ON A DISTINGUISHED CLASS OF INFINITE DIMENSIONAL REPRESENTATIONS OF sp(2n; C ) SVATOPLUK KR YSL Abstract. In this paper we have shwn hw a tensr prduct f an innite dimensinal representatin within a certain distinguished class f innite dimensinal irreducible representatins f sp(2n; C ) with the dening representatin decmpses. Further we have prved a therem n cmplete reducibility f a k fld tensr prduct f the dening representatin (tensred) with a member f the distinguished class. 1. Intrductin The main aim f the paper is t study a distinguished class f irreducible innite dimensinal representatins f the symplectic algebra C n (s called bunded mdules) and tensr prducts f elements in this class with the dening representatin. The mtivatin fr such a study is cming frm a study f invariant dierential peratrs n maniflds with a given parablic structure. Invariant peratrs in questin are acting n vectr bundles assciated t nite dimensinal representatins f suitable parablic subgrups f a semisimple Lie grup G. In the case G = Spin (n) a particular rle is played by peratrs acting n bundles n Spin maniflds assciated t spinr-tensr representatins. On a class f maniflds with a prjective cntact structure ([3]), the crrespnding vectr bundles are assciated t representatins f symplectic grup. But there are n analgues f spinr representatins amng nite dimensinal mdules. It was suggested by Kstant (see [6]) that certain innite dimensinal representatins frm a suitable analgue f spinr representatins f the rthgnal Lie algebra D n in this case. They were used, fr example, in a denitin f a symplectic versin f the Dirac peratr by K. Habermann (see [4]). These tw innite dimensinal representatins are mdules f the metaplectic grup (a duble cver f the symplectic grup), they are called the Segal-Shale-Weil representatins. By analgy with the rthgnal case, it is interesting t understand the structure f tensr prduct f these spinr representatins with nite dimensinal mdules. It leads t the family f bunded representatins intrduced in [1]. 2000 Mathematics Subject Classicatin: MISSING!!!!! The paper is in nal frm and n versin f it will be submitted elsewhere. 1
2 S. KR YSL In the secnd sectin, we shall review basic facts abut the symplectic algebra and we shall intrduce its spinr representatins. Then we shall present sme result n decmpsitin f tensr prducts f nite and innite dimensinal representatin prved by B. Kstant ([7]) in the third sectin. In the furth sectin, tensr prducts f spinr mdules and nite dimensinal representatins are described fllwing [2], which leads t a denitin f a distinguished class f bunded representatins. The fth sectin cntains new results n the decmpsitin f bunded mdules with the dening representatins and its pwers. 2. Spinr representatins f sp(2n; C ) Let us recall rst sme basic facts n the symplectic algebra C n = sp(2n;c ): This algebra cnsists f 2n 2n matrices ver cmplex numbers f the frm A1 A A = 2 A 3 A 4 where A 1 = A T ; A 4 2 = A T and A 2 3 = A T : The Cartan algebra 3 h f C n cnsists f all diagnal 2n 2n matrices. If i dentes the prjectin nt the (i; i) element f the matrix, then the set f all rts equals = f( i j ); 1 i < j ng [ f2 i ; i = 1; : : : ; ng : The set f all simple rts equals The Chevalley basis f C n = f 1 = 1 2 ; : : : ; n 1 = n 1 n ; n = 2 n g : is given by X i j = E i;j E n+j;n+i ; 1 i < j n ; X 2i = E i;n+i ; i = 1; : : : ; n X i + j = E i;n+j E j;n+i ; 1 i < j n ; Y = X T ; 2 ; H i = E i;i E j;j + E n+j;n+j E n+i;n+i ; i = 1; : : : ; n 1 ; H n = E n;n E 2n;2n ; where E i;j is a matrix having 1 at the place (i; j). The algebra C n has a very useful realisatin cnsisting f dierential peratrs n C [x 1 ; : : : ; x n ]: It is shwn in [5] that the Lie algebra generated by fx i @ i+1 ; x i+1 @ i ; i = 1; : : : ; n 1g [ f@ 2 ; 1 (x1 ) 2 g (where @ i is the partial dierentiatin in x i ; i = 1; : : : ; n) is ismrphic t the algebra C n via the ismrphism : C n!end(c [x 1 ; : : : ; x n ]); dened by (X i i+1 ) = x n i@ n i+1 ; i = 1; : : : ; n 1 ; (X (i i+1 )) = x n i+1@ n i ; i = 1; : : : ; n 1 (X 2n ) = 1 2 @2 1 ; (X 2n ) = 1 2 (x1 ) 2 :
ON A DISTINGUISHED CLASS OF INFINITE DIMENSIONAL REPRESENTATIONS OF sp(2n; C ) 3 The requirement that the basis f i g n is an rthnrmal basis denes the inner prduct ( ; ) n h: Using the standard ntatin = 2 (; ) ; the fundamental weights f!g n are dened as the basis dual t the basis f ig n, i.e. (! i ; j ) = ij. There is a very clse analgy between representatins f C n = sp(2n;c ) and D n = s(2n;c ): Finite dimensinal representatins f C n have their cunterpart in tensr representatins f D n (i.e., representatins f D n with highest weights cnsisting frm integers). On the ther hand, there is n nite dimensinal representatin f C n similar t spinr representatins f D n. It was suggested by Kstant ([6]) that a prper analgy f spinr representatins f rthgnal grups are certain innite-dimensinal representatins f symplectic grups called the Segal-Shale-Weil representatins. They bth share the prperty that they are representatins f the duble cver f the crrespnding grups. The analgy can be nicely seen using the fllwing realisatin f these representatins. Cnsider rst the rthgnal algebras s(2n; C ) and chse a maximal istrpic subspace V subspace f C 2n ; it has dimensin n. Spinr representatins f D n can be realized n the Grassmann algebra S= V (V ) = n V i (V ). It decmpses int tw parts S= S + S ; where S + = j22z V j (V ) and S = j22z+1v j (V ); the s called half-spinr representatins. In the case f symplectic algebra, there is a similar cnstructin. Cnsider the dening representatin C 2n f s(2n;c ) with the crrespnding symplectic frm and chse again a maximal istrpic subspace V ' C n. The innite dimensinal space as described abve (using the C [x 1 ; : : : ; x n ] = 1 (C i n ) is a representatin f C n ismrphism ). It als decmpses as 1 (C i n ) = S + S : As in the rthgnal case, the rst representatin is the direct sum f even dimensinal symmetric pwers and the secnd ne f the dd dimensinal nes. This is a nice example f a supersymmetry, where the space f plynmials in n cmmuting variables (the symplectic case) has as an analgy the space f plynmials in n anticmmuting variables (the rthgnal case). This analgy explains why spinr representatins fr C n are innite dimensinal. Finite dimensinal representatins f D n can be all realized as spinr-tensrs, i.e., as submdules f tensr prducts f ne f tw spinr representatins with a tensr representatin. Cnsequently, an analgue f these nite-dimensinal representatins f D n is a class f innite dimensinal representatins f C n cnsisting f submdules f tensr prducts f ne f tw innite dimensinal spinr representatins f C n with a nite dimensinal representatin f C n. This is a class f representatins we are ging t study in the paper. 3. Tensr prducts f finite and infinite dimensinal representatins In this sectin we shall review sme basic facts n tensr prducts f nite and innite dimensinal representatins, details can be fund in [7]. Let g be a cmplex
4 S. KR YSL semisimple Lie algebra, h its Cartan subalgebra and U(g) its universal envelping algebra. Let us chse a space + f psitive rts and the crrespnding decmpsitin g = n h n + ; where n are nilptent subalgebras. Dente by Z the center f the universal envelping algebra U(g) and by Z 0 the set f all characters : Z! C : Cnsider a a representatin : g! End(V); where V is a nite r innite dimensinal cmplex vectr space. Assume that admits a central character : h 2 Z 0, i.e., (X)v = (X)v fr all v 2 V and X 2 Z. This is the case, e.g., if is irreducible. There is a map h 0! Z 0 given by 7!, where (u) = f u (), u 2 Z. The element f u is the unique element f the universal envelping algebra U(h) f the algebra h, fr which u f u 2 Un + ; where n + is the nilptent subalgebra f g and Un + is the left ideal generated by n +. Let W dente the Weyl grup f the algebra g and let ~ dente the ane actin f a Weyl grup element 2 W n the space f weights, i.e., where ~() = ( + ) ; X = 1 2 h 2 2 + It is well knwn that the map h 0! Z 0 sending! is an epimrphism and = if and nly if and are cnjugate with respect t the actin ~. Let us cnsider a representatin : g!end(v ) f the algebra g n a nite dimensinal cmplex vectr space V with the highest weight 2 h 0 : The main result needed frm [7] is the fllwing therem. Therem 1. Let f 1 ; : : : ; k g dentes the set f all weights f the representatin and Y i = fy 2 V V ; uy = +i (u)y; u 2 Zg ; i = 1; : : : ; k : Assume that the characters +i 0 : are all distinct. Then V V = Mrever, if Y i is nt zer, then Y i is the maximal submdule f V V admitting +i. km Y i : 4. Cmpletely pinted mdules In this paragraph we review sme basic facts n bunded and cmpletely pinted mdules frm [2]. Mre details can be fund there. The set f bunded mdules is a set f innite dimensinal representatins f C n, which is an analgue f the set f nite-dimensinal representatins f D n with half-integer highest weights. Let g be a cmplex simple Lie algebra and L h its Cartan subalgebra. Let us cnsider an h-diagnalisable g-mdule V, i.e., V= 2wt(V)V, where wt(v) h 0 is the space f all weights f the mdule V. We say that it is a mdule with bunded multiplicities
ON A DISTINGUISHED CLASS OF INFINITE DIMENSIONAL REPRESENTATIONS OF sp(2n; C ) 5 if and nly if there is a k 2 N such that dimv k fr all 2 wt(h). The minimal k is called the rder f the mdule. The mdule is called cmpletely pinted prvided the rder f this mdule is 1. The bunded mdules have sme nice prperties. Fr example, it is knwn that a simple cmplex Lie algebra has an innite dimensinal irreducible mdule with bunded multiplicities if and nly if it is a either a special linear algebra r a symplectic algebra. In the paper, we shall cnsider irreducible highest weight mdules. Fr any weight 2 h 0 ; we shall dente by L() the unique irreducible mdule with highest weight. Every such mdule can be realized as a qutient f the Verma mdule with the highest weight : The spinr representatins (r the Segal-Shale-Weil representatins) belng t this class. It is easy t cmpute that the weight f a cnstant plynmial is + = 1! 2 n and the weight f the mnmial x 1 is =! n 1 3! 2 n. Hence S + ' L( + ) and S ' L( ). Bth these representatins are cmpletely pinted (dierent mnmials have dierent weights). It can be shwn that the ppsite claim is als true. If a highest weight mdule L() is a cmpletely pinted C n mdule, then = + r (see [1]). The fllwing key facts describe the structure f the tensr prduct f a spinr representatins with a nite dimensinal mdule (fr details see [1, 2]). P n Therem 2. Let = i! i be a dminant integral weight f C n and let L() be the crrespnding irreducible nite dimensinal highest weight mdule. Let n T + = d i i ; d i 2 Z 0 ; d i 2 2Z; 0 d i i ; i = 1; : : : ; n 1; 0 d n 2 n + 1 n T = d i i ; d j + n;j 2 Z 0 ; d i 2 2Z; 0 d i i ; i = 1; : : : ; n 1; 0 d n 2 n + 1 : Then L( ) L() = M 2T L( + ) : Let us dente by A the fllwing set f weights. A = and n n X i! i ; i 0; i 2 Z; i = 1; : : : ; n 1; n 2 Z+ 1 2 ; n 1 + 2 n + 3 > 0 A = fl(); 2 A g : Therem 3. The fllwing cnditins are equivalent 1. L() 2 A 2. L() is a direct summand in the decmpsitin f L() L( 1 2! n) fr sme dminant integral 3. L has bunded multiplicities.
6 S. KR YSL Let shall write the set A as a unin f tw subsets A = A + A, where n 2 A ; = + + m i i ; m i 2 2Z ; A + = A = n 2 A ; = + m i i ; m i 2 2Z : Weights frm the set A + (i.e. we cnsider weights frm A shifted by ) are all included in tw Weyl chambers nly { the unin f the dminant Weyl chamber and its image under the reectin with respect t n. All that can be nicely illustrated in the case f C 2 : At the next picture, we can see the crrespnding tw Weyl chambers in the Cartan-Stiefel diagram f C 2 belw. Elements f A are shifted by ; elements f A + are dented by dts and elements f A by squares. 2! 2 1 =! 1 5. Tensr prducts with the defining representatin In this sectin, we are ging t study tensr prducts f any bunded mdule with the dening representatin L(! 1 ). We shw that these prducts are cmpletely reducible and that there are n multiplicities in the decmpsitin. By inductin, we get cmplete reducibility als fr a prduct with pwers f L(! 1 ). These are exactly facts needed fr future applicatins in a study f invariant dierential peratrs n prjective cntact maniflds (see [3]). Therem 4. Let 2 A and let (! 1 ) = f i ; i = 1; : : : ; ng dente the set f all weights f the dening representatin L(! 1 ).
ON A DISTINGUISHED CLASS OF INFINITE DIMENSIONAL REPRESENTATIONS OF sp(2n; C ) 7 Then L() L(! 1 ) is cmpletely reducible and L() L(! 1 ) = M 2A L() where A f = + ; 2 A ; 2 (! 1 )g. Prf. Suppse that L() is a direct summand in the decmpsitin f sme L( 1 2! n) L() fr sme integral dminant (the ther case can be treated in a same way). Thus L() L(! 1 ) (L( 1! 2 n) L()) L(! 1 ) = L( 1! 2 n) (L() L(! 1 )) and therefre we knw that direct summands in L() L(! 1 ) are in A. We shall prve that the characters + and + are distinct fr any ; 2 (! 1 ). This is equivalent t the fact that + and + are nt cnjugated. This can be seen as fllws: Let + 2 W 1 and + 2 W 2 where W 1, W 2 are tw Weyl neighbur chambers. This chambers are described in the f i g n basis as fllws W 1 = W 2 = n n X n i i ; 1 > : : : > n > 0 i i ; 1 > : : : > n 1 > n > 0 : This tw Weyl chambers are mapped t each ther by the reectin in the plane rthgnal t n. Frm the structure f the set (! 1 ) it is evident that 2 A implies that + P 2 A P n fr each 2 (! 1 ): Thus the dierence (+) (+) = m i i, n where m i 2 2Z. Tw elements + 2 W 1 and + 2 W 2 are cnjugated (by the reectin in the plane rthgnal t n ) when their dierence is (+) (+) = (2k + 1) n { a cntradictin. At the picture f the Cartan-Stiefel diagram fr C 2, we can see that if + and + are cnjugated by the reectin in the plane rthgnal t 2 then ne f them is represented by a dt and the secnd ne by a square. But + and + fr ; 2 (! 1 ) ere bth represented either by dts r by squares. Nw, we use the therem n decmpsitin f direct prduct f a nite and innite representatin t cnclude that = + ; fr sme 2 (! 1 ). Cnsider the representatin k L(! 1 ) Vfr sme V 2 A. We knw that L(! 1 ) V is cmpletely reducible and its direct summands are in A. Let us label these summands by integers, denting its chsen psitin in the direct sum. Tensring L(! 1 ) V = n 1 b 1 =1V b1 by L(! 1 ) we btain an direct sum again since each V b1, b 1 = 1; : : : ; n 1 is in A and therefre decmpses when tensred by L(! 1 ) due t the previus Therem 4. We dente the b 2 therm f the direct sum V b1 by V (b1 ;b 2 ). Cntinuing in this prcess (r by inductin) we btain L(! 1 ) k V = b1 ;::: ;b kv (b1 ;::: ;b k ). Thus we have prved Crllary. The representatin f L(! 1 ) V k is cmpletely reducible and decmpses as L(! 1 ) k V = b1 ;::: ;b kv (b1 ;::: ;b k ) : ;
8 S. KR YSL Example. In this example we shall dente a mdule L() with highest weight simply by () written in the basis f fundamental weights and we shall describe the set A fr = (10 : : : 01 3 2 ). We knw that We als knw that (0 : : : 01 3 2 ) (10 : : : 0) = (0 : : : 0 1 2 ) (10 : : : 01 3 2 ) : (10 : : : 0) (10 : : : 0) = (20 : : : 0) (010 : : : 0) (0) : We can decmpse the fllwing tensr prducts using the prescriptin f the abve therem t btain that (0 : : : 01 3 2 ) (20 : : : 0) = (20 : : : 01 3 2 ) (10 : : : 01 3 2 ) (0 : : : 01 3 2 ) (0 : : : 01 3 2 ) (010 : : : 0) = (010 : : : 01 3 2 ) (10 : : : 0 1 2 ) We als knw that (0 : : : 01 3 2 ) (0) = (0 : : : 01 3 2 ) (0 : : : 0 1 2 ) (10 : : : 0) = (0 : : : 01 3 2 ) (1 : : : 0 1 2 ) : Frm this we can deduce that: (10 : : : 01 3 2 ) (10 : : : 0) = (20 : : : 01 3 2 ) (0 : : : 01 3 2 ) (010 : : : 01 3 2 ) (10 : : : 0 1 2 ) : Hence we can see here that in this case, the set A is given by A = f = + ; 2 A ; 2 (! 1 )g : Many ther examples lead t the same result, hence we cnjecture that the same fact will be true in general. References [1] Britten, D. J., Hper, J., Lemire F. W., Simple Cn-mdules with multiplicities 1 and applicatin, Canad. J. Phys. 72 (1994), 326{335. [2] Britten, D. J., Lemire, F. W., On mdules f bunded multiplicities fr the symplectic algebra, Trans. Amer. Math. Sc. 351 N. 8 (1999), 3413{3431. [3] Cap, A., Slvak, J., Sucek, V., Bernstein-Gelfand-Gelfand Sequences, Ann. f Math. 154, n. 1, (2001), 97{113, extended versin available as ESI Preprint 722. [4] Habermann, K., Symplectic Dirac Operatrs n Kahler Maniflds, Math. Nachr. 211 (2000), 37{62. [5] Dixmier, J., Algebras envelppantes, Gauthier-Villars, Paris, (1974).
ON A DISTINGUISHED CLASS OF INFINITE DIMENSIONAL REPRESENTATIONS OF sp(2n; C ) 9 [6] Kstant, B., Symplectic Spinrs, Sympsia Mathematica, vl. XIV (1974), 139{152. [7] Kstant, B., On the tensr prduct f a nite and an innite dimensinal representatins, J. Funct. Anal. 20 (1975), 257{285. Mathematical Institute, Charles University Sklvska 83, 186 00 Praha, Czech Republic E-mail: krysl@karlin.mff.cuni.cz