JOURNAL OF ALGEBRA 202, 512᎐540Ž. 1998 ARTICLE NO. JA977279
Dual Pairs in PinŽ. p, q and Howe Correspondences for the Spin Representation
M. J. Slupinski*
Departement´´´ de Mathematique, Uni¨ersite de Louis Pasteur et CNRS() URA 01 , 7, rue View metadata, citation and similarRene papers´ Descartes, at core.ac.uk 67084 Strasbourg Cedex, France brought to you by CORE Communicated by Peter Littelmann provided by Elsevier - Publisher Connector
Received January 16, 1997
In this paper we obtain examples of dual pairs in the group PinŽ. V m W by considering the inverse images of the subgroups OVŽ.mIdWVand Id m OW Ž . under the double covering map : PinŽ.Ž. V m W ª OVmW. The main technical results are the definition of the group of determinant graded double covers of y1 OVŽ.and the fact that the map W ¬ ŽŽ.OV mIdW .defines a homomorphism 2 from the Witt group to this group when the ground field k satisfies k*rŽ.k* ( ޚ2 and y1 is not a square in k. We also show that the spin representation of PinŽ. V m W sets up Howe correspondences for each of these dual pairs when k s ޒ. ᮊ 1998 Academic Press
INTRODUCTION
If Ž.X, is a real, symplectic vector space and Sp is its group of symplectic automorphisms, a pair of subgroups Ž.A, B of Sp is said to be a dual pair in Sp if the commutant of A in Sp is B and vice versa. This notion was introduced by R. HoweŽ seewx H1, H2 and especially wx H3. who classified all irreducible, reductive dual pairs in Sp. The inverse images & A˜˜,Bof A and B in Sp, the non-trivial double cover of Sp, also form a dual pair, and understanding the decomposition of the WeilŽ. or oscillator & representation of Sp into irreducible components under the action of A˜˜=Bprovides a unified way of understanding many equations in mathe- matical physicsŽ cf.wx H3. and also a number of topics in pure mathematics Žcf.wx H2. .
* E-mail: [email protected].
512
0021-8693r98 $25.00 Copyright ᮊ 1998 by Academic Press All rights of reproduction in any form reserved. DUAL PAIRS IN PinŽ. p, q 513
Many features of the theory of symplectic vector spaces have analogues in the theory of non-degenerate inner product spaces. For example, the Clifford algebra and spin representation associated to a non-degenerate inner product space can be thought of as analogues of the Weyl algebra & and the Weil representation. The analogue of the double cover Sp of Sp is the non-trivial double covering group Pin of the orthogonal group O and in this article we will study dual pairs in PinŽ. V m W obtained, more or less, as inverse images under : PinŽ.Ž. V m W ª OVmW of the dual pair ŽOV Ž .mIdWV, Id m OW Ž ..in OV ŽmW . Žsee alsowx A. . Here V and Ware non-degenerate inner product spaces over aŽ. not arbitrary field k. y1 The first main result is the identification of the group ŽŽ.OV mIdW .. If W is the one-dimensional inner product space Žk, x 2 ., then y1 ŽŽ.OV mIdW .Ž.is isomorphic to Pin V but in general this is not true. We show y1 Ž.cf. Theorem 2.9 that the map W ¬ ŽŽ.OV mIdW .defines a group det homomorphism GrŽ. V ؒ: Witt Ž. k ª GrCo¨ 2Ž.V from the Witt group of k to the group of ‘‘determinant graded double covers’’ of OVŽ.Žsee Section 1 for the definition.Ž. . The simple structure of Witt k and an easy induction y1 then allow us to identify ŽŽ.OV mIdW .Žcf. Corollary 2.13. . y1 In contrast with the symplectic case, the groups ŽŽ.OV mIdW .and y1 ŽŽ..IdV m OW are not necessarily mutual commutants in PinŽ. V m W ᎏone may have to go to subgroups to get examples of dual pairs. In Section 3, we give the complete list of dual pairs in PinŽ. V m W obtained in this wayŽ. cf. Corollary 3.5 . In Section 4 we show that if k s ޒ, the spin representation of PinŽ V m W.Žcf. Definition 4.1 . sets up a Howe correspondence between irreducible representations of these dual pairs just as the Weil representation does for & certain dual pairs in Sp. This result isŽ. in the language of R. Howe a ‘‘Ž. skew duality version’’ of Theorem 4.3.4.2 inwx H4 . Throughout the article, except in Section 4, we work with non-degener- 2 ate inner product spaces over a ground field k satisfyingŽ. i k*r Žk* . ( ޚ2 andŽ. ii y1 is not a square in k Žexamples are k s ޒ and k s ކq , the finite field with q elements, if q s 3Ž.. mod 4 . The reason for thisŽ cf. the Appendix. is that these are exactly the fields to which the standard construction of ‘‘Pin’’ groups over ޒ Žcf.wx ABS. can be generalised. Over such a field, inner product spaces have a ‘‘signature’’ , which defines an isomorphism of the Witt group of k with ޚ when y1 is not a sum of squares in k, or with ޚr4ޚ when it isŽ. cf. the Appendix . The author thanks R. Howe for pointing out that the examples of dual pairs and Howe correspondences ofwx S were probably part of a general family of similar results. This remark was one of the main motivations for writing this article. I also thank R. Stanton for many useful conversations, A. Huckleberry for an invitation to the Ruhr Universitat¨ at Bochum where 514 M. J. SLUPINSKI most of this paper was written, and the referee for his comments.
1. THE GROUP OF ޚ2-GRADED DOUBLE COVERS
In this section, we will define the group of ޚ2-graded double covers of a group G and prove the basic properties of this group needed in the rest of the paper.
Let G be any group and let Cov2Ž.G be the group of equivalence classes of double covers of G. In abstract language, this is the Maclane cohomol- 2 ogy group HGŽ.,ޚ2 but let us give an explicit description. A double cover of G is a group Gˆ together with a surjective group homomorphism : Gˆ ª G which fits into an exact sequence
ˆ 1 ª ޚ2 ª G ª G ª 1.
The group y1Ž.e is necessarily central and by abuse of notation we will always denote the non-trivial element y1. Two double covers G˜˜12and G ˜˜ are equivalent if there is a group isomorphism f: G12ª G such that 12sf. The group structure can be described as follows: the product of the $ $ equivalence class of 11: G ª G with the equivalence class of 22: G ª G is the equivalence class of the group Äwxg12, g g Gˆˆ 1= G 2rÄŽ.1, 1 , Ž.y1, y1:411Ž.g s 2 Ž.g 24with projection wxg1, g 2¬ 11Ž.g . The identity in Cov2Ž.G turns out to be theŽ. class of the direct product n n G=ޚ2 and since the map Ž.y1 wxˆˆg, g ¬ ŽŽ.Ž ˆg , y1 .. defines an iso- morphism Gˆˆ. G ( G = ޚ22, it follows that Cov Ž.G is an abelian group of exponent 2.
1.1. DEFINITION.Aޚ22-gradedŽ we will usually omit the ‘‘ޚ ’’ in what follows.Ž double cover of a group G is a pair Gˆˆ, ., where .: G ª G is a double cover of G and : G ª ޚ2 is a group homomorphism. Two graded double covers Ž.Ž.Gˆˆ11, and G 22, are equivalent if and only if 1s 2 and Gˆˆ12and G are equivalent as double covers. In the category of graded groups there is a natural product and using it we define the product of two graded double covers of G.
1.2. DEFINITION. Let Ž.Ž.Gˆˆ11, and G 22, be two graded double cov- ˆˆ ers of G. The product Ž.Ž.G11, ) G 22, is the graded double cover ŽÄg,g Gˆˆ=G ˆ:Ž.g Ž.g4,., where Gˆˆ= G ˆis the wx12g 1ޚ2211s 2212 1ޚ22 group whose underlying set is Gˆˆ12= G rÄŽ.Ž1, 1 , y1, y1 .4 and whose mul- DUAL PAIRS IN PinŽ. p, q 515 tiplication is
ѨŽ11 Žh ..Ѩ Ž 2 Žg 2 .. wxwxg12,gh 12,hsgh 11,Ž.y1 gh22.
Notation. Here, Ѩ: ޚ2 ª Ä40, 1 is the map ѨŽ.1 s 0 and Ѩ Žy1 .s 1. gg ѨŽŽg.. Ѩ Ž Ž g .. We will often write Ž.y112 for Ž.y1 11 2 2 when it is clear which characters 12, : G ª ޚ 2are involved. For brevity, we will denote the ˆˆ underlying group of the graded double cover Ž.Ž.G11, ) G 22, also by ˆˆ G12)Geven though the definition of the product in this group depends on the choice of characters 12and . ѨŽŽh..Ѩ Ž Ž g .. Remark 1.2.1. The factor Ž.y1 11 2 2 is introduced to make the following proposition true, which ultimately will imply Theorem 2.9, the ѨŽŽh..Ѩ Ž Ž g .. main result of the next section. Note that Ž.y1 11 2 2 s ŽŽ.11h, 22 Ž..gޒޒ, where Ža, b .is the Hilbert symbol of ޒ.
1.3. PROPOSITION. LetŽ.Ž. Gˆˆ11, , G 22, be graded double co¨ers of G1 ˆ and G22, respecti¨ely. Let A, Bbeޚ-graded algebras and let ␣: G 1ª A and ˆ :G2 ªB be homomorphisms which preser¨e degree and such that
␣ Ž.y1 s  Ž.y1 sy1. Then ⌽: Gˆˆ= G ˆA ˆB defined by ⌽Žg , g .Ž.Ž.␣ g ˆ gisa 1 ޚ221212ª m wxs m homomorphism which preser¨es degree. Proof. Abstract nonsense.
1.4. THEOREM.iŽ. The product ) defines the structure of an abelian group on the set of equi¨alence classes of graded double co¨ers of G. This group will be denoted GrCo¨ 2Ž.G . Ž.ii There is an exact sequence of abelian groups,
1 ª Co¨ 22Ž.G ª GrCo¨ Ž.G ª Hom Ž G, ޚ 2 .ª 1 and GrCo¨ 2Ž.G is of exponent 4. 2 Ž.iii The map : HomŽ G, ޚ22 .ª Co¨ Ž.Ggi¨en by Ž.ss Ž. , where s: HomŽ. G, ޚ22ª GrCo¨ Ž.ŽG is a set-theoretic . section of Ž.ii , does not depend on the choice of s and defines a group homomorphism. Proof. To prove that the product is abelian, one checks that the map ˆˆ ˆˆ ѨŽ2Žg2..Ѩ Ž 1Ž g1.. ⌽:G12)GªG 21)Ggiven by ⌽Žwxg12, g .Ž.sy1 Žwxg21,g . defines an equivalence. Associativity is proved in a similar way and the details are left as an exercise. It is easily seen that E s Ž.G = ޚ22, e , where e: G ª ޚ is the trivial homomorphism, satisfies E)Ž.Ž.Gˆˆ, ( G, , and thus we have an iden- tity element in GrCo¨ 2Ž.G . 516 M. J. SLUPINSKI
To construct an inverse of Ž.Gˆˆ, one proceeds as follows. Let Ž.Gy1, be the graded double cover defined by: ᎏthe underlying set of Gˆy1 is Gˆ; 1 Ѩ Ž Ž g ..Ѩ Ž Ž h.. ᎏthe multiplication of Gˆy is g: h syŽ.1 gh; ᎏthe projection of Gˆy1 is the projection of Gˆ. It is a straightforward verification that the product g : h defines a group structure of the set Gˆ. Thus the group of the graded double cover 1 1 Ž.Ž.ŽGˆˆ,)Gy,sGˆˆ)Gy,e .is given by
1 ˆˆy 1 ˆ G)GsÄ4wxg,h:GGˆˆŽ.gsy Ž.hsÄ4"wxg,g:ggG with projection Ž" g, g .Ž. g . Define ⌿: GˆˆGy1 G = Gˆˆ)GGy1wxs ˆ ) ª Ä4"1by
nn ⌿Ž.Ž.y1wxg,gsŽ.GŽ.Ž.g,y1. Then
mn ⌿Ž.Ž.y1wxg,g Ž.y1 wxh,h
mqnmgh qn s⌿ž/Ž.y1 gh, Ž.y1 g : h s ⌿Ž.Ž.y1 wxgh, gh mqnmn sŽ.GGGŽ.Ž.gh , y1 s Ž. Ž.Ž.g , y1 Ž. Ž.Ž.h , y1 and so ⌿ is a group homomorphism which is evidently surjective and injective. We also have
n EG(⌿Ž.Ž.y1wxg,gs Ž.g and n y1 1g,g g Gˆˆ)GGŽ.Ž.ywxs Ž.
1 so that as graded covering spaces, Gˆˆ)Gy( E and the result is proved. ˆ To proveŽ. ii , one defines an inclusion Co¨ 22Ž.G ª GrCo¨ Ž.G by G ¬ ˆˆ Ž.G,eand a surjection GrCo¨ 22Ž.G ª Hom Ž G, ޚ .by ŽG, .¬ . This gives the exact sequence of the statement and since Ž.Ž.Gˆˆ, ) G, s ˆˆ ˆ 4 Ž.Ž.G)G,egCo¨ 2 G is of order 2, it follows that Ž.G, s E. 1.4.1. Remark. The exact sequence Theorem 1.4Ž. ii has a canonical section s: HomŽ. G, ޚ22ª GrCo¨ Ž.G and a canonical retraction r: GrCo¨ 22Ž.G ª Co¨ Ž.G where ˆˆ sŽ.s ŽG=ޚ2, . and rGŽ.Ž., sG, DUAL PAIRS IN PinŽ. p, q 517 but since these maps are not group homomorphisms, the sequence is not in general split.
1.5. PROPOSITION. Let g HomŽ. G, ޚ242and let ޚ ª ޚ be the non- tri¨ial double co¨er of ޚ2 . Then
Ž.s*ޚ4, where *ޚ42g Co¨ Ž.G is the fibre product of G with ޚ4o¨er ޚ 2.
Proof. By Theorem 1.4Ž.Ž.Ž.Ž. iii , (G=ޚ22,)G=ޚ,. The underlying group of Ž.is therefore the set of equivalence classes
Ä4Ž.Ž.g, z , g, w : g g G; z, w g ޚ2
Žwhere wŽ.Ž.g, z , g, w xws Ž.Ž.g, yz , g, yw x.with multiplication
gg1 Ž.Ž.Ž.Ž.g, z , g,wg11,z,g 1,w 1 s Žgg1, zz 1 .Ž.Ž, y1 gg1, ww 1 ..
Define F: Ž.ªG=ޚ4 by
FgŽ.Ž.,z,g,wsž/g,zw' Ž.g , where ޚ24s Ä41, y1 and ޚ s Ä1, y1, i, yi 4are subsets of ރ, where the covering map ޚ42ª ޚ is the square and where' : ޚ24ª ޚ is the map ѨŽa.ѨŽb. ''1s1, y1 s i, and satisfies 'ab syŽ.1 ''ab. Then the map F 2 is a group isomorphism onto the set ÄŽ.g, z g G = ޚ4: Ž.g s z 4, i.e., onto *ޚ4.
Remark 1.5.1. The square root used here can be written 'x s ␥ ޒŽ.x, , y2ix where ␥ ޒŽ.x, is the Weil factor of ޒ and the character Ž.x s e Žsee, for example, the appendix ofwx R. . The formula 'ab s ѨŽa.Ѩ Žb. Ž.y1 ''ababove is then a special case of Theorem A4 inwx R .
EXAMPLE 1.5.2. If G s ޚ22, then Cov Ž.ޚ22( ޚ and the non-trivial element is the covering ޚ42ª ޚ . The elements of GrCo¨ 2Ž.G can then be represented by Ž.Ž.Ž.Ž.ޚ22= ޚ , e , ޚ 22= ޚ , Id , ޚ 4, e , and ޚ 4, Id . By Propo- sition 1.5, Ž.Ž.ޚ44, Id ) ޚ , Id ( Id*ޚ 44( ޚ and thus GrCo¨ 24Ž.G ( ޚ . The fundamental exact sequence Theorem 1.4Ž. ii is then not split and is isomorphic to
1 ª ޚ242ª ޚ ª ޚ ª 1 and the corresponding canonical sectionŽ. cf. Remark 1.4.1 is a square root. 518 M. J. SLUPINSKI
1.6. The Group of -Graded Double Co¨ers.
1.7. DEFINITION. Let be an element of HomŽ. G, ޚ2 .A -graded double cover of G is a ޚ2-graded double cover of the form Ž.Gˆ, or Ž.Gˆ,e. The group of all such is denoted GrCo¨ 2Ž.G . If is non-trivial, from Theorem 1.4Ž. ii we have an exact sequence
2 1 ª Co¨ 22Ž.G ª GrCo¨ Ž.G ª Ä4 , ( ޚ 2ª 1.
1.8. PROPOSITION.iŽ. The group GrCo¨ 2Ž.G is isomorphic to the group whose underlying set is Co¨ 22Ž.G = ޚ and whose multiplication is
ѨŽz.ѨŽw. Ž.Ž.Gˆˆ,zH,wsŽ.GHd ˆˆ , zw ,
where d s *ޚ4. Note thatŽ.Ž.Ž. Gˆˆ, y1 G, y1 s d,1 for any Gˆg Co¨ 2Ž.G . Ž.ii The exact sequence
1 ª Co¨ 222Ž.G ª GrCo¨ Ž.G ª ޚ ª 1.
is split if and only if *ޚ4 is isomorphic to the tri¨ial double co¨er. Proof. One checks easilyŽ. there are 4 cases that the canonical section Ž.cf. Remark 1.4.1 s: ޚ22ª GrCo¨ Ž.G and the canonical retraction r: GrCo¨ 22Ž.G ª Co¨ Ž.G satisfy
ѨŽz.ѨŽw. szwŽ.sd szsw Ž.Ž. and ѨŽ Ž␣..Ѩ Ž Ž  .. rŽ.␣ s d r Ž.Ž.␣ r 
2 using the fact that sŽ.y1 s d Ž.by Proposition 1.5 . The map f: GrCo¨ 222Ž.G ª Co¨ Ž.G = ޚ given by fŽ.␣ s ŽŽ.r ␣ , ␣ Ž..defines a bi- jection and by transporting the group structure and using the above identities, partŽ. i follows. PartŽ. ii is now immediate. In the rest of this paper, we will be interested in the case where G is OVŽ.,B, the group of isometries of a non-degenerate bilinear form Ž.V, B Ž.over a field of characteristic / 2 , and : OVŽ,B.ªޚ2 is the determi- nant homomorphism.
1.9. COROLLARY. IfŽ. V, B is a non-degenerate bilinear form o¨er a field of characteristic / 2, then the exact sequence
det 1 ª Co¨ 222Ž.OVŽ.,B ª GrCo¨ Ž.OVŽ.,B ªޚy1,
is not split. DUAL PAIRS IN PinŽ. p, q 519
Proof. The restriction of the double cover det*ޚ4 to a subgroup of OVŽ.,B generated by a hyperplane reflection is isomorphic to ޚ42ª ޚ and therefore non-trivial.
2. CLIFFORD ALGEBRAS AND GRADED DOUBLE COVERS OF ORTHOGONAL GROUPS
In this section we will be concerned with finite dimensional, non-degen- erate inner product spaces Ž.V, BV over certain fields k Žsee the next paragraph for the precise specification.Ž . The group of isometries of V, BV . will be denoted by OVŽ.,BV and the group of determinant-graded double det covers by GrCo¨ 2ŽŽ..OV as defined in Subsection 1.6. The aim will be to define a homomorphism from the Witt group of k to the group det GrCo¨ 2ŽOV Ž .. and identify its image. RecallŽ see Chapter 2 ofwx L for details.Ž. that Witt k , the Witt group of k, is the quotient of GrothŽ. k , the Grothendieck group of equivalence classes of non-degenerate quadratic forms over k, by the additive subgroup generated by the hyperbolic plane 22 2 Žk,xyy.. Throughout, we will refer to a det-graded double cover of OVŽ.of the form Ž.Gˆ, det as odd, and to a det-graded double cover of OVŽ.of the form ŽGˆ, e .as even. It is well known that one can construct non-trivial double covers of real orthogonal groups using Clifford algebrasŽ cf.wx ABS. . This construction can be generalised to fields k satisfying the properties
22 k*rŽ.k*(ޚ2 and y1 f Ž.k*, Ž.) and only to fields satisfying these conditions as explained in the Appendix. The real numbers ޒ satisfy Ž.) but also all real closed fields and the finite fields ކq with q elements, where q s 3Ž.Ž mod 4 seewx L. . Throughout this section the ground field k is supposed to satisfy Ž.) . Let us now recall the details of the above construction.
2.1. DEFINITION. Let Ž.X, B be a finite dimensional, inner product space. The Clifford algebra CXŽ.,B is the quotient algebra of the tensor k algebra TXŽ.s[Xm by the two-sided ideal T generated by elements of the form x m y q y m x y 2 BxŽ.,yI, where x, y g X are considered as elements of TXŽ.of degree 1. It is a ޚ2-graded algebra, where even Ž resp. odd.Ž elements are the images of even resp. odd. elements in TXŽ.via the natural projection.
One shows that the composition of natural maps X ª TXŽ.ªCX Ž,B . is an injection and thus we can consider X as embedded in the Clifford 520 M. J. SLUPINSKI
algebra, which we will do from now on. If Ä4e12, e иии enis an orthonormal basis of X, the 2n elements Ä4e . e иии e form a basis of ii12 ik1Fi1-иии-ikFn CXŽ.,B and the rules of calculation are eeijqee jis2BeŽ. i,e j. The Clifford algebra has the following universal property: if A is a real, associative algebra with identity, then any linear map f: X ª A such that fxfyŽ.Ž.qfyfx Ž.Ž.s2Bx Ž,y .1᭙x,ygXextends to a unique algebra homomorphism f˜: CXŽ.,BªA. This implies the
2.2. PROPOSITION Žcf.wx ABS.Ž . If X11, B .Ž and X 22, B . are non-degen- erate, inner product spaces then CŽ.Ž.Ž X11, B mˆCX 22,B (CX 1[X 21,B [Bas22.ޚ-graded algebras.
Proof. We apply the universal property to extend the isometries i1: X112ª X [ X and i 2212: X ª X [ X to obtain graded algebra homomor- ˜˜˜ˆ phisms i jjj: CXŽ.Ž,B ªCX1212[X,B[B.Ž.. Then i 12m i ␣ m  s ˜˜i12Ž.Ž␣i  .is the required isomorphism. The Clifford algebra CXŽ.,B comes with some natural automorphisms and antiautomorphisms and using them one can construct a non-trivial double cover of OXŽ.,B.
2.3. DEFINITION.Ž. i The automorphism of TX Ž . given by x12m x k иии m xk ¬ Ž.y1 x12m x иии m xkpreserves the ideal T and so induces an g automorphism of CXŽ.,B, which will be denoted c ¬ c . Then CXŽ.,B Ž. Ž.Ä Ž. g4Ä Ž. sCXqy,B[CX,BscgCX,B:csc [cgCX,B:cs g yc4defines the ޚ2-grading of the Clifford algebra.
Ž.ii The antiautomorphism of TVŽ.given by x12m x иии m xkk¬ x mиии x21m x preserves the ideal T and so induces an antiautomorphism T of CXŽ.,B, which will be denoted c ¬ c . 2.4. DEFINITIONrPROPOSITION. LetŽ. X, B be a finite dimensional, non-degenerate inner product space and let CŽ. X, B be its Clifford algebra. The group PinŽ. X, B is defined as
T g 1 PinŽ. X, B s Ä4c g CX Ž.,B:cisin¨ertible, cc s "1 and c Xcy s X .
gy1 We define PinŽX, B.: PinŽ. X, B ª GL Ž. X by PinŽX, B. Ž.Ž.cxscxc . Then the following hold:
Ž.i PinŽX, B. Žc . is an isometry of Ž X, B .. Ž. Ž. Ž. Ž. ii The group Pin X, B is a disjoint union Pin X, B s Pinq X, B Ž. jPiny X, B where
Pin""Ž. X, B s Pin Ž. X, B l CX Ž.,B. DUAL PAIRS IN PinŽ. p, q 521
ŽŽ..Ž. We ha¨e det PinŽX, B. c s 1 iff c g Pinq X, B . This defines a non-tri¨ial ޚ2-grading of PinŽ. X, B . Ž. Ž. Ž . Ž. iii If x g X satisfies B x, x s "1, then x g Piny X, B and x gOXŽ.,B is the reflection in the hyperplane orthogonal to x.
Ž.iv The map PinŽX, B.: Pin Ž. X, B ª OX Ž.,Bisa2to 1 co¨ering y1 map of OŽ. X, B and PinŽX, B.Ž.1 s "1. Ž. Ž .Ž Ž .. v E¨ery element of Pinqy X, B resp. Pin X, B is a product of an e¨enŽ. resp. odd number of x satisfyingŽ. iii . Thus the groupŽŽ Pin X, B .. with the grading induced by its inclusion in CXŽ.,B is the odd graded double co¨er ŽŽ.. Pin X, B , det , which will be denoted PinŽ. X, B . Proof. PartsŽ. i , Ž ii . , and Ž iii . are true over any field Ž the proofs inwx ABS for the case k s ޒ do not use any special properties of ޒ.. The proofs of wxABS for partsŽ. iv and Ž. v in the case k s ޒ depend on the fact that ޒ has the properties Ž.) and therefore also apply here as explained in the Appendix.
2.4.1. PROPOSITION.iŽ.Let Ž V, BV . be a non-degenerate inner product space such that V contains two orthogonal ¨ectors ¨ 12, ¨ such that BŽ.¨ 11, ¨ Ž. Ž. Ž. sB¨22,¨ /0. Then the double co¨er Pinq V ª SO V is non-tri¨ial ŽŽi.e., not isomorphic to SO V .= ޚ2 .as anŽ ungraded. double co¨er. A fortiori, the double co¨er PinŽ. V ª O Ž. V is non-tri¨ial. Žii .The double co¨er Pin Ž²1 :Hy ²1 :.ªO Ž²1 :Hy ²1 :. is non- tri¨ial. Ž . Ž² : ² :. Ž² : ² :. iii The double co¨er Pinq 1 Hy1ªSO 1 Hy1 is isomorphic to the tri¨ial co¨er. Ž.iv We ha¨e the following group isomorphisms: ²: ޚ ޚ ²: ޚ PinŽ.1 ( 22= ; PinqŽ.1 ( 2; ²:ޚ ²:ޚ PinŽ.y1 ( 4; Pinq Ž.y1 ( 2.
Proof. Ž.i One can always suppose BŽ.Ž.¨ 11, ¨ s B ¨ 22, ¨ s "1 since H H ksatisfies the property Ž.) . Let R Ž¨ 12.Ž, R ¨ .Ž.g OV be the reflections in the hyperplanes orthogonal to ¨ 12and ¨ , respectively. The subgroup HH ÄIdV, RŽ¨ 12.ŽR ¨ .4of SOŽ. V is covered by the subgroupÄ 1, y1, ¨¨12, 4 Ž. Ž .2 ޚ y¨¨12 of Pinq V and since ¨¨12sy1, this is 4and the covering is non-trivial. Ž.ii If ¨ is a vector such that B Ž¨, ¨ .sy1, the subgroup H Ä1dV , RŽ¨ .4 of OŽ²1 :Hy ²1 :. is covered by the subgroupw 1, y1, ¨, y¨4 2 of PinŽ²1 :Hy ²1 :. and since ¨ sy1, this is isomorphic to ޚ4 and the double cover PinŽ²1 :Hy ²1 :.ªO Ž²1 :Hy ²1 :. is non-trivial. 522 M. J. SLUPINSKI
Ž.iii Choose two vectors e, i g V such that BeŽ.Ž.,esBi,i s0 and 1 22 BeŽ.,i s2 so that in the Clifford algebra e s i s 0 and ei q ie s 1. It Ž² : ² :. ÄŽ. 0 4 is easy to see that in this basis SO 1 Hy1s01r:gk* . One ⌰ Ž² : ² :. Ž² : ² :. verifies that the map : SO 1 Hy1ªPinq 1 Hy1 given by
0 11 ⌰1sŽ.Ž.eyiyeqis Žei q ie . 0 ''<<<< 00 Ž² : ² :. is a homomorphism and section of the covering map Pinq 1 Hy1 ª SOŽ²1 :Hy ²1. :. Ž.iv This is straightforward. Remark 2.4.2. The group SpinŽ. X, B is defined Ž seewx ABS, Sch. by Ž. Ž. Ž. Ž . Spin X, B s Pinq X, B .If ksatisfies ) , one shows as inwx ABS that xgSpinŽ. X, B if and only if x is the product of an even number of vectors of length "1. With the hypotheses of Proposition 2.2 and using simplified notationŽ so that CXŽ.X,B B becomes C, becomes , etc.. , we 1212[ [ PinŽX11, B . 1 have the
2.5. LEMMA.iŽ. If pjjj: Pin ¨ C are the natural inclusions, then ⌽: Pin =ˆPin Cgien by ⌽ Ž.˜˜i i p ˆp is an injecti e group 1 ޚ 221ª ¨ s m 2( 1m 2¨ homomorphism, compatible with gradings. Ž.ii The image ⌽ŽPin =ˆ Pin. is contained in Pin. 1 ޚ 2 2 Ž.iii The following diagram of group homomorphisms commutes,
⌽ 6 Pin =ˆ Pin Pin 1 ޚ 2 2
12=
6 6
6 O12=OO
Ž.Ž.a0 Ž . where : O12= O ª O is the injection a, b sg0bOsOX12[X . Proof. Ž.i This follows immediately from Proposition 1.3 and Proposi- tion 2.2.
Ž.ii An element x11g X such that Bx111Ž.,x s"1 can be consid- ered as an element of Pin . Then x ,1 Pin =ˆPin and by definition 11wxg 1ޚ22 DUAL PAIRS IN PinŽ. p, q 523
Ž.cf. Proposition 2.2 ,
⌽Ž.wxx11112,1 sixŽ.gX[X;C.
2 But BiŽŽ11 x .,ix 11 Ž ..sBx 11 Ž,x 1 .so that ix11 Ž .s"1inC. Hence ix11Ž. gPin by Proposition 2.4Ž. ii . Similarly ⌽Žwx1, x22.g Pin for x g X2such that Bx22Ž.,x 2s"1. The groups Pinjare both generated by products of unit vectors and therefore Pin1 =ˆޚ Pin2 is generated by products of 2 elements of the form wxx12, x . Hence ⌽Žwxg, h .g Pin for any wxg, h g Pin1 =ˆPin . ޚ 2 2 Ž.iii Consider again an element x ,1 Pin =ˆ Pin where x wx11g ޚ221g X11such that BxŽ.1,x1s"1. Then ŽŽ⌽ wxx11,1..s Žix Ž1 ..as in Ž ii . ; but ix11Ž.is a B-unit vector in X1[ X 2and so ŽŽ..ix11 is the reflection in the hyperplane B-orthogonal to ix11Ž.by Proposition 2.4Ž. ii . In other words,
⌽x,1 x ,Id . Ž.Ž.wx11s Ž.Ž.1X2
Similarly for a unit vector x22g X ,
⌽1, x Id , x . Ž.Ž.wx2s Ž.X1 22Ž.
Now let g g Pin11be a product of unit vectors g s xy1иии z1. Then
Ž.⌽Ž.wxg,1 sŽ.⌽Ž. wxy11иии z 1,1 x
sŽ.⌽Ž.wxwxx11,1 иии z ,1 x ,Id иии z ,Id sŽ.Ž.11Ž. X2211Ž. X g,Id . sŽ.1Ž. X2
Similarly for h g Pin2 ,
⌽1, h Id , h . Ž.Ž.wxs Ž.X12Ž.
Hence for g, h Pin =ˆPin , wxg 1 ޚ 2 2 Ž.⌽Ž.wxg,h sŽ.ŽŽ⌽Ž.Ž.Ž.wxwxg,1 1,h s ⌽ wxg,1 s ⌽ wx1, h g,Id Id , h g, h . sŽ.Ž.1Ž. XX21212Ž.s Ž. Ž. Ž.
Now let us fix a finite dimensional, non-degenerate inner product space
Ž.V,Band denote the orthogonal group OV Ž,BV .by O. The above 524 M. J. SLUPINSKI
remarks provided us with a non-trivial, odd graded double cover of O, denoted by Pin. We will now show how to obtain other graded double covers of O from a generalisation of this construction.
Let Ž.W, BW be a non-degenerate inner product space of arbitrary finite dimension. Then Ž.V m W, BVWm B is a non-degenerate inner product space and by the above, we can construct PinŽ. V m W, BVWm B , which is an odd, graded double cover of OVŽ.mW,BVWmB . But the group OVŽ.mW,BVWmB contains O as the subgroup O m IdWand so : y1 Ž.OmIdWWª O s O m Id is a double covering and the group y1 Ž..OmIdW is naturally graded as a subgroup of PinŽ V m W, BV m dimW BW .ᎏthe grading is just det .Ž Here for brevity, denotes the covering map .. In the same way, the graded subgroup PinŽVmW, BVWmB . y1 ŽŽ..IdV m OW is a graded double cover of OWŽ.ᎏthe grading is just det dim V.
y1 dimW 2.6. DEFINITION. The graded, double cover Ž Ž.O m IdW , det . of O OVŽ.,B will be denoted by GrŽ. V, B , or more economi- s VVŽW,BW. cally by GrŽ. V W when there is no danger of confusion. y1 dim V The graded, double cover Ž ŽŽ..IdVWm OW , det.Ž. of OW,B will be denoted by GrŽ. W, B , or more economically by GrŽ. W ŽV, BV. WV when there is no danger of confusion.
Remark 2.7. The graded double covers GrŽ. V WWand Gr Ž. V are isomorphic: the natural isomorphism of graded algebras CVŽ.mW ( CWŽ.mV restricts to the required isomorphism.
Remark 2.8. The underlying group of GrŽ. V W is clearly isomorphic to Ž. the fibre product O =OŽVmW . Pin V m W . The following result will be the main tool used in identifying the graded
double covers GrŽ. V W .
2.9. THEOREM. IfŽ.Ž. W11, B and W 22, B are non-degenerate, inner prod- uct spaces of finite dimension, then
Gr V, B Gr V, B Gr V, B . Ž.VVVW1212[W,B[BW( Ž.Ž.11,BW) Ž.Ž.22,B
Thus, the map W ¬ GrŽ. VW defines a homomorphism GrŽ. V ؒ: Groth Ž. k det ªGrCo¨ 2ŽŽ..O V from the Grothendieck group of quadratic spaces to the det group of determinant-graded double co¨ers GrCo¨ 2ŽŽ..OV . Proof. We will use abbreviated notation whenever the meaning is clear.
Applying Proposition 2.5 with Xjjs V m W , we have the commutative DUAL PAIRS IN PinŽ. p, q 525 diagram
⌽ 6 PinŽ. V W =ˆ Pin Ž. V W Pin V W W m 1ޚ2m 212Ž.m Ž.[
12=
6 6
6 OVŽ.Ž.mW12=OVmWOVŽ.mŽ.W1[W2
By definition,
Gr Gr Ž. = y1 g Id , g Id : g OVŽ. WW12) s 12ž/Ä4Ž.m WW1m 2g and Gr y1Ž.O Id . Thus to prove the theorem, we just W12[WWs m 12[W have to verify that
g Id , g Id : g OVŽ. O Id , ž/Ä4Ž.m WW12m g s m W1[W2 which is true by definition. Any non-degenerate inner product space is isomorphic to a direct sum of one dimensional inner product spaces so that, by induction, the identifi- cation of GrŽ. V, B is now reduced to the case where dim W 1. VW,BW s
2.10. EXAMPLE. If we take for Ž.W, BW a 1-dimensional, inner product space of signatureŽ. 1, 0 , then Gr Ž V, B . PinŽ. V, B . This is be- VW,BVW( cause the linear isomorphism ¨ ¬ ¨ m e Ž.where e is any unit vector in W extends to aŽ. graded algebra isomorphism CV Ž.Ž,BVV(CVmW,B m BWV.Žby the universal property of CV,B .. This extended map restricts to a graded isomorphism PinŽ.Ž. V, B Gr V, B . VV( W,BW
2.11. EXAMPLE. If we take for Ž.W, BW a 1-dimensional inner product space of signatureŽ. 0, 1 , then the odd graded double cover GrŽ. V, BVW,B W (PinŽ. V, yBV . This is because the linear isomorphism ¨ ¬ ¨ m e Žwhere eis any unit vector in W .Ž.extends to a graded algebra isomorphism
CVŽ.Ž,yBVV(CVmW,B mBW .by the universal property of CVŽ.,yBV . This extended map restricts to a graded isomorphism PinŽ.Ž. V, B Gr V, B . y VV( W,BW
As pointed out in, for example,wx K , the groups PinŽ. V, BV and PinŽ. V, yBV may or may not be isomorphic. However, viewed as odd graded double covers they are inverses with respect to the multipli- cation ). 526 M. J. SLUPINSKI
2.12. PROPOSITION. PinŽ. V, BVV) PinŽ. V, yB ( E. Proof. This is a consequence of the following proposition of Karoubi:
2.12.1. LEMMA Žcf.wx K.Ž.Ž.Ž. . The map : TV ªTV gi¨en by x s 1 rŽr 1. r Ž.y12 y xifxgTVŽ.,factors to gi¨e a graded, linear isomorphism ⌰: cd CVŽ.Ž,BVVªCV,yB . such that ⌰ Ž.Ž.cd sy1 ⌰Ž.c⌰ Ž.d and such that ⌰is the identity on V. Since ⌰ maps unit vectors to unit vectors, it maps products of unit vectors to products of unit vectors and therefore by Proposition 2.4, ⌰
maps PinŽ. V, BVVbijectively onto Pin Ž V, yB .. For e g V such that BeŽ.,e "1, we have ⌰Ž.e Ž.e because both VPs inŽV,yBVV.( s PinŽV, B . are equal to reflection in the hyperplane orthogonal to e by Proposition 2.4. Hence ⌰Ž.g Ž.g ᭙g Pin Ž V, B .since g PinŽV,yBVV.( s PinŽV, B . g V g PinŽ. V, BV is a product of unit vectorsŽ. Proposition 2.4 and ⌰ is a homomorphism up to sign by the lemma. Thus, comparing with the y1 definition of PinŽ. V, BV given in the proof of Theorem 1.4, we see that the odd graded covering space PinŽ. V, yBV is isomorphic to y1 PinŽ. V, BV . det : One can now identify the image of GrŽ. V ؒ: Groth Ž. k ª GrCo¨ 2ŽŽ..OV
2.13. COROLLARY. LetŽ. W, BW be a non-degenerate inner product space o¨er k such that W ( r²:1 H s ²y1. :Then: Ž.i We ha¨e the following isomorphisms of graded double co¨ers of OVŽ.:
GrŽ. V W( OV Ž.=ޚ2 if r y s s 0Ž. mod 4 . GrŽ. V W( Pin Ž V, BifrV . yss1Ž. mod 4 .
GrŽ. V W( det*ޚ4 if r y s s 2Ž. mod 4 . GrŽ. V W( Pin Ž V, yBifrV . yss3Ž. mod 4 .
Ž.ii The image of the homomorphism GrŽ.Ž. V ؒ: Groth k ª det GrCo¨ 2ŽŽ..O V is the subgroup generated by PinŽ. V, BV and is isomorphic to ޚ4. Proof. By repeated application of Theorems 2.9 to 2.12, one has
rys GrŽ. V W( Pin Ž V, BV . .
4 Since PinŽ. V, BV ( E by Theorem 1.4, this depends only on r y s Ž.Ž.mod 4 . Part i now follows from Example 2.10 if r y s s 1Ž. mod 4 , from Example 2.11 if r y s s 3Ž. mod 4 , and from Proposition 1.5 if r y s s 2 Ž.mod 4 . DUAL PAIRS IN PinŽ. p, q 527
det The image of the homomorphism GrŽ. V ؒ: Groth Ž. k ª GrCo¨ 2ŽŽ..OV is clearly generated by PinŽ. V, BV and is isomorphic to ޚ4 since by 2 Proposition 1.5 and Corollary 1.9, PinŽ. V, BV ( det*ޚ4 is not the trivial double cover OVŽ.,BV =ޚ2.
COROLLARY. The homomorphism GrŽ. V ؒ: Groth Ž. k ª .2.14 det GrCo¨ 2ŽŽ..O V factors through a homomorphism GrŽ. V ؒ: Witt Ž. k ª det det GrCo¨ 22ŽŽ..O V from the Witt group of k to GrCo¨ ŽŽ..OV . Proof. By definitionŽ see Chapter 2 inwx L. , the Witt group is the quotient of the Grothendieck group of quadratic forms over k by the additive subgroup generated by theŽ. class of H, the 2-dimensional inner product space of signatureŽ. 1, 1 . By Proposition 2.12 Ž. i with r y s s 1 y det 1s0, GrŽ. V H is the identity element in GrCo¨ 2ŽŽ..OV and the result follows.
2.15. Remark. As shown in the Appendix, WittŽ. k ( ޚ if y1 is not a sum of squares in k and WittŽ. k ( ޚ4 if it is.
3. HOWE DUAL PAIRS IN PinŽ. V m W
Let V and W be finite dimensional, real, non-degenerate inner product
spaces. The pair of subgroups ŽŽ.OV mIdWV, Id m OW Ž ..in OV ŽmW .is an example of a dual pair in the sense of R. HoweŽ cf.wx H1, H2. , i.e., each subgroup is the commutant of the other in OVŽ.mW. The groups Gr Ž. V W and GrV Ž. W are by definitionŽ cf. Definition 2.6. the inverse images in PinŽ.Ž. V m W of OV mIdWVand Id m OWŽ., respectively, under the double covering map : PinŽ.Ž. V m W ª OVmW. In this section we investigate to what extent they are mutual commutants in PinŽ. V m W . Throughout this section, the ground field k is supposed to satisfy the property Ž.) of Section 2. Let us first collect some properties of orthogonal orthogonal groups over k which we shall need. In the following proposition, V is a non-de- generate inner product space over k, SOŽ. V s Äg g OVŽ.: det g s 14 and c GsÄ fgEndŽ. V : fg s gf ᭙g g G4Ž.this is a k-algebra . c 3.0. PROPOSITION.iŽ.OV Ž . skId except if k s ކ3 and V ( ²:1 H ²:y1.
Ž.ii If k s ކ3 and V ( ²:1 Hy ²1, :then O Ž V . is abelian, isomor- phic to ޚ22= ޚ , and SOŽ. V s Ä4"Id . As a OŽ. V -representation, V is the c sum of two non-tri¨ial, irreducible, non-isomorphic representations and OŽ. V ( k = k. c Ž.iii If dim V / 2, then SOŽ. V s kId. 528 M. J. SLUPINSKI
c Živ .If dim V s 2, then SO Ž V . is abelian. We ha¨eSO Ž²1 :H ²1 :. ( c kŽ'y1 . Žalways . and SO Ž²1 :Hy ²1 :. (k=kifk\ކ3. Proof. Exercise. This has the following immediate corollariesŽ in which V, W are non-de- generate inner product spaces over k, SOŽ. V , OV Ž.,SO Ž. W , and OW Ž. are considered as subgroups of OVŽ.mW, and GЈ denotes the commu- tant in OVŽ.mW of a subgroup G .:
3.1. COROLLARY. Suppose k / ކ3. Then we ha¨e the following identities together with those obtained by interchanging V and W in the statements.
Ž.i OV Ž .ЈsOW Ž .. Ž.ii If dim V / 2, then SO Ž V .Ј s OW Ž .. Ž.iii If V ( ²:1 Hy ²1, :let V s L [ L* be the decomposition as a sum of isotropic subspaces so that V m W s L m W [ L* m W. We set ty1 GLdLLŽ. W s ÄId m g q Id *m g : g g GLŽ. W 4. This is a subgroup of OVŽ.mW isomorphic to GL Ž. W . Then
SOŽ. V Ј s GLdd Ž. W and GLŽ. W Ј s SO Ž. V .
Ž.iv If V ( ²:1 H ²:1, let J be one of the two isometric complex structures on V, let J J Id , and let UŽ. V W Äf OVŽ.W: VWs m JVm s g m fJVVs Jf4.Then
SO V Ј UV W and U V W Ј SO V . Ž.Ž.s JJVVm Ž.Ž.m s
ŽWe impose the condition k / ކ3 only to get a uniform statement. The calculation of commutants if k s ކ3 presents no difficulty but the statement of the results is rather cumbersome..
3.2. COROLLARY. The following pairs of subgroupsŽ together with those obtained by interchanging V and W in the statements.Ž are dual pairs i.e., mutual commutants.Ž in O V m Wifk ./ކ3: Ž.i ŽOV Ž .,OW Ž ... Ž.ii ŽSO Ž V ., UV ŽWifV .. ²:1 ²:1. JV m ( H
Ž.iii ŽSO Ž. V , GLd Ž W .. if V ( ²:1 Hy ²1. : The next lemma is a generalisation of Proposition 2.4Ž. iii :
3.3. LEMMA.iŽ.Let ¨ g V be a unit ¨ector Ž i.e., BV Ž¨, ¨ .s "1, . let H H RŽ¨ . g O be reflection in the hyperplane ¨ , and let w12, w иии wbedimW H an orthonormal basis of W. Then the element R˜Ž.¨ s ¨ m w12. ¨ m w иии ¨ DUAL PAIRS IN PinŽ. p, q 529 mwdimW of the Clifford algebra CŽ.Ž. V m W, BVWm B is in Gr VW and H Ž.¨mw12.¨mwиии ¨ m wdimWs RŽ¨ .m Id W . E¨ery element of GrŽ. VW is a product of such elements.
Ž.ii If x g Gr Ž V .WV and y g GrŽ. W , then Ž.dim V dimW 1 Ѩ xѨ y xy syŽ.1 q yx, where Ѩ xŽ. resp. Ѩ y is zero if Žx . Ž resp. ŽyisinSOV .. Ž .mIdW Ž resp. IdV m SOŽ.. W and equals 1 otherwise.
Proof. Ž.i Each ¨ m wi is a unit vector in V m W and therefore belongs to PinŽ. V m W by Proposition 2.4Ž. iii . Hence their product is in PinŽ. V m W . By Proposition 2.4Ž. iii , Ž¨ m wiV .is reflection in the hyperplane B m BWi-orthogonal to ¨ m w and taking the product we get Ž¨ m w12. ¨ m w H иии ¨ m wdimW .Žs R ¨ .m IdW . Every element of GrŽ. V W is a product of such elements since every ggOVŽ.is a product of reflections.
Ž.ii By part Ž. i , every element of the group GrŽ. V W is a product of H elements of the form R˜Ž.¨ s ¨ m w12. ¨ m w иии ¨ m wdimW , where ¨ g V is a unit vector and w12, w ,...wdimW is an orthonormal basis of W. Similarly, every element of the group GrV Ž. W is a product of elements of H the form Rw˜Ž.s¨12mw.¨ mwиии ¨ dimVm w, where w g W is a unit vector and ¨ 12, ¨ , иии ¨ dim V is an orthonormal basis of V. Thus to prove partŽ. ii it is sufficient to prove that
dim V dimW R˜˜Ž.Ž.¨HHRw sy Ž. y1 Rw˜ Ž.Ž. HH R ˜¨ for all ¨ g V, w g W such that BVWŽ.¨, ¨ s "1, Bw Ž,w .s"1. Given unit vectors ¨ g V and w g W, choose orthonormal bases ¨ s ¨ 12, ¨ , иии ¨ dim V and w s w12, w , иии wdimW of V and W, respectively. Then Ä4¨ijmwis an orthonormal basis of V m W and so in the Clifford algebra CVŽ.mW, we have
¨ijklklijm w . ¨ m w q ¨ m w . ¨ m w s 2 B VikWjlŽ.Ž.¨ , ¨ Bw,w. Hence
R˜Ž.Ž.¨ H Rw˜ H
s¨1112mw.¨mwиии ¨ 1m wdimW ¨ 1121dimm w . ¨ m w иии ¨ Vm w1 dim V dimW syŽ. y1 ¨1121dimmw .¨ mw иии ¨ Vm w11¨ m w 1. ¨ 1
mw21dimиии ¨ m w W dim V dimW syŽ. y1 Rw˜˜ Ž.Ž.HH R¨ . 530 M. J. SLUPINSKI
3.4. THEOREM. LetŽ.Ž. V, BVW, W, B be finite dimensional, non- degenerate, inner product spaces o¨er k / ކ3. For bre¨ity, we write GrŽ. V Žresp. Gr Ž W .. for the subgroup GrŽ V .WV Ž resp. Gr Ž W .. of Pin Ž V m W .. Let 0 y1 0 GrŽ. W be the group ŽŽ..IdV m SO W and GrŽ. V be the group y1 ŽŽ.SO V m IdW .. If G is a subgroup of PinŽ. V m W , GЈ will denote its commutant in PinŽ. V m W . Then the following identities holdŽ together with those obtained by inter- changing V and W in the statements..
Ž.i If dim V and dim W are odd, then Gr Ž V .Ј s Gr Ž W .. 0 Ž.ii If either dim Vordim Wise¨en, then Gr Ž V .Ј s GrŽ. W . 0 Ž.iii If dim V / 2, then GrŽ. V Ј s Gr Ž W .. 0 y1 Ž.iv If V ( ²:1 Hy ²1, : then GrŽ. V Ј s ŽŽ..GLd W and y1 0 ŽŽ..GLd W Ј s GrŽ. V . Ž.vIf V ²:1 ²:1, then Gr0Ž. V Ј y1ŽŽUV W .. and ( H s JV m y1 ŽŽUV W ..Ј Gr0Ž. V . JV m s Proof. PartsŽ. i , Ž ii . , and Ž iii . are all proved in the same way. Let us prove partŽ. iii as an example. By Lemma 3.3 Ž. ii , xy s yx if Ѩ x s 0 and 0 0 hence GrŽ. W : GrŽ. V Ј.If zgGrŽ. V Ј, then Ž.z g SO Ž. V Ј s OW Ž . 0 by Corollary 3.1Ž. ii . Hence z g Gr Ž W .and so GrŽ. V Ј : Gr Ž W .. To prove partsŽ. iv and Ž. v we use well-known representations of the Clifford algebra CVŽ.mW as endomorphism algebras.
Proof of Ž.iv . Let ŽV, BV .be a two dimensional inner product space such that V ( ²:1 Hy ²1 : and let ŽW, BW .be an n-dimensional non-de- generate inner product space over k of signature Ž.r, s . Choose basis vectors E and I of V such that BEVVŽ.,EsBI Ž.,Is0 and BE V Ž.,I 1 s2 . In this basis ¡ 00¦ ¡ ¦ OVŽ. 11: k* : k* and s~¥~¥00g j g ¢§¢§0 0 ¡0¦ SOŽ. V 1: k*. s~¥0 g ¢§0
If Ä4w12, w иии wnis an orthogonal basis of unit vectors of W, then, writing Eaaaafor E m w and I for I m w considered as elements of CVŽ.mW, we have the relations
EEabqEE basII abqII bas0 and
EIabqIE b as␦ a,bWBwŽ. a,w b. DUAL PAIRS IN PinŽ. p, q 531
Now let ⌳Ž.W be the exterior algebra over W and define a linear map c: VmWªEndŽŽ⌳ W ..by
ceŽmw .Ž. seww Ž. and ciŽmw .Ž. si Ž. , where ewwŽ. s w n and i : ⌳Ž.W ª ⌳ Ž.W is the interior product along w, that is, the unique anti-derivation such that iwwWŽ.ЈsBw ŽЈ,w . Ј ⌳1Ž. Ž.2 Ž. if w g W s W. It is easily checked that cx sBxVmW ,xIdand so cextends to an algebra homomorphism c: CVŽ.mW ªEnd ŽŽ..⌳ W , which is in fact an isomorphism is as well known. Via the isomorphism c: CVŽ.mW ªEnd ŽŽ..⌳ W , the antiautomor- T phism x ¬ x of CVŽ.ŽmW cf. Definition 2.1 . defines an antiautomor- TTT phism of EndŽŽ⌳ W .., also denoted x ¬ x , such that eaas e and i aas i . Therefore there exists a non-degenerate bilinear form I: ⌳Ž.W = ⌳ Ž.W ªkŽ.unique up to scalar multiplication such that
T IxŽ.Ž.,sIŽ.,xŽ. ᭙xgEndŽ.⌳Ž.W , ᭙ , g ⌳ Ž.W .
p q This bilinear form satisfies IŽ⌳ Ž.W , ⌳ Ž..W s 0if pqq/nsince then, taking a Ä4i , иии i , j , иии j , we have IeeŽŽ.Ž..иии e 1,ee иии e 1 f 1 p 1 qiiijij12 p 12 q s BwŽ,wIei . ŽŽie . e иии e Ž1, .e иии e Ž1 ..Bw Ž,wI . ŽŽ1 . pee aa aaaaiq 1 ip j1 jq s aa y ai1 иии eiŽ.1,e иии e Ž..1 Bw Ž,wIee .Žиии e Ž.1,ie иии e Ž..1 0 us- iajjp 1 qq aaaiiajj1 p 1 qs ing i aaŽ.1 s 0 and iebbsyeiaif a / b. Similarly, one shows that IŽ. , 1nŽn1. pnp syŽ.12yI Ž, .and that I <⌳ = ⌳ yis non-degenerate. y1 We claim that the double cover ŽGLdd Ž W ..of GL Ž W .in Pin Ž V m W.Žis isomorphic to GL W .= ޚ22. To prove this define ␣: GLŽ W .= ޚ ªEndŽŽ⌳ W ..by 1 ␣ Ž.g, z s z ⌳Ž.g ,1Ž. < 3.4.1. LEMMA. For w g W and forŽ. g, z g GL Ž. W = ޚ2 , Ž.Ž.i ␣ g, zewgseŽw.␣ Ž.Ž.Ž.g,z,ii␣g,ziwgsity1Žw.␣ Ž.g,z, T det g and Ž.Ž.Ž.iii ␣ g, z ␣ g, z s . < Proof. The first two equalities are immediate consequences of the fact that ⌳Ž.g is an automorphism of the exterior algebra ⌳ ŽW .. 532 M. J. SLUPINSKI To proveŽ. iii , choose e иии e Ž.1 ⌳ pŽ.W and e иии e Ž.1 ⌳ pŽ.W ii1 pg jj1 qg with p q q s n. Then I ␣ Žg, z .T ␣ Žg, ze .иии e Ž.1,e иии e Ž.1 ž/ii1 p jj1 q I␣g,ze иии e 1,␣ g,ze иии e 1 sŽ.Ž.ii1 p Ž.Ž.jj1 q Ž. 1 sI⌳Ž.geiiиии e Ž.1,⌳ Ž.ge jjиии e Ž.1 < Since I <⌳ p Ž.W = ⌳nypŽ.W is non-degenerate it follows that T ␣Ž.Ž.g,z␣g,zsdet gr< y1 ␣ Ž.g, zewgŽ.␣ Ž.g,z seŽw.and y1 ␣Ž.g,ziwgŽ.␣ Ž.g,z sity1Žw., y1 which is equivalent to Žc Ž␣ Žg, z ... s gd. ByŽ. iii of the lemma, 1 1 cy (␣Ž.g,zgPin Ž V m W .since det gr< Remark 3.4.2. By Corollary 2.13, GrŽ. V is isomorphic to OV Ž.=ޚ2, PinŽ. V, BV , det*ޚ4 or Pin Ž V, yBV ., respectively according to whether ryss0, 1, 2, or 3Ž. mod 4 , respectively. This can be seen explicitly as Ž. Ž. Ž01 . follows. The group OV is generated by SO V and10 subject to the Ž.Ž.01 01y1 y1 Ž. relations 10x 10 s x . Calculation shows that g Pin V m W given by syŽ.Ž.Ž.E11qI yE 22qI иии yEnnq I , Ž.01 Ž. TnŽ.qs covers 10g OV and satisfies sy1, 1 2 Ž.Ž.rsr s 1 syŽ.112 yyq Ž. and n ˜˜xy1 xy10Ž.where ˜x GrŽ. V .2 Ž. sž/<< g 0 The group GrŽ. V is then the subgroup of CVŽ.mW generated by GrŽ. V and subject to the relationsŽ. 1 and Ž. 2 which only depend on r y s Ž.mod 4 . Note further that the element commutes with Gr0Ž. W and y1 therefore c( (c commutes with the action of SOŽ. W = ޚ2 in ⌳ Ž.W . 1 p np The relationŽ. 2 implies that c( (cy maps ⌳ Ž.W to ⌳ yŽ.W and thus is proportional to the Hodge star operator on these spaces; the constant of proportionality, however, depends on p. Proof of Ž.v . Let us first introduce some notation and recall one elementary fact. We set kc s kŽ.'y 1 and if X is a k-vector space we set ˜ 22 XckcsXmk.If ksatisfies Ž.) , the group UkŽ.1 s Ä␣ q i g kc: ␣ q  s"14 is a non-trivial double covering group of the group UkŽ.1 s Ä␣ q 22 ˜ igkck: ␣ q s14 , where the covering map U Ž.1 ª Uk Ž.1 is the square. If Y is a hermitian vector space over kc and if UYŽ.is the associated unitary group, we denote UY˜Ž.the double covering group of UYŽ.defined 2 by UY˜˜Ž.s Ždetck . *U Ž.1 s ÄŽ.u, z g UY Ž.Ž.=U ˜k1 : det cu s z 4. This ˜˜ group comes with a natural homomorphism' detck : UYŽ.ªU Ž.1 given by 'det ccŽ.u, z s z, where det is the kc-valued determinant. Now let V s ²:1 H ²:1 and W be an n-dimensional non-degenerate inner product space over k of signature Ž.r, s . Let J be one of the two 2 isometric complex structures of V Ži.e., J syId and J g OVŽ..and choose e g V such that BeV Ž.,es1. Then Ä4e, Je is an orthonormal basis 534 M. J. SLUPINSKI of V and in this basis ␣ y OVŽ. :␣, k,␣22 1 s½5ž/␣ g q s ␣ 22 j :␣,gk,␣qs1 ½5ž/y␣ and 2 2 SOŽ. V s Ä4␣ q  J : ␣ ,  g k, ␣ q  s 1. If Ä4w12, w иии wnis an orthogonal basis of unit vectors in W, we set 11 Zaaaaaas22Ž.Ž.emwyiJe m w and Z s e m w q iJe m w and then these vectors form a basis of Ž.V m W ccover k . We have the following relations in the complex Clifford algebra CVŽŽmW .c .: ZZabqZZ basZZ abqZZ bas0 and ZZabqZZ bas␦ a,bWBwŽ. a,w b. Now let ⌳Ž.Wccbe the exterior algebra of W . Define a kc-linear map c: Ž.VmWccªEnd ŽŽ.⌳ W by cZŽaa .Ž. se Ž. and cZŽ.aaŽ. si Ž. , where eaaŽ. s w n and i acc: ⌳Ž.W ª ⌳ Ž.W is the interior product along waw, that is, the unique anti-derivation such that iwŽ.ЈsBwW ŽЈ,w . Ј ⌳1Ž. Ž.2 Ž. if w g Wccs W . It is easily checked that cx sBxŽVmW.,xIdand so c extends to an algebra homomorphism c: CVccŽ.mWªEnd ŽŽ..⌳ W , which is in fact an isomorphism of kc-algebras. We claim that the double cover y1 ŽŽUV W ..of UV ŽW .in JJVVm m PinŽ. V W is isomorphic to UV˜Ž.W. First identify V W with W m JcVm m by mapping e m w to w and Je m w to iw. By transport of structure this defines a map u ¬ udcwhich identifies UWŽ.with uVJ ŽmW .. ˜V Define : UWŽ.ccªEnd ŽŽ..⌳ W by 1 Ž.u˜˜s⌳Ž.u, 'det c u˜ ˜ where ⌳: UWŽ.ccªEnd ŽŽ..⌳ W is the composition of the covering map with the usual exterior algebra representation of UWŽ.c . DUAL PAIRS IN PinŽ. p, q 535 The following lemma implies Theorem 3.4Ž. v just as Lemma 3.4.1 implies Theorem 3.4Ž. iv . We leave the details to the reader. 3.4.3. LEMMA. For w g Wcc and for u˜s Ž.u, z g UW˜ Ž., Ž.i Ž.ue˜˜˜wuseŽw. Ž.u,ii Ž. Ž.uiwusity1Žw.Ž.u ˜, and T Ž.iii Ž.u˜˜ Ž.u s 1. 3.5. COROLLARY. With the hypotheses of Theorem 3.4, the following pairs of subgroupsŽ together with those obtained by interchanging V and W in the statements.Ž. are dual in Pin V m W : Ž.i ŽGr Ž V ., Gr Ž W .. if dim V and dim W are both odd. 0 Ž.ii ŽGrŽ. V , Gr Ž W .. if either dim Vordim Wise¨en and dim V / 2. 0 y1 Ž.iii ŽGrŽ. V , ŽGLd Ž W ... if V ( ²1 :Hy ²1. : Ž.iv ŽGr0Ž. V , y1ŽU Ž W ... if V ²1 : ²1. : JV ( H Proof. This is immediate from Theorem 3.4. Using Corollary 2.13, one can now identify the groups appearing in these dual pairs. There are only 4 possibilities for the underlying group of GrŽ. V s Gr Ž. V W , OVŽ.=ޚ2, PinŽ. V, BV, det*ޚ4, and PinŽ. V, yBV 0 0 and 4 possibilities for the underlying group of GrŽ. V s GrŽ. V W , SO V ޚ , Pin V, B , det*ޚ , and Ž.= 2 qŽ.Ž.V 4q PinqŽ. V, yBV depending on whether the signature of Ž.W, BW is 0, 1, 2, or 3 Ž. mod 4 , respectively. There may be some coincidencesŽ. as groups in these two Ž.ޚ Žޚ .Ž . Ž . lists, for instance SO V = 24( det* qqobvious and Pin V, BV( Ž.Ž . Pinq V, yBV by Lemma 2.12.1 . As shown above, y1 ŽŽ..Ž.GL W GL W = ޚ and y1ŽŽ..UW is dd( 2 JV the non-trivial double cover of UWŽ.obtained by ‘‘taking the square root JV of the determinant.’’ 4. HOWE CORRESPONDENCES FOR pinc REPRESENTATIONS The purpose of this section is to show that if k s ޒ, the ‘‘pinc’’ representation of PinŽ. V m W sets up a Howe correspondence for the dual pairs of Corollary 3.5. This will be a more or less direct consequence of Theorem 4.3.4.2 inwx H4 . Let us recall the relevant definitions. 536 M. J. SLUPINSKI 4.0. DEFINITION. Let AB: A ª GLŽ. X and : B ª GL Ž. X be repre- sentations of the groups A and B in the complex vector space X such that ABŽ.Aand Ž.B commute. Then X is said to set up a Howe correspon- dence for Ž.A, B if there is an A = B-module isomorphism X ( Riim S , [igI where the RiiŽ.resp. S are distinct irreducible complex representations of AŽ.resp. B . Remark 4.0.1. It is easily verified that the irreducible OVŽ.mW repre- sentation Ž.V m W m ރ sets up a Howe correspondence for the dual pairs of Corollary 3.2. If B is a real, non-degenerate bilinear from on X then it is well known that if dim X is even the complex Clifford algebra CXccŽ.sCX Ž,Bc .is simple and that if dim X is odd, CXcŽ.sI12[I, where I 1and I 2are its twoŽ. two-sided simple ideals. Let pic: CXŽ.ªI ibe the projections in the Ž. latter case. For future reference we note that the restrictions pic: CXq ªIi are isomorphisms. 4.1. DEFINITION. Let Ž.X, BX be a real non-degenerate inner product space. Ž.i If dim X s 2m,a pinc structure on X is the choice of a 2 m-dimensional complex vector space S and an algebra isomorphism : CXcŽ.(End Ž. S . The restriction of this isomorphism to Pin Ž. X : CXc Ž. is the pinc representation of PinŽ.Ž X . It is well known that S is an irreducible representation of PinŽ.. X . Ž.ii If dim X s 2m q 1, a pinc structure on X is the choice of a pair m of 2 -dimensional complex vector spaces Ž.S12, S and an algebra isomor- phism CXcŽ.(End Ž. S12[ End Ž. S . The restrictions of the projections 1: CXcŽ.ªEnd Ž. S12and : CXc Ž.ªEnd Ž. S2to Pin Ž. X are the pinc representations of PinŽ.Ž X . It is well known that S12and S are distinct irreducible representations of PinŽ.. X . 4.1.1. Remark. The pinc representations are often called ‘‘spinor’’ representations of PinŽ. X . The reason for the slightly different terminol- ogy here is that if Ž.M, g is an oriented Riemannian manifold and CM Ž.c is the complex Clifford algebra bundle over M, the existence of global complex vector bundles such that CMŽ.c (End Ž. S when dim M is even or such that CMŽ.c (End Ž. S12[ End Ž. S when dim M is odd, is equivalent to the existence of a global ‘‘spinc ’’ structure in the sense of differential geometry. 4.2. THEOREM. A pinc representation of PinŽ. V m W sets up a Howe correspondence for the dual pairs of Corollary 3.5. DUAL PAIRS IN PinŽ. p, q 537 Proof. One uses the explicit pinc structures given in the proofs of Corollary 3.5Ž. iii and Ž. iv to prove this for the corresponding dual pairs. k k The point is that the ⌳ Ž.W in Corollary 3.5Ž.Ž iii resp. ⌳ Ž.Wc in Corollary y1 3.5Ž.. iv are distinct irreducible representations of ŽGLd Ž W .. Žresp. y1 ŽŽUV W ..and also of their centres. JV m For the dual pairs of Corollary 3.5Ž. i and Ž ii . we need the following lemma which in Howe’s terminologyŽ cf.wx H4. is a ‘‘double commutant’’ version of Theorem 4.3.4.2 inwx H4 . 4.2.1. LEMMA. Let V and W be real non-degenerate inner product spaces and let : PinŽ. V m W ª End Ž.⌺ be a pinc-representation. If : PinŽ V m y1 W.ŽªOVmW . is the double co¨ering map, let GrŽ. W s ŽIdV m 0 y1 OŽ.. W and GrŽ. V s ŽŽ.SO V m IdW .. Then the commutant of 0 ŽGr Ž W .. in End Ž⌺ . is generated as an associati¨e algebra by #Ž grŽ.. V , where gr0Ž. V is the Lie algebra of Gr0Ž. V . Proof. Since the action of GrŽ. W by conjugation factors through OWŽ., GrŽW. OŽW. we have CVccŽ.mW sCVŽ.mW ; by 4.3.4.2 inwx H4 , the algebra OŽW. 0 CVcŽ.mW is generated by soŽ. V s grŽ. V and in particular, CVcŽm .GrŽW. Ž.GrŽW. 0Ž. Ž . W sCVcq mW since gr V : CVcq mW. If dim V m W is even, : CVcŽ.Ž.mW(End ⌺ is an algebra isomor- phism and the lemma follows. If dim V m W is odd, : CVcŽ.mWªEnd Ž.⌺ restricts to an algebra Ž.Ž.⌺ Ž isomorphism : CVcq mW(End and therefore maps CVcq m W.ŽGrŽW. isomorphically onto End ⌺.ŽGrŽW.. Now according to Howewx H4 , a ‘‘double commutant’’ theorem like this implies a ‘‘duality’’ theorem for a dual pair ŽŽ..Gr W , G acting in ⌺ if the Lie algebra of the group G is gr0Ž. V . This proves the theorem for the dual pairs of Corollary 3.5Ž. i and Ž ii . since the Lie algebra of either Gr Ž V .or Gr00Ž. V is gr Ž. V . Remark 4.3. Theorem 4.2 can be false for other fields satisfying Ž.) . For instance if k is a finite field with q elements and q s 3Ž. mod 4 , q 1 y s1 for all g k; then, as in Theorem 3.4Ž.Ž iv , , z .in the centre of y1 p nr2 ŽGLd Ž W ..( GL Ž W . = ޚ2 acts as multiplication by zŽ r<< .on ⌳pŽ.W. In particular, ⌳ pŽ.W and ⌳ pqqy1Ž.W are isomorphic representa- 0 y1 tions of GrŽ. V but non-isomorphic representations of ŽŽ..GLd W . APPENDIX It is well known that one can construct non-trivial double covers of real orthogonal groups using Clifford algebrasŽ cf.wx ABS, K, L. . The same recipe applied to an orthogonal group over an arbitrary field fails unless 538 M. J. SLUPINSKI the field satisfies certain conditions. In this appendix we show that these conditions are precisely the condition Ž.) of Section 2 and calculate the Witt group for fields satisfying Ž.) . A.1. DEFINITION Žseewx ABS, K, Sch.Ž. . Let X, B be a finite dimen- sional, non-degenerate inner product space over a field k and let CXŽ.,B be its Clifford algebra. The Clifford group ⌫Ž.X, B is defined as g 1 ⌫Ž.X, B s Ä4c g CX Ž.,B:cis invertible and c Xcy s X . The main properties of this group are summarised in the following A.2. PROPOSITION Žcf.wx ABS, Sch.Ž. . i If c g ⌫ ŽX, B ., then X,B Ž.cis g y1 an isometry of X, where X, BŽ.Ž.cxscxc ;there is an exact sequence X, B 6 1 ª k* ª ⌫Ž.X, BOX Ž.,Bª1. Ž. Ž. ⌫ Ž.Ž. ii If x g X satisfies B x, x / 0, then x g yX, B and X,Bxis ⌫ Ž. the reflection in the hyperplane orthogonal to x. E¨ery element of q X, B ŽŽ..⌫ Ž. resp. y X, B is a product of an e¨en resp. odd number of such x. T Thus, by DefinitionrProposition 2.4, PinŽ. X, B s Äc g ⌫Ž.X, B : cc s "1.4 If k s ޒ, the map X, B: PinŽ. X, B ª OX Ž.,B is a double covering mapŽ cf.wx ABS. but this is not true in general: y1 A.3. COROLLARY.iŽ. X, BŽ.1lPin Ž X, B .s Ä41, y1 if and only if y1 is not a square in k. Ž.ii The map X, B: Pin Ž. X, B ª OX Ž.,B is surjecti¨e if and only if 2 for all x g X such that BŽ. x, x / 0, there exists g k* such that BxŽ.,x s"1. Proof. PartŽ. i follows immediately from Proposition A.2Ž. i . Part Ž ii . follows from Proposition A.2 and the fact that every element of OXŽ.,Bis a product of hyperplane reflections. A.4. COROLLARY. The map X, B: PinŽ. X, B ª OX Ž.,B is a double co¨ering for all non-degenerate inner product spacesŽ. X, Bo¨er a field k if and only if k satisfies 22 k*rŽ.k*(ޚ2 and y1 f Ž.k*. Proof. If X, B: PinŽ. X, B ª OX Ž.,B is a double covering for all non-degenerate inner product spaces Ž.X, B over a field k, then taking DUAL PAIRS IN PinŽ. p, q 539 2 Ž.ŽX,Bsk,x.where g k*, we see that there is a g k* such that 2 s"1 by Corollary A.3Ž. ii . The converse is immediate from Corollary A.3. 2 For such a field there is a homomorphism<< : k ª Ž.k* given by < q(r²:1 Hs ²y1, : where ²:"1 is the quadratic form "x 2on k, and r and s are positive integers, a priori not uniquely determined by q. This implies that for such fields one has a notion of ‘‘signature’’: A.5. PROPOSITION.iŽ.If the field k satisfies Ž) ., then either y1 is not a sum of squares in k* or y1 is a sum of two squares in k*. Ž.ii If y1 is not a sum of squares, then there are Ž up to isometry . exactly two anisotropic quadratic forms in each dimension n g ގ*. These are n²:1 and n ²y1. :The map : Groth Ž k .ª ޚ gi¨en by Žq .s r y s is well defined and factors to gi¨e an isomorphism : WittŽ. k ( ޚ. Ž.iii If y1 is a sum of two squares, then there areŽ up to isometry. exactly four anisotropic quadratic forms. These are 0,²: 1 , ²y1, : and ²:1 H ²:1.The map : GrothŽ. k ª ޚr4ޚ gi¨en by Ž.Ž.q s r y s mod 4 is well defined and factors to gi¨e an isomorphism : WittŽ. k ( ޚr4ޚ. Ž. ism ␣ 2 ␣ Proof. To prove part i , suppose that y1 s Ýis1 ii, where g k*, and that y1 is not a sum of fewer than m squares. If m ) 2, then 2 2 ␣12q ␣ is itself a squareᎏit cannot be minus a square since then, by division, y1 is a sum of two squares which contradicts the minimality of m. But then y1 is a sum of m y 1 squares which also contradicts the minimality of m. Thus we must have m s 2. A quadratic form is anisotropic if qxŽ.s0«xs0. By Proposition 1.4 of Chapter 2 inwx L , the elements of WittŽ. k are in bijection with the isometry classes of all anisotropic forms. PartŽ. ii now follows as in Proposition 3.2 of Chapter 2 inwx L . PartŽ. ii is clear once we have remarked that in this case, the quadratic forms²: 1 H ²:1 and ²y1 : Hy²:1 are isometric so that there are no anisotropic forms in dimension greater than two. Remark. Both cases occur since y1 is not a sum of squares in ޒ but is a sum of squares in the finite fields ކq if q s 3Ž. mod 4 . 540 M. J. SLUPINSKI REFERENCES wxA J. Adams, Duality for the spin representation, preprint, University of Maryland, 1991. wxABS M. F. Atiyah, R. Bott, and A. Shapiro, Clifford modules, Topology 3 Ž.1963 , 3᎐38. wxH1 R. Howe, -series and invariant theory, in Proc. Sympos. Pure Math., Vol. 33, Part 1, pp. 275᎐285, Amer. Math. Soc., Providence, 1979. wxH2 R. Howe, Remarks on classical invariant theory, Trans. Amer. Math. Soc. 313 Ž.1989 , 539᎐570. wxH3 R. Howe, Dual pairs in physics: Harmonic oscillators, photons, electrons and single- tons, in Lectures in Appl. Math., Vol. 21, pp. 179᎐206, Amer. Math. 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