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Home , Spin representation

Proc. NatL Acad. Sci. USA Vol. 78, No. 6, pp. 3308-3312, June 1981

Spin and. wedge representations of infinite-dimensional Lie. algebras. and groups (/spin representation/affine Kae-Moody /highest weight representation/line bundle) VICTOR G. KAC* AND DALE H. PETERSONt *Massachusetts Institute ofTechnology, Cambridge, Massachusetts 02139; and tUniversity of Michigan, Ann Arbor, Michigan 48109 Communicated by Bertram Kostant, November 24, 1980 ABSTRACT We suggest a purely algebraic construction ofthe o(V, U;4)) on the space s(V, U) by spin representation of. an infinite-dimensional orthogonal Lie al- gebra (sections 1 and 2) and a corresponding group (section 4). ovvu(a)[x + (CfV)U] = *na)x + xd + (COV)U. From this we deduce a construction ofall level-one highest-weight representations of orthogonal affmne Lie algebras in terms ofcre- Note that if a E Ofin(V) = CC2V, then *(a)x = ax - xa, and ation and annihilation operators on an infinite-dimensional Grass- we may take d = a, so that we obtain the usual definition (ref. mann algebra (section 3). We also give a similar construction of 2): ovvu(a)[x + (CtV)U] = ax + (CfV)U. Hence, on Ofin(V), OV U the level-one representations of the general linear affine Lie al- is the usual spin representation. gebra in an infinite-dimensional "wedge space." Along these lines Let a be the involutive automorphism ofCIV defined by a(x) we construct the corresponding representations of the universal - -x,x E V and CCV = Ce+v ff Cf V be the corresponding central extension of the group SL.(ktt,t-1]) in spaces of sections eigenspace decomposition. This induces the decomposition ofline bundles over infinite-dimensional homogeneous spaces (sec- s(V,U) = s+(V, U) s-(V,U) into adirect sum ofirreducible half- tion 5). spin representations, Ocvwu and -v As soon as the choice a a-+ d is made, we obtain a k-valued 1. Let V be a over a k with 2 # 0 (we do not two-cocycle y of the Lie algebra o(V, U;O0): assume that dim V < 00) and 4 be a nondegenerate k-valued symmetric on V. Define Lie algebras (with the -Xa,b)Is(vmO = [oVu(a), ovu(b)] - avu([a,b]). usual bracket): To make this choice, suppose that U' is a subspace of U with o(V;4) = {a E Endk V 4(ax,y) + O(x,ay) = 0, x,y E V}, dim U/U' < 0 and that U" is a subspace ofV such that we have (nonorthogonal) direct sum decompositions V = U' (D U", V Ofin(V;4)) = {a E o(V;c) dim a(V) U", p"': V-k U"' be the associated projections. Then for a E o(V,U;O), Suppose U is an isotropic subspace of V satisfying we choose U"L = U; dim U'/U s 1. [1] a=pap [3] Introduce a Lie subalgebra of o(V;O): Then we have o(V,U;O) = {a E o(V;O) dim [U + a(U)]/U < x0}. = - trace p'(ap"b - bp"a)p". Y(a,b) 2 [4] We shall construct the projective spin representation a u of the Lie algebra o(V, U;O). Now suppose in addition that U"' is isotropic and that there Recall (ref. 2) that the Clifford algebra CeV associated to exists a (indexed) subset {u,: i E I} C U' and a basis {ui: i E I} (V, 4)) is an associative k-algebra with unit 1, defined as the quo- of U"' such that O(ui,,u) = SY. Let po be a projection of V along tient of the algebra over V by the two-sided ideal gen- U' + U"'E onto the finite-dimensional subspace U" n U" of V. erated by elements of the form x 0& y + y 09 x - O(xy)1, xy Then for any a E o(V, U;O)), there exists a set {aj:j E J} C CfV E V. We identify V with a subspace of Cfv. Ifa FE o(V;O), the such that for any x E CfV, there exists a finite setJ(x) C J such action of a on V extends uniquely to a derivation *(a) of CfV, that ajx E (CfV)U forj V J(x), and o(a)[x + (CfV)U] = IjeJ(x) and ir is a representation of o(V;)) on Cfv. ajx + (CfV)U. In this case, we write ota) = Yjj aj. Then for Let Cf2V be the linear span in CeV of elements of the form a E o(V,U;4)), we have [x,y]: = xy - yx for x,y CE V. Then Cf2V is a Lie subalgebra of CeV. If a E Cf2V, x E V, then [a,x] E V, and x '-+ [a,x] lies in o(v u(a) = (apo + poa) + E (au')ui - u'(au,). [5] ofi(V;4)), allowing us to identify Cf2V and ofi.(V). For a E Cf2V, 2 2 ~~~~ir~ x E cV, one has *(a)x = [a,x]. Let (CfV)U be the left ideal of CfV generated by U, and set s(V, U) = CeV/(CfV)U. For a 2. Let k[t,t-'] be the ring offinite Laurent series over k. For- E o(V,U;O), there exists EaCf2V such that P = YjEZcjti E k[t,t-1], set Res P = c 1. Let V be a finite di- mensional vector space over k, and let V = k[tt-] %kV = *(a-&)U C U. [2] ESez(t'0 V) be the associated loop space, regarded as a vector Furthermore, a is defined modulo (CfV)U + k by Eq. 2. This space over k. allows us to define the projective spin representation ovu of Call A E Endk V homogeneous of degree m ifA(t' 09 V) C ts+m 0 V for s E Z. In this case, we assign to A the sequence The publication costs ofthis article were defrayed in part by page charge A(') E Endk V, i E Z. defined by A(t' 09 x) = ti+m 0 Al" (x), x payment. This article must therefore be hereby marked "advertise- E V. Let gl,(V) be the Lie subalgebra ofEndk V spanned by the ment" in accordance with 18 U. S. C. §1734 solely to indicate this fact. homogeneous A E Endk V. 3308 Downloaded by guest on September 27, 2021 Mathematics: Kac and Peterson Proc. Natl. Acad. Sci. USA 78 (1981) 3309

Fix r E Z, and set r' = [r/2]. Let 4 be a nondegenerate k- i }, called the set ofdual simple roots. Define reflections r, valued on V. 'Define a nondegenerate 8 GL()*), 0 . i < e, by ri(A) = A - A(h2)a, and let W C k-valued symmetric bilinear form 4)r on V by GL(b*) be the group generated by the ri. We regard W as the = tr of W generated by rl...re. Set nf+ = n+ ED (tk[t] ),r(P 0 x, Q 0 y) Res PQ O spin representation ar ofo.(V;r) on s(V,Ur) to be the restriction Let V be a vector space over k with basis ul,,.. IUn_ n 3, of oVOr to o*(V;r). and let 4 be the symmetric bilinear form on V defined by Set U V__ U' = V' r and define dby Eq. 3 for a E 0(uiun-J+1) = 8,,. Then g = o(V;4)) is identified with the Lie o(V,Ur;4r). Then the cocycle y is given on o*(V;r) by Eq. 4 as algebra of n X n matrices skew symmetric with respect to the follows. If A and B are homogeneous of degrees m and s, re- side diagonal. Let b be the set of diagonal matrices in g and spectively, .then y(A,B) = 0 unless m + s = 0, and ifm 2 0, n+ be the set of strictly upper triangular matrices in g. Set e rm-1 = [n/2], and let U be the ne-stable maximal isotropic subspace , 0 or s = 0, and i 2 n - + 1},J+ = {(n - i + Is) (i,-s) EEJ-},J = J UJ+. ar(A) =- E > As (t--r- u') (ts 0 U) r+I s2-r' lj=i For (i,s) E8, set 4i8 = t 2 09 ui if n is even; =s n r+ _ r+1 -28;r-2r,1 > A(.g'r-1) (t-r'-l ui) (t-r'-l 0 Uj) [8] st2 0)uj) (t 2 09ue+i) ifn is odd. Then {f4sofi 's'} = 3i+i',n+l 2r2r, :J 8, '5#1 where {a,b} = ab + ba. Let Xr, n be the (Grassmann) subalgebra of CfVe generated by the anticommut- ifm = 0. ing elements fi,s (i, -s) E _. Set Xr,, = Xn n Ce V. For 3a. We assume in section 3 that k is algebraically closed of (i, -s) 8_,E define the creation operator ui+-, and the annihi- characteristic 0. lation operator - by: Ci4 S(x) = 4i- xI 4s(1) = 0, Given a Lie algebra g over k, we define the associated loop Cis(fri -s=x)=ii ss x - -s_ ,Cs(x) for x E Xrn, (i X-s') algebra g = k[t,t-'] ®&kg with the obvious bracket, regarded E1J_ as If nr is even (respectively nr is odd), then CfV = (CeV)Ur a Lie algebra over k. Given a finite-dimensional represen- $ (respectively Cf+V = tation v: g EndkV, we define in the obvious way a repre- Xrn (C-fV)Ur E Xr,n). Therefore, we sentation ir: -g EndfV. 'In particular, the orthogonal loop al- may identify Xrvn with s (respectively X,, = X+, with gebra = (tS o(V;4)) acts on V, and because S$0gr). Then for (i,s) E J, the action of left multiplication by fi, 6(V;4)) ESEZ 0 6(V;O) on (respectively sa is given C o.(V,r), we may restrict o-r to 6(V;4)). We obtain a projective sovor or) on Xr,n by the operator 4i(s), representation that becomes a linear representation of the cen- where for (i, -s) 8EJ-, fi(-s) = ri48-, fn-i(s)=I tral extension o(V;4)) of 6(V;O) defined below. The following theorems describe the action of 6(V;) on the Let g be a finite-dimensional simple Lie algebra over k, f half-spin modules Xvr,. a ofg, A the set ofroots ofg in b, Wthe Weyl THEOREM 1. Let n and r be integers with nr even and n 2 group of A, A, a set. of positive roots, n+ the corresponding 3. Then (i) thefollowingformulas define a linear representation maximal nilpotent subalgebra of g, II = {a1,...,ae} the set of &r of 6(V;4)) on X,,n, which is a linearization of the projective simple roots, 0 the highest root, and (, ) a nondegenerate in- spin representation ar: variant symmetric bilinear form on g (and, hence, on g* and rr(c) = I; form # 0, ,*). For aE with (a, a) # 0, define fE = b* H,,8 by f3(H,,) 2(,B, a)/ I r r (a,a). for 18 Ij*. The affine Lie algebrag associated to g is an Ar(tM0 a) +aj4 4~ii~~ +2 J extensiong = g kc ofthe loop algebrag by a one-dimensional 2 sEZ i 2 2 center kc, with bracket Ur(1 a) = aij4(-s- 2 >n-j+l(S + 2 ) [A,B]d = [A,B]a + I(6,0)Res (dAB)c, for A,B EE . [9] +2 r,r2'n+1 aij(°)n-j+1)- Let h = G) kc, with basis {ho: = c - He; hi: = Hai., 1 ' i,j Downloaded by guest on September 27, 2021 3310 Mathematics: Kac and Peterson Proc. Natl. Acad. Sci. USA 78 (1981) (ii) For m E Z, define operators Dm on X,,, by Parts a follow immediately from Eqs. 7, 8, and 9. 1 r_ To prove part b, consider the Lie algebra A: = k[tt-,d/ D -l (2s + m + r + 1) -s-2) dt] (®kgl(V) of differential operators on V. Let Ar = sfn Z o(V6;+) be the subalgebra of 4r-skew self-adjoint elements of A, spanned by elements of the form x &-j+1s +m++ r ) form #O. 2~~~~~~~~~ dm,e,a = tt+m(( ) + t-(r++m)() tr+e+m) a, Do=-2 E E(2s +r +1) 2s--r' i where 4(ax,y) + (- 1)e 4(x,ay) = 0 for x,y E V. In particular, d,, = dmi11 = tm+l d/dt + 1/2(r + m + 1)tm lie in Ar' and Dm x ei -S- 2~ 2 ) 2)n-i+1~ ~~2 1 = qr,(din). Define a two-cocycle yo on sA. by: Then SYOV~tdt!te+md) 0$ a,' te +m(t)9dt! a') [DmDm,] = (m' - m) Dm+m? + 2 6,,-,' m(m2 24 = 88mm(tracevaa')(-1)eet!(e'! m + -1 + 36r,2r'+1)I; Then if r = 0, the cocycle y defined by Eq. 7 is the restriction [Dm,) r(A)] = 6r(dm(A)) for A E o(V;4). of Vo to AO, and is given by (iii) As 6(V;0)-modules, Xr L(Ao), X7,n L(A )for r even, 1 and X:,n L(A5), Xr L(A-1)for r odd. YO(dm,t,a, dm',V,a') = 8m,-m' (tracev aa') In the statement of Theorem 2, X' denotes a sum to be ex- tended only over summands not involving &+J(0), which is X [(e + el)! + M-~ee!(+ e) undefined. THEOREM 2. Let n and r be odd integers with n . 3. Then This and Eq. 8 suffice to verify part b for r = 0; the general case (a) Thefollowingformulas define a linear representation a+ of is similar. 6(V;4)) on Xr n, which is a linearization of the proyective half- Finally, part c is proved by using the so-called Weyl-Kac spin representation r+: character formula for L(A). One checks the equality ofthe char- acters by applying the "principal specialization" sD to both sides r (c) = I; for m # 0, and using formula 3.29 from ref. 3. 3c. For m, a semisimple Lie algebra, we define the affine 6,r (tm 08a) 2 i,eiji s 2 Lie algebra m' to be the direct sum of the affine Lie algebras associated to the simple summands of m. Then the notions of section 3a generalize in an obvious way to m and 'm. Given a x en-j+l S + m + 2) + aaie+l e(m); homomorphism r. m -* o(V;), we obtain an induced homo- morphism rfm 6(V;4). Then, using the composition ° 1 s-rE ( ~23 72 2i and some choice of maximal isotropic subspace U of V, we (4(1®a)= - ~~r;1>+ r +1) regard Xv: = X_ ,dim v as an m-module, called the spin module of m associated to T. ' + 1 Let g be a simple Lie algebra, g = f D p a Cartan decom- +2 aj(i(O)en-j+1(0) a jes(°) position of g such that f is semisimple and contains a Cartan subalgebra b of g. Then, because f preserves the Killing form (b) For m E Z, define an operator Dm on Xr n by: on)p, we have an inclusion f C o(p). Let A (respectively Af) be the sets of roots of g (respectively f) in A,l a set of positive 1 r_ __ D =-- (2s + m + r + 1)ei s 2V roots, Af+ = A+ nAf, A (respectively A) the sum of the fun- E E A damental weights of (respectively ?), W the Weyl group of g in b, W1 = {w E W: w(A+) D A+}. Choose the maximal isotropic X en-i+1S + M+ 2 ) + f + 1(m) subspace U = p+: = p n n+ ofP. PROPOSITION 1. The spin module Xp of? associated to the inclusion f C has the decomposition form 0, o(P) XP= L(w(fi) - . Do -2 (2s + r + 1) 2s--r' i The highest weight vectors are pure lying in CeP. The proof is essentially the same as that ofthe finite-dimen- sional analogue in ref. 4. x{iks2+2ni+ (s 2 / We shall describe in detail the most: beautiful case Then: g = sp(V E V') D sp(V) @ sp(V') = f, [DmDm'] = (' -m)Dm+m + 24 -Ml_ m(m2 + 2)1; where in V ® V' as a f-module. Let dim V = 2n, dim V' = 2m. Recall that a composition of n into m + 1 parts is an (m [Dm,64r (A)] = (6+(dm(A)) for A E 0(V;45). + 1)-tuple (ko,...,nk) of nonnegative integers with ko + ... + km = n; denote by Pnm the set of such compositions. Then, (c) As th(V;)-module SvfoL(As)., - there is a natural bijection between Pnm and the set of all m- These theorems are proved as follows. element subsets of {1,2,.. .,m+n}, so that taking the comple- Downloaded by guest on September 27, 2021 Mathematics: Kac and Peterson Proc. Natl. Acad. Sci. USA 78 (1981) 3311

mentary subset induces a bijection 7r '- ir from Pnm to Pmn. A,B E , the map UIA ... AuI-* U1 ... un + (CkV)B from We label the fundamental weights of s~p(U), dim U = 2(, ac- Amax(A/A n B) to (AIB) is an . Let Ai E8 V, i E cording to the diagram 8 : o ... %<= o For Z/3Z. Then by considering the alternating two-form 4 on A1 XT = (ko,3. km) E Pnm, so that ir = (k,. k) E Pm n, set fA2 (3) A3 given by f(xl f x2 fx3, Yi ED Y2 @ Y3) = Y*XiYi, ) one obtains a canonical element of Homk ((AlA2) = + + kmAm,A'(*) = + ... + -4(yi,xi+j), A(XT) k0A0 koA' knA 09 (A2IA3),(A1IA3)), which is the product # 09 Y + #y. (b) PROPOSITION 2. The spin module Xvv,®~ associated to sp(V) 6(vV,U; is a double cover of a corresponding subgroup of eD sp(V') C o(V 0 V') has thefollowing s'p(V) E s'p(V')-module GL(V), which is constructed in section 5c. (c) Results similar to decomposition: those of section 5 hold for the spin representations ofo(V, U;O) and O5(V,U;O,). (d) One can write down the representation Xv,u ~i3 L(A(ir)) 09 L(A'(fT)). and the corresponding cocycle in terms of the Fermi integral Xvosv, VITPnm (cf. ref. 1). (e) One can show that Iv u 09 Ivu is isomorphic to We note that as a special case of Proposition 2 we obtain the Av u (see section 5a) restricted to O(V, U;O;), provided that U' decomposition of the s72 E sAp2-module: - U. 4b. Recall the notation of section 2. Let G be an algebraic n group overk. The group G: = G(k[t,t-']) is called the associated (D3 L(sA0 + (n - s)AI) 0 L(A). X-1,4n s0= loop group. Given a representation v: G Endk V, one as- sociates the representation k: G -- Endk V. In particular we Remarks. (a) One gets similar decompositions for the spin have the general linear and the orthogonal loop groups m(V) module associated to o(V) + o(V') C o(V 0 V'); (b) Proposition and O(V). Fix r E Z. Then 6(V) 8 O(V, so that we may 1 can be generalized to the case of f and g of equal rank, for Ur;or), restrict 'V Or to O(V), obtaining a projective representation instance Xr of 0(V) with differential 0r. g = sl(V @ V') D sl(V e V') n (gl(V) @ gl(V')) = f. From Theorems 1 and 2c we deduce: PROPOSmON 3. The projective representations or of the (c) The spin representation associated to the adjoint represen- group SOA(kJ[t,']) in s-(V,Ur) are irreducible, provided that tation ofa simple Lie algebra g is decomposed into a direct sum char k = 0 and n 2 3. ofseveral copies ofthe '-module L(j)). This holds for an arbitrary 5a. Let V be a vector space over a field k (we do not assume Kac-Moody Lie algebra and follows from the obvious product that dim V < 00). Let A(V) = eW Ak(V) be the exterior algebra decomposition for the character of L(p). over V. Ifdim V = n < 00, set Amin(V) = An(V). For subspaces 4a. Let the assumptions on k,V,f,U be as in section 1. Set A C B ofV with dim A <00, we have a canonical inclusion A(B/ O(V;O) = {g E GL(V)10(gx,gy) = O(x,y) for x,y EE V}. For g A) 09 Am'(A) C A(B); ifalso dim B <00, this gives a canonical 8 O(V; ,), extend the action of g on V to an automorphism f3 isomorphism Am'(B/A) 09 Amax(A) Amax(B). ggB- ofCeV. Set Two subspaces A,B of V are called commensurable if O(V,U; ,) = {g 8 O(V; ,) dim (U +

For a E (U'IV), g E GL(V,U;V), Avu (4uu (A)) ° a = a. k and [p,q] = (Res dp/dt q)c' for p,q Eik. Writing gl(V) = sl(V) ° Avu(A). Therefore, the projective wedge representation of f kI, we set: gl(V) = sA(V) f k. GL(V;TN) on A(V, U;W) depends only on V and WV. Let b be a Cartan subalgebra of sl(V), and set D= kc Define a subgroup ofGL(V;.U) (which is independent of the ff kI ED kc' C gA(V). Define A' A'l Ei 6* by A'(b E$ kc) = 0 choice of U E V) by =A'( ED kc), A'(I) =0=O A(c'), A0(c') = 1 = A'(I). Then we have a theory of irreducible highest weight modules L(A) of GL0(V;W?) = {g E GL(V;lf) dim (U + g(U))/U g?(V) (cf. section 3a). Let U = k[t] ®k V C V, and let be the set of subspaces = dim (U + gM(U))/U}. -k A of V such that for some k E Z, rkU D A D tkU. Then it is Then the corresponding subgroup GLO(V,U;) C GL(V, U;9V) easy to see that gl(V) C gl(V;To). consists of the A such that Avu(g) is degree-preserving on THEOREM 3. For 0 ' i ' n-1, the gl(V)-modules A(i)(V,U;VO) A(V,U;V). and L(Ai + nA' + iA'") are isomorphic. Set P = {U' CE lI dim U/(U n U') = dim U'/(U n u')}, 5d. Now we again turn to the corresponding group. Set U0 and define a line bundle 2 on P by 2 = {(a',U')I U' E P, a' k[t-'] 0k V, and let TP' be the set of subspaces A of V such E (UIU')}. Then GLO(V;Tl) acts transitively on P, and G4,(V, U;) that for some k E Z. t UO D A D t1UO. Then SL(V) C acts on 2 by (a,g) * (a',U') = (a(g * a'), g(U')). Let Q* - GLO(V;TO), so that we obtain a central extension SL(V) C {(a',U')I U' E P, a' E (U'IU)} be the dual line bundle. GLo(V,Uo;Wo) of SLCV). Then forj > 0, there is a unique in- ForA,B E- P, A C B, setPAB ={U' E PIA C U' C B}. Then clusion H.: = {g E SL(V)I g I mod tik[t]} C S'L(V) such that we regard PA,B as a Grassmanian, thus as a smooth algebraic A,u0(a) is locally unipotent for a E H,j >0; a subgroup ofSL(V) variety. Then the restrictions of Q,Q* to PAB are algebraic va- is called open if it contains Hj for some j > 0. rieties. We call a section of Q* over P regular if its restriction FixiEZ,0 is n-1. Set to each PAB is a regular map, and denote by Ho(P,2*) the space n of regular sections of Q* over P. Then the obvious map 2 U, = (t-'k[t-'] ®k V) ( (+ kus A(o)(V,U;lJ) induces a A(o)(V,U;3?)* Ho(P,2-*). s=i+l It is easy to see that this map is an isomorphism ofGLO(V, U;W?)- modules. Let Pi be the set of subspaces A of V such that A D t-A, A and 5b. Let dA denote the natural representation of the Lie al- Ui are commensurable, and dim A/(A n U) = dim Ui/(A n gebragl(V) in A(V). Define the Lie algebragl(V;D?) = {a E gl(V)I U,). Define a filtration PiO) C P(il) C ... of Pi by for any A E SD? there exists B CE V such that A D B + a(B)}. P(s) = {A E Pil ts-lk[t-l] 0 V D A D t-s+lk[trl] 09 V}. Let a E gl(V;J), A CE 9P. Choose B E 9D, with A D B + a(B), and a decomposition V = B e B'. Consider the sequence of We regard the Pjs) as projective varieties. maps: A(V/A) 0 (U|A) C A(V/B) 0 (UIB) = A(B') 0 (U|B) Consider the line bundle 2S = {(a,A)I A 8 P., a E (AJUi)} dA(a) $ 1 A(V) 09 (U|B) -* A(V/B) 09 (UIB). Their composition over Pi. For s E8 Z+, we regard Q* restricted toIis as an algebraic defines a map a': A(V/A) 0 (UIA) -- A(V, U;T?). Up to addition variety, and call a section of9_' regular if its restriction to each of scalars times the inclusion, aA is independent of the choices P(is) is a regular map. Then the group SL(V) acts on the space of B,B'. From this we may define a degree-preserving projec- H0(Pi,2Q) of regular sections of QP. tive representation a -+ a* = limACVa* of gl(V;T?) on THEOREM 4. The SL(V)-module Ho (Pi,2Q*)O of regular sec- A(V,U;T). tions with open stabilizer of the line bundle 2' is irreducible. To make a definite choice of a*, choose a direct sum decom- As sl(V)-modules, we have H0(Pi,, *)o L(Ai). position V = U ) U', with associated projections p: V - U, p' : V- U'. Then we require a*(A0(V/U)0 (UIU)) C U'A(V, U;TN), Note Added in Proof. Recently, Frenkel (5) independently obtained, where the right-hand side is defined by using the action ofA(V) by a different method, some of the results of section 3b of the present on A(V, U;V). We define the projective wedge representation paper. He referred to a paper by Bardakei and Halpern (6) in which they dAvu ofgl(V;9J) on A(V, U;P?) by dAv,u(a) = a*. One can show actually construct the restriction of the spin representation of o(6) to that its two-cocycle is given by (cf. Eq. 4): y(a,b) = trace p(ap'b gl(3). - bp'a)p. We owe to G. Lusztig the idea of the construction of the group The representations Avu and dAv u satisfy the following re- SLn(k[tt-']), which inspired ourwork on the wedge representation. We lations. Let a E gl(V;TI), g E GL(V;V?), g = (a,g) E would like to thank him for discussions on this matter. The research has GL(V, U;W?), and set P = g .Pg Then been partially supported by National Science Foundation Grants MCS- 8005813 and MCS-800286. Avu(A)dAvu(a)Avu(4') 1. Berezin, F. A. (1966) The Method of Second Quantization (Aca- = dAv u(gag-1) + [trace (p - p)(ap + pa)]I. demic, New York). 2. Chevalley, C. C. (1954) The Algebraic Theory of Spinors (Colum- = bia Univ. Press, New York). 5c. Let k be a field ofcharacteristic 0, V kul ED ... EDku$ , 3. Kac, V. G. (1978) Adv. Math 30, 85-136. a vector space over k. Recall the notation of section 2. 4. Parthasarathy, R. (1972) Ann. Math. 96, 1-30. Regard k as a commutative Lie algebra, so that k = k[tt-]. 5. Frenkel, I. B. (1980) Proc. Natl. Acad. Sci. USA 77, 6303-6306. Define the affine Lie algebra k = k 1 kc', where c' centralizes 6. Bardakci, K. & Halpern, M. B. (1971) Phys. Rev. D 3, 2493-2506. Downloaded by guest on September 27, 2021