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

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Minimal Representations of E6, E7, and E8 and the Generalized Capelli Proc. Nati. Acad. Sci. USA Vol. 91, pp. 2469-2472, March 1994 Mathematics Minimal representations of E6, E7, and E8 and the generalized Capelli identity (Joseph ideal/unitary representa/geonec quantzatn/Jordan algebra/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 Bertram Kostant, 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 Lie group 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 simple Lie group 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 field) 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) subgroup, Capelli identity and the fact that V' has degree 4, we produce Gross and Wallach show that, for certain subgroups, 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 compact group 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 Lie algebra 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 integers. 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 ,$ = symmetric space 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).
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