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Minimal Representations of E6, E7, and E8 and the Generalized Capelli

Minimal Representations of E6, E7, and E8 and the Generalized Capelli

Proc. Nati. Acad. Sci. USA Vol. 91, pp. 2469-2472, March 1994 Minimal representations of , , and and the generalized Capelli identity (Joseph ideal/unitary representa/geonec quantzatn//hnpremel fmctfon) RANEE BRYLINSKI*t AND BERTRAM KOSTANTt *Pennsylvania State University, University Park, PA 16802; tHarvard University, Cambridge, MA 02138; and tDepartment of Mathematics, Massachusetts Institute of Technology, Cambridge, MA 02139 Contributed by , December 8, 1993

ABSTRACT We explicitly construct, in a uniform fashion, 'r(e) and ir0() rests upon an application of a generalized the (unique) mnimal and spherical representation ir, of the Capelli identity. The generalization is established in a paper split real ofexceptional type E6, E7, orEs. We obtain of Kostant and Sahi (5) and is a statement about operators several algebraic and analytic results about if. which arise from the norm of a Jordan algebra. A relation between E6, E7, and E8 and Jordan algebras is a well-known In this paper the term minimal representation refers to an result ofTits (9). In fact, Tits has a construction ofE6, E7, and infinitesimally irreducible representation ofa E8 in terms of the Jordan algebras ,I = Herm(3, F), where F whose anihilator, in the enveloping algebra, is the Joseph is respectively C, H, and the octonians. But these are degree ideal (1). Results about minimal representations ofexceptional 3 Jordan algebras. We define a construction ofE6, E7, and E8 Lie groups appear in refs. 2-4. Vogan (2) obtained results on using the three (all classical) degree 4 Jordan algebras ,I' = the K-type structure of any minimal representation as well as Herm(4, F), where F is respectively R, C, and H. In fact, a several existence results, including the existence of a unitary construction ofE6, E7, and E8 using degree 4 Jordan algebras spherical minimal representation of the split form of E8. is not new (see refs. 6 and 7). Curiously, one has Kazhdan and Savin (3) constructed aunitary spherical minimal representation ofthe split forms (over a local ) of E6, E7, dim ,I' = dim 9i + 1, and E8. A very big step forward is made in ref. 4. The main interest in ref. 4 is not the split forn-although even here the and a relation between ,$' and $ is a theorem in ref. 7. Let d K-type structure is determined-but in the rank 4 ("quater- = dimRF so that for us d takes the values 1, 2, and 4. nionic" case) real forms. Making, among other things, a The representation ir, is unitary. Using the generalized brilliant use ofthe admissibility ofaparticular SU(2) , Capelli identity and the fact that V' has degree 4, we produce Gross and Wallach show that, for certain , there is a family of polynomials of degree 4 which lead to the a discrete reduction-in some cases of Howe type-and determination of the g,-invariant pre-Hilbert space structure determine the reduction. The representations considered in B on R(Y). In fact, we explicitly express B in terms of the ref. 4 are constructed by cohomological induction (e.g., Zuck- coefficients ofthe hypergeometric function F(1 + d/2, 1 + d; erman functors). Their existence and the existence of the 2 + 3/2d; z). The representation ire, is spherical, and if L0 C corresponding unitary structure rest on some rather sophisti- G0 is the SL(2, R) root subgroup corresponding to a split form cated derived functor machinery. of the span of (h, e, f), then 4IL, is the spherical function of In the present paper we are concerned with the (unique) the subrepresentation of xr0IL0 generated by the K0-spherical spherical and minimal representation ir, ofthe split form G0 of = + E6, E7, and E8, and our approach is fairly direct. Very much vector ld. Explicitly, if x e f, then in the spirit of a usual construction of the metaplectic repre- sentation, we explicitly exhibit the operators (which turn out 4(exp tx) =F(1 +,2 1+d; 2+2d; -sinh2(f)), to be pseudodifferential) ir(x) where x E g, the complexifica- tion of gO = LieG0, and ir: U(g) -- EndV is a representation ofthe enveloping algebra U(g) on the Harish-Chandra module which is just the Jacobi function (W'/+2,/2d"0). For Jacobi V of ir,,,. The construction is explicit enough to readily yield a function notation see, e.g., ref. 8. number of algebraic and analytic results about irf. Remark: The split group L0 here and the Let g = f + p be a complexified Cartan decomposition of SU(2) in ref. 4 are real forms of the same complex root SL(2, g corresponding to go. Let G be a simply-connected Lie C) group. In the ref. 4 case one has admissibility and a group with g and let K be the subgroup corre- discrete decomposition for the SU(2). In our case there will sponding to f. Also let C, C g be the minimal (nontrivial) instead be a direct integral decomposition for L, That this G-nilpotent orbit in g. Motivated by precepts of geometric direct integral is computable is a consequence of the fact the quantization, we find that we can take the Harish-Chandra Jacobi function 4j+2i3/2d,0) is not only integrable on (0, ) but module V for ire, to be the ring R(Y) of regular functions on exponentially decays so fast that it admits a Jacobi transform. a Lagrangian submanifold Y of Om.. In fact, we can take Y See, e.g., ref. 8. = °mifn p so that Y is K-homogeneous. The action of K on In ref. 4 the SU(2) group and complex conjugates of it R(Y) is the obvious one, and the problem of constructing ir appear as one member ofa Howe pair-thereby giving rise to becomes the determination of ir(p). But in fact this is readily a discrete reduction for the pair. In our case Lo is unique-up reduced to the determination of two operators, ir(e) and to real conjugation-and it too is a member of a Howe pair. ir(f), where (h, e, f) is the SL(2, C) triple corresponding to However, in our case the reduction is continuous rather than the highest root of g. The determination (see Theorem 4) of discrete. The nontrivial element r of the center of L, defines an involution on G0 whose fixed point subgroup F0, "mod- The publication costs ofthis article were defrayed in part by page charge ulo" T, is a direct product payment. This article must therefore be hereby marked "advertisement" in accordance with 18 U.S.C. §1734 solely to indicate this fact. F0 = Lo x SO.

2469 Downloaded by guest on September 28, 2021 2470 Mathematics: Brylinski and Kostant Proc. Nad. Acad. Sci. USA 91 (1994) Furthermore, SO is semisimple and split (e.g., in the E8 case, so that the span u C f is a Lie algebra isomorphic to LieSL(2, S is the split form of E7). Then SO is a Howe pair partner to C). In such a case the eigenvalues ofad z on f are necessarily L,-,in a particularly strong way. Let X0 be the Lo cyclic . If (z, u, v) is an S-triple in f we will say that it is of subspace of the Hilbert space XC of iro which is spanned by Jordan type if the eigenvalues of lad z are exactly the 1rT(L0)ld-so that 4,+I/+3/2dO) defines a direct integral decom- numbers {1, 0, -1}. The corresponding eigenspaces then position (with respect to Lo) of X0. Let K' = Ko n So so that define a decomposition Ko is a maximal compact subgroup of SO. Then f = fo + f1 + f-1. Using the action ofto on f- , the Tits-Koecher theory defines so that if the (continuous) "components" of XC0 admit S0 the structure of a simple (formally real) Jordan algebra ,0 infinitesimal characters they would, as spherical vectors, whose complexification is given by generate spherical irreducible representations of 5,, leading to an Fo direct integral decomposition of the Fo cyclic '*C = f-1, subspace generated by 1d. That this is in fact the case is a consequence of the following result. where v E f-1 becomes the identity element. A Tits-Koecher Let EndoR(Y) be the algebra ofall ad g-finite operators on theorem is that, modulo isomorphisms, the correspondence the Harish-Chandra module R(Y). Of course lr(U(g)) C (z, u, v) tIC EndoR(Y). Let c(Lo), c(So) E U(g) be the quadratic Casimir elements associated to Lo and S0. Let I be the complexifica- defines a bijection yielding all complexified simple Jordan tion of LieLo and P the complexification of LieSo. algebras. Furthermore, THEOREM 1. We have ir(U(g)) = EndR(Y). Furthermore, any operator in End0R(Y) which commutes with nr(I S)@ is degree ,$ = rank (d,f,). a polynomial in c(Lo) and also a polynomial in c(SO). In particular, this applies to any element in the center of the Note that with this construction we can regard the sym- and metric algebra Sff') as the ring ofpolynomial functions on the enveloping algebras U(I) U(B). Jordan algebra f-1. Its structure as a f0-module is particularly It follows, for example, from the above theorem, that in a beautiful. First of all, S(f1) is multiplicity free, so that ifS(fl)n direct integral decomposition the infinitesimal characters of denotes the space ofhighest weight vectors in Sff1), then each U(f) pair offwith certain infinitesimal characters of U(B). The highest weight in SfO1)" occurs with multiplicity 1. In fact, if explicit pairing off may be determined from results in ref. 4. n is the degree of the Jordan algebra there exists n linearly We note here, however, that in the pairing off there is a role independent highest weights y,,j = 1,... , n, occunring in for the Tits Jordan algebra 1. We have 5(fl)n such that if Pj E S(f1)n a highest-weight vector corre- sponding to yj, then S(f1)n is the polynomial ring and Furthermore, i' C f. In addition, the weights associated to the infinitesimal char- degree Pj =j, acters of the discrete components of irIKR are directly so that one immediately locates all irreducible f-modules in expressible in terms of the weight of the Jordan norm of ,' S(f1). Of particular importance is the nth-degree polynomial and the weights associated to the infinitesimal characters of P,. The corresponding to module is one-dimensional and in the "continuous" components of ir0IS,, are in a similar way fact {Pk}keN is the set of all t semi-invariants in S(f 1). But expressible in terms of the weight of the Jordan norm of Ot. most important from the point ofview ofJordan algebras and Restricting oneself to S,, spherical vectors, the trivial for the generalization of the Capelli identity, Pa is the Jordan representation of L, occurs with multiplicity 1-not in the norm (the generalization of ) of ft1. Now the t Hilbert space for ir,, but as a distribution. Remarkably, highest weights occurring in the polynomial ring 5(f1) can however, it is represented as a function on Y. also be regarded as highest weights for f itself. The corre- THEOREM 2. There exists a unique (up to scalars) r(Lo) sponding irreducible f-modules define the multiplicity free invariant distribution v which is also K.-fixed. Furthermore, f-module structure on the ring R(O) of regular functions on v is a holomorphicfunction on Y. In fact, if y E R(Y) is the the nilpotent orbit 0 = K-v, recalling that v is the identity restriction to Y ofthe symbol (as afunction on Omen) ofc(LO), element ofthe Jordan algebra f-I. All ofthese representations then, where 2F3 denotes the generalized hypergeometric are spherical with respect to the symmetric pair (f, fV). This function, we have is in particular true for the weight y,, of the Jordan norm P,. But / 3 3 3 1 3 \ v=2F3(1 +-d, -+-d; 1 +-d, 1 +d, 1 +-d; e 1 2 Simple Jordan algebras enter Lie theory in a remarkable way, mainly due to work ofTits and Koecher (see refs. 9 and is still an integral highest weight and the corresponding 10). We recall this theory. Let K be a complex simple Lie irreducible f-module V#, is not spherical. Consider the direct group and let f = Lie K. A 3-element subset (z, u, v) C f is sum called an S-triple if one has the commutation relations = f + V* [z, u] = 2u Now there are exactly three simple Jordan algebras I' of [z, v] = -2v degree 4. All three are classical. They are given as

[u, v] = z Y = Herm(4, F)c, Downloaded by guest on September 28, 2021 Mathematics: Brylinski and Kostant Proc. NatI. Acad. Sci. USA 91 (1994) 2471 where now F = R, C, or H. To determine ir(e) and ir(f) we first recall the generalization THEOREM 3. Consistent with the action off, g admits the ofthe Capelli identity as established in a paper ofKostant and structure of a in case n = 4. Moreover, Sahi (5). corresponding to F = R, C, or H, we have g = E6, E7, or E8. The classical Capelli identity is an equality of two differ- Furthermore, putting , = V#, the sum ential operators on the ring R(Mn(C)) ofpolynomial functions on the space Mn(C) of all n x n complex matrices. If xV E G= f + P R(Mn(C)) are the coordinate functions corresponding to is the complexified Cartan decomposition ofthe splitform go entries, then one of the differential operators is det xV of g and fi is the highest root. deta/Oxu. This degree-preserving differential operator clearly The fact that n must equal 4 enters at a number of points, depends only on the underlying additive structure of Mn(C). one of which is that, in general On the other hand, left translation defines a representation a of GL(n, C) on R(M.(C)). Let {yd} be the of LieGL(n, 1 C) corresponding to the matrix units. Even though the Lie algebrais ofcourse noncommutative, one can make sense out of det(yV + (n - 1)4,) as an element in the center of the where the Pi are strongly orthogonal roots. Thus we can have enveloping algebra. The Capelli identity asserts the necessary condition r(det(yU + (n - 1)8v.)) = det(xU)det(a/8xV). (i, ) = (I, () Kostant and Sahi regarded this as a statement about M(n, C) for a root if = we when it is regarded as a Jordan algebra. In terms of our highest only n 4. For the three cases have previous notation the generalization concerns the Jordan g f dim i' norm Pn E Sn(fl). Any element q in the symmetric algebra S(f-1) over the Jordan algebra ,i' = f-1 defines a constant coefficient differential operator 8(q) on the polynomial ring E6 SP(8) 10 S(f'). On the other hand, there is a natural involution of f which carries f1 to f-1. Let Qn, E Sn(f-1) correspond to the E7 SL(8) 16 Jordan normP. Then PnO(Qn) generalizes the right side ofthe E8 50(16) 28. Capelli identity. The generalization of the identity is a state- ment of the existence and the construction of an element W1 Note that in the quotient field of CentU(f) such that, on S(P),

1 ad W. = Pna(Q.). dim' + 1 =- dim °mi.. 2 In any case, since S(f') is multiplicity free as a P-module, the operator Pna(Qn) reduces to a constant on each of the Now by definition Y= oi,, n p. If e = eis the highest root f0-irreducible components. The generalized Capelli identity vector corresponding to 4i, then e is also the highest weight implies that the constant must be a rational function of the vector for the action of f on p, and one readily has that highest weight. For the case which concerns us here, namely n = 4, we have W4 E Cent U(0), so that the rational function Y = K e is a polynomial of degree 4. Explicitly we have is the orbit ofthe highest weight vector in p. Now if Vn* is the K-irreducible submodule in the symmetric algebra S(p) gen- P4a(124) = In j + * * * +i4 +-2(k- 1) erated by , then no is the highest weight of Vnp,. Then, as K-modules, on the irreducible f0-submodule in S(f1) with highest weight 00 vector R(Y) >2 Vas P4 P2 P4 P4. where R(Y) is the ring of regular functions on Y. Proposing the Harish-Chandra module V for ir, to be R(Y), we find the We recall d = 1, 2, 4 according to whether we are in the cases problem is to get , (and hence g) to operate on R(Y) and to g = E6, E7, E8. determine the Go invariant unitary structure B. To accom- To convert R(Y) into a Harish-Chandra module for ire, we plish both ofthese objectives we readily see that it suffices to will use (i) the commutative ring structure with which R(Y) just determine-consistent with the action of K-how the is naturally endowed and (ii) the generalized Capelli identity. S-triple (h, e, f) operates on R(Y) where in root notation The latter enters the picture mainly because of the following remarkable fact. Even though R(Y) is a f-module and S(f1) is (h, e, f) = (h#, ej,, em). only a f0-module, we have that

But h E f and we know how f operates. More explicitly, R(Y)e = 5(f1) C C[e, e-1] choices can made so that as [Pf-modules and as algebras, where subscript e denotes localization. Recall that dim Y = dim F' + 1 and also that e h =z = e* where 4i is one-half the weight of the Jordan norm P4. 2 THEOREM 4. With respect to the commutative ring struc- where (z, u, v) is the Jordan S-triple which defines the Jordan ture on R(Y) the ideal generated by e is exactly the of algebra the operator r(P4). Hence (1/u(e))ir(P4) is a well-defined algebraic differential operator on R(Y), where g4e) is mul- '= Herm(4, F)c. tiplication by e. Then Downloaded by guest on September 28, 2021 2472 Mathematics: Brylinski and Kostant Proc. Nad. Acad. Sci. USA 91 (1994)

3 2-~e(( 3 )( 3 d )-1 i O(t) = F 1 + 2 d, 1+d;2+ d; -sinh/(t)

is a well-defined pseudodjiferential operator on R(Y), where or in standard Jacobi function notation E is the Euler operator. Furthermore, extending the action of f, there exists a unique structure of an irreducible g-module =(1+-1+ d~O)O on R(Y) such that =d

ir(e) = We) + v(e) In particular, unlike the sphericalfunctions on SL(2, R) (i.e. and i(f) is defined similarly where Q4 replaces P4. Finally, iX Jacobifunctions oftheform 4T.O)) which arisefrom irreduc- is the Harish-Chandra moduleforthe minimal representation ible unitary representations of SL(2, R) and are never inte- iro of Go. grable, we have that 4,tL0 is not only integrable but infact, It is clear that ir, is a spherical representation with the identity element Id as spherical vector. The description of a using the notation of ref. 8, the Jacobi transform g0-invariant pre-Hilbert space structure B on R(Y) is normal- ized by requiring that B(ld, Id) = 1. We then find that B can = be described in terms ofthe coefficients ofthe hypergeometric P(A) 0(t)C(AI°(t)A(t)dt Ill function F(1 + 1d, 1 + d; 2 + 'd; z). The generalized Capelli identity for the case where n = 4 plays a key role in providing is defined. such a description. For convenience identify R(Y) with Xn=o Our methods here extend to give analogous results on the V,,*. Obviously V,,* and Vm* are orthogonal with respect to B construction, unitary structure, and harmonic analysis of a if n # m. Furthermore, since V.# is f-irreducible, BIVn# is minimal representation iro of a simple (simply-connected) determined up to a scalar so that it suffices only to know that real Lie group Go in each case where the associated sym- scalar on a single nonzero element of V"+. A natural candidate metric space GO/KO is non-Hermitian and ir1 exists. This will is the highest weight vector en Of V"^. THEOREM 5. In hypergeometric notation the normalized be explained in our next note. structure the GO-invariant Hilbert space for minimal repre- This research was supported in part by an Alfred P. Sloan sentation CO ofG0, where Go is the splitform ofE6, E7, or E8, Foundation Fellowship to R.B. and by National Science Foundation is such that Grant DMS-9307460 to B.K. l1 + 2d (1 +d)n 1. Joseph, A. (1976) Ann. Sci. Ec. Norm. Super. 9, 1-30. 2. Vogan, D. (1981) SpringerLecture Notes (Springer, Berlin), Vol. B(en/n!, en/n!) = 880, pp. 506-535. 3. Kazhdan, D. & Savin, G. (1990) Isr. Math. Conf. Proc. 2 +-~d n! Piatetski-Shapiro Festschrift 2, 209-223. 4. Gross, B. & Wallach, N. (1994) Lie Theory and Geometry: In Honor ofB. Kostant, eds. Brylinski, J.-L., Brylinski, R., Kac, Returning to notation which was introduced earlier, let be V. & Guillemin, V. (Birkhauser, Boston), in press. the span of the root S-triple (h, e, f) and let Lo C Go be the 5. Kostant, B. & Sahi, S. (1991) Adv. Math. 87, 71-92. subgroup corresponding to L, = g',nf so that L4, SL(2, R). 6. Kantor, I. (1973) Dokl. Akad. Nauk SSSR 208, 254-258; The minimal representation IrT is determined by the spherical English transl. (1973) Soviet Math. Dold. 14, 254-258. function 0O on Go corresponding to the spherical vector Id. 7. Allison, B. & Faulkner, J. (1984) Trans. Am. Math. Soc. 283, We cannot as yet explicitly determine TOO. But we can 185-210. determine its restriction to L,. In fact, if x = e + f so that x 8. Koornwinder, T. H. (1984) Special Functions: Group Theoret- E fo and we put +(t) = B(iT0(exp tx)ld, Id), the following ical Aspects and Applications (Reidel, Dordrecht, The Neth- result encodes B: erlands), pp. 1-85. THEOREM 6. is the Jacobifunction defined in hypergeo- 9. Tits, J. (1%2) Indagationes Math. 24, 530-535. metric notation by 10. Koecher, M. (1967) Invent. Math. 3, 136-171. Downloaded by guest on September 28, 2021