IC/93/55 H4TH INTERNATIONAL CENTRE FOR THEORETICAL PHYSICS \ A REDUCTION OF THE GLOBALIZATION AND U(1)-COVERING INTERNATIONAL Tran Dao Dong ATOMIC ENERGY AGENCY UNITED NATIONS EDUCATIONAL, SCIENTIFIC AND CULTURAL ORGANIZATION MIRAMARE-TRIESTE 1C/93/55 INTRODUCTION International Atomic Energy Agency In order to find irreducible unitary representations of a connected and simply and connected Lie group G, the Kirillov's orbit method furnishes a procedure of quantization, starting from linear bundles over a G-homogeneous symplectic manifold (see [5]). In United Nations Educational Scientific and Cultural Organization [1] and [2], Do Ngoc Diep has proposed the procedure of multidimensional quantization for genera! case, starting from arbitrary irreducible bundles. This procedure could be INTERNATIONAL CENTRE FOR THEORETICAL PHYSICS viewed as a geometric version of the construction of M.Duflo [4], In 1988, W. Schmid and J.A. Woif [3] described in terms of local cohomology the maximal globalization of Harish-Chandra modules to realize the discrete series representations of semi-simple Lie groups by using the geometric quantization and the derived Zuckerman functor modules. In [9], we modified the construction suggested by W. Schmid and J.A. Wolf to the case A REDUCTION OF THE GLOBALIZATION of U(l)-covering by applying the technique of P.L. Robinson and J.H. Rawnsley [6]. AND U(1)-COVERING Our purpose is to give an algebraic version of the multidimensional quantization with respect to U(l)-covering. In this paper, we reduce the same problem to the case of reductive Lie groups. Using the unipotent positive distributions we construct geometric complexes and their corresponding Harish-Chandra modules. Then we will describe the Tran Dao Dong * maximal globalization of Harish-Chandra modules in terms of local cohomology with respect to U(l)-covering. International Centre for Theoretical Physics, Trieste, Italy. 1. UNIPOTENT POSITIVE DISTRIBUTIONS ABSTRACT Let G be a real algebraic Lie group. Denote by Q the Lie algebra of G and Q* its dual space. The group G acts in Q* by the coadjoint representation. Denote by GF We suggest a reduction of the globalization and multidimensional quantization to the stabilizer of F € Q* and by QF its Lie algebra. Let UF be the unipotent radical the case of reductive Lie groups by lifting to U(l)-covering. Our construction is connected of Gf and UF be its Lie algebra. Denote by QF the reductive component of Gp in its Cartan-Levi's decomposition Gp = UFQF- with the M. Dufio's third method for algebraic groups. From a reductive datum of the given real algebraic Lie group we firstly construct geometric complexes with respect to 1.1. U(l)-covering of the unipotent radical. U( 1 )-covmng by using the unipotent positive distributions. Then we describe in terms of Let G be the U(l)-covering of GF and Up be the inverse image of UF local cohomology the maximal globalization of Harish-Chandra modules which correspond F under the projection <7j : Gp —* GF, where Oj is the homomorphism defined in to the geometric complexes. [8,82]. Since _ v»w ^UF^I is a short exact sequence then we have the split short exact sequence of corresponding MIRAMARE - TRIESTE Lie algebras 0 _ W(l) —• Lie UpW —*li —>0 March 1993 F ll) W . Thus Up is the U(l)-covering of UF and we have Lie Up SWfffiM(l). From the local triviality of the (Jf-principal bundle QF >—> UF\G -» GF \ G there exists a connection on the bundle. Then the Kirillov 2-form Ba of JiT-orbit £1 passing F induces a nondegenerate closed G- invariant 2-form Bn on the horizontal part \ G) defined by the formula Permanent address: Department of Mathematics, Hue College of Teacher Education, 32 Le Loi Street, Hue, Vietnam, c/o Institute of Mathematics, P.O. Box 631, Bo Ho, 10 000 where / € Up\G, k(f) = F e ft, and Jfe, is the linear lifting isomorphism induced Hanoi, Vietnam. from k (see [7]). As in [8], the symplectic group SP{TU)H{UF \ G); &a(f)) has an Un- c connected covering Mp (T{f)fj{UF \ G)) and we obtain the following isomorphisms u ll) Then B ^ = Gp « Bo is the t7(l)-covering of B = GF • Ba and we have \ G)) = Sp{QIQF) and c 1 uw Mp (TU)H(UF\G))z. Denote by Bo the inverse image of Bo in B under the [/(l)-covering projecti ( As in [8] we have Using these isomorphisms we can view VF "asa Lie subgroup of the Cartersian product c of Lie groups UF X Mp (T(f)H{UF \ G)). Proposition 1.3. in a small neighbourhood of the identity of Upil) we obtain 1.2. Unipotent positive distributions. We do not assume that the orbit tl passing F € Q* is an integral orbit, i.e. there does not exist a unitary character XF of GF. where <p 6 M satisfying the relation \2DetC^-i = exp(-j^). Definition 1.1. A point F € Q* is called (u, U(l))-<idmisi>ible (u for unipotent radical, Tile integral kernel of8F^ is given by the formula 1 U(l) for U(l)-covering) iff there exists a unitary character 9F : UF —• S such that ^r < z,w > -— w,tv ..[/(i \ . __ , it in 4fi where z,w £ (0 + B)/(B nB). where (X,^) We see that if F is U(l)-admissible (see [8]) then it is (u, {/(Inadmissible, but Denote by 2ir^. (F) the set of all equivalent classes of irreducible unitary rep- the reverse does not hold in general. resentations of GF such that the restriction of the composition of a} and each of them to UpW is a multiple of the character 6^W. When F is (u, [/(Inadmissible and Definition 1.2. A smooth complex tangent distribution L C (T(UF \ G))c is called a r 6 Z^iF), the pair (F,r) is called a reductive datum. unipotent positive distribution iff Let o be some fixed irreductive unitary representation of GF in a separable (i) L is an integrable and G-invariant subbundle of (TH(UF \ G))c- Hilbert V such that the restriction of (d 0 a A to (7F is a multiple of the character (ii) L is invariant under the action Ad of Gp^ ttF . (iii) V/ £ UF\G, the fibre Lf is a positive polarization of the symplectic vector space ({TU)H(UF \ G))c,Bn(/)), i.e. Definition 1.4. The triplet (L,p,u0) is called a (a, 6F)-unipotent positive polariza- ( a) dim I, = \d\mT{J)H(UF \ G),_ tion, and L is called s. weakly Lagrangian distribution iff = 0 for all X,YeLf, (i) aa is an irreducible representation of the subgroup Bo in a Hilbert ) > 0 for all X dZj, space V such that the point <7o in the dual BQ is fixed under the natural action of G and where A' is the conjugation of X. We say that L is strictly positive iff the inequality (7) F T is strict for nonzero X £ Lf. (<ra o < ;)lGi;(i>nBi'(i> -{ao ffj)lG£<i>nBt'<i> We see that if £ is a unipotent positive distribution then the inverse image B of (ii) p is a representation of the complex Lie algebra AT © t/(l)c in V LF — ktLf under the natural projection p : Qc —* QC/{QF)C is a positive polarization which satisfies E.Nelson's condition and in Qc (see [8]). Let X be a unipotent positive distribution such that LCiL and L + L are the com- plexifications of some real distributions. Then the corresponding complex subalgebra B = p~'(fcti/) satisfies the following conditions: B nF and B + B are the complexifi- Applying the proof of Theorem 2 in [1] we obtain cations of the real Lie subalgebras B 0 Q and (B + S) n Q. Denote by Bo and JV<j the Proposition 1.5. Let F e£l be (u,U(l))-ndinis$ible and suppose that (L,p,o0) corresponding analytic subgroups. We suppose that the subalgebra B is algebraic. is a (5, flp )-unipo(ent positive poiarization. Then there exists a unique irreducible The unipotent positive distribution L is called dosed iff all the subgroups fl0, representation o of Bvw in space V = V ® V such that No and the semi-direct products B =; GF • Bg and N = GF • JV0 are closed in G. In what follows, we assume that L is closed. We know that Bo is a normal subgroup in B and GF has adjoint action on BQ. Moreover, GF acts on Bo and we can define the semi-direct product GF ex Bo. 2. THE CONSTRUCTION OF GEOMETRIC COMPLEXES Proposition 2.1. There exists a canonicaJ isomorphism vw 2.1. Classical constructions. AK*)) * Hr(UF\G;O(£ )), p>0 Suppose that G is a connected real reductive Lie group. We fix a Cartan subal- 2.2. Three geometric complexes. gebra H of Qc and consider F € Q" such that {QF)C = H. Then H = GF >S a Cartan uw subgroup of G. Let (F, a) be a reductive datum and a : B —* U(V) be the rep- We note that the differential 5 of (2.3) extends naturally to hyperfunction uw v(v> resentation obtained in Proposition 1.5. Denote by E and Z the homogeneous sections, so we obtain a complex vector bundles on H \ G and UF \ G respectively associated with the restrictions of a and UF . In the category of smooth vector bundles we have the bundles dE (2.4) and £u(1) are equivalent.