where p E {1, . . . , 4} and i E {1, 2}. Then H is the subalgebra of S generated by the 16 products X1,iX2jX3,kX4,l, where i, j, k, I E {1, 2} so that =

=

H = @ naoCn[xl,, x1,2}-Cn[x2,l,

X2,2}Cn¶X3,1, x3,2] {Cn[x4,1, x4,2] C S,

where Cn[u, v] is the space of degree n polynomials in u and v. Notice then that H is the space of invariants in S under a scaling action of C* x C* x C*. Let (3 be the differential operator on S given by

X1,1

X

+

ax11'j X1,2 aX1,2 -

-

+ 1.

Applying Theorem 3, we find that the following 28 pseudodifferential operators on S preserve Hand, as operators on H, they form a basis of a complex Lie algebra g isomorphic to ?o(8, C). a a a a '~Xp,2 s~Xpjl Xp,2 Xpjl aXp,2 axpl axpl - xp,2where p E{1, 2, 3, 4},

(_.)i+j+k+l

a4

x1,x243,k4,l-AP((+ aX1X.aX2j~'aX3,k'aX4,1'

XlwiX2jX3,kX4,1

p

1))ali8XJ8~k84l

where {i, i'} = {J, j'} = {k, k'} = {l, l'} = {1, 2}.

The normalized pre-Hilbert space structure B on H is such that

B(xp j/n!, xpJ/n!) = 1/(n + 1), where p E {1, . , 4} and i E {1, 2} (in agreement with ref. 6). Example 2: Let g& be of type G2. Let S be the polynomial ring in 4 variables ul, u2, x1, x2 and let S' R(Y) be the

Mathematics: Brylinski and Kostant subalgebra generated by the 8 products uj'xj and u~ui xj, where {i, i'} = {1, 2} andj E {1, 2}. A model of His the S'-submodule H = (D2oC3"+[uj,

u2WC"[xl, x2] C S.

Let (3 be the differential operator on S given by

a a (3=X1-+X2-+ ax2 ax2 1.

U2

aU2

au, Ul au, -U2 au2 -

-

-

a a a a X2 -, XI X2aX2 ax, ax, aX2

X1 -,

(3..1)+i+ 1) audx 27x(3 UiX-

- -

a4

where {i, i'} = {j, }l= {1, 2}

iaXJ

''

a4 (-l)'+j 'j 278(,B + 1) auiauiaxjs where {i, il} = ti, j'} = {1, 2}.

The normalized pre-Hilbert space structure B on H is such that (3n + 3)!

B(U~n2 I' u'.,n+2xln!) u?" 2x'/n!) - 33n3!n!(n + 1)!(n + 1)! B(IXixi/n!, n

where i, j E {1, 2}.

6029

Full details and proofs of the results announced here and in ref. 2 will appear elsewhere. It is a pleasure to thank David Vogan for helpful discussions concerning half-forms and cases vin and x in Table 1. Part ofthis work was carried out while R.B. was visiting Harvard University, and she thanks the Harvard mathematics department for its hospitality. This research was supported in part by an Alfred P. Sloan Foundation Fellowship to R.B. and National Science Foundation Grant DMS-

9307460 to B.K.

Theorem 3 implies that the following 14 pseudodifferential operators on S preserve H and, as operators on H, they form a basis of a Lie algebra isomorphic to the complex simple Lie algebra of type G2. -1

Proc. Nati. Acad. Sci. USA 91 (1994)

1. Joseph, A. (1976) Ann. Sci. Ecole. Norm. Sup. 9, 1-30. 2. Brylinski, R. & Kostant, B. (1994) Proc. Natd. Acad. Sci. USA 91, 2469-2472. 3. Kazhdan, D. & Savin, G. (1990) in The Smallest Representations of Simply-Laced Groups, Israel Mathematics Conference Proceedings, Piatetski-Shapiro Festschrift, eds. Gelbart, S., Howe, R. & Sarnak, P. (Weizmann Science, Jerusalem) Vol. 2 pp. 209-223. 4. Gross, B. &Wallach, N. in Lie Theory and Geometry:In Honor oflB. Kostant, eds. Brylinski, J.-L., Brylinski, R., Guillemin, V. & Kac, V. (Birkhaeuser, Boston), in press.' 5. Vogan, D. A. (1994) Invent. Math. 116, 677-791. 6. Kostant, B. (1990) in Operator Algebras, Unitary Representations, Enveloping Algebras, and Invariant Theory, eds. Connes, A., Duflo, M., Joseph, A. & Rentschler, R. (Birkhaeuser, Boston), pp. 85-124. 7. Binegar, B. & Zierau, R. (1991) Commun. Math. Phys. 138, 245-258. 8. Vogan, D. A. (1981) in Non-commutative Harmonic Analysis and Lie Groups, Springer Lecture Notes, eds. Carmona, J. & Verge, M. (Springer, Berlin), Vol. 880, pp. 506-535. 9. Vogan, D. A. (1991) in Harmonic Analysis on Reductive Groups, eds. Barker, W. & Sally, P. (Birkhaeuser, Boston), pp. 315-388. 10. Brylinski, R. & Kostant, B. (1994) J. Am. Math. Soc. 7, 269-298. 11. Brylinski, R. & Kostant, B. (1992) Bull. Am. Math. Soc. 26, 269-275. 12. Kostant, B. & Sahi, S. (1991) Adv. Math. 87, 71-92.