A Generalized Weil Representation for the finite split orthogonal group O q (2n,2n) , q odd greater than 3
aa r X i v : . [ m a t h . R T ] J un A Generalized Weil Representation for the finite split orthogonalgroup O q (2 n, n ), q odd greater than 3. Andrea Vera Gajardo. ∗ Abstract
We construct via generators and relations a generalized Weil representation for the split orthogonalgroup O q (2 n, n ) over a finite field of q elements. Besides, we give an initial decomposition of therepresentation found. We also show that the constructed representation is equal to the restriction of theWeil representation to O q (2 n, n ) for the reductive dual pair (Sp ( F q ) , O q (2 n, n )) and that the initialdecomposition is the same as the decomposition with respect to the action of Sp ( F q ). Weil representations have proven to be a powerful tool in the theory of group representations. Theyoriginate from a very general construction of A. Weil ([12]), which has as a consequence the existence of aprojective representation of the group Sp(2 n, K ), K a locally compact field. Weil built this representationtaking advantage of the representation theory of the related Heisenberg group, as described by the Stone-von Neumann theorem in the real case ([7]). In particular, these representations have allowed to buildin a universal and uniform way all irreducible complex linear representations of the general linear groupof rank 2 over a finite field ([11]), and later over a local field, except in residual characteristic two ([6]).Weil representations can be constructed in various ways. For instance, they can be constructed viaHeisenberg groups, via constructions of equivariant vector bundles ([4]), via presentations or via dualpairs ([1]), as we shall see later. The method using presentations is accomplished by having a simplepresentation of the group, and then defining linear operators on a suitable vector space which preservethe relations among the generators of the presentation. This idea was originally suggested by Cartier,and used successfully by Soto-Andrade, for symplectic groups Sp(2 n, F q ) ([11]) and complex irreduciblerepresentations of SL(2 , F q ). In the case of Sp(2 n, F q ) he considered this group as a group ”SL(2)” butwith entries in the matrix ring M n ( F q ), and thus he obtained a suitable presentation for the group. Inthis way, he constructed the Weil representation for the symplectic groups Sp(2 n, F q ). In the case ofSp(4 , F q ), Soto-Andrade obtained all the irreducible representations of this group by decomposing thetwo Weil representations associated to the two isomorphy types of quadratic forms of rank 4 over F q .This point of view was generalized and gave rise to the groups SL ε ∗ (2 , A ) for ( A, ∗ ) an involution ringand ε = ± ∈ A . These groups are a generalization of special linear groups SL(2 , K ), where K is a field.They were defined for ε = − ε = 1 in [10].The groups SL ε ∗ (2 , A ) include, among others, symplectic and split orthogonal groups. For instance,if K is a field, A = M n ( K ), ⋄ the transposition of matrices then SL ε = − ⋄ (2 , A ) is the symplectic groupSp(2 n, K ) and SL ε =1 ⋄ (2 , A ) is the split orthogonal group O K (2 n, n ). Thus, these groups allow us to lookat higher rank classical groups as rank two groups, considering them with coefficients in a new ring. ∗ The author was partially supported by Fondecyt Grant 1120578. he procedure used by Soto-Andrade in [11] was approached even in the case of a non semi-simpleinvolutive ring ( A, ∗ ) with non-trivial nilpotent Jacobson radical. In fact, in [2] Guti´errez found a Bruhatpresentation and a generalized Weil representation for SL ε = − ∗ (2 , A m ), where A m = F q [ x ] / h x m i .In [3] Guti´errez, Pantoja and Soto-Andrade build Weil representations in a very general way viagenerators and relations, for the groups SL ε ∗ (2 , A ) for which a “Bruhat” presentation analogue to theclassical one holds. In this way, in order to use this method it is very important to have an adequatepresentation of the group. In [9], Pantoja generalized the classical Bruhat presentation of SL(2 , K ) toSL ε = − ∗ (2 , A ), when A is simple artinian ring with involution.In this work, we construct a generalized Weil representation of finite split orthogonal groups O q (2 n, n ),using the method described in [3]. As mentioned above, this group is naturally the group SL ε =1 ⋄ (2 , M n ( F q )).However one of the results of this paper is to realize this group as a SL ε = − ∼ (2 , M n ( F q )) group, where ∼ isa certain specific involution in M n ( F q ), different from ⋄ . This allows to use the Bruhat-like presentationthat is exhibited in [9] and facilitate significantly technical aspects of the construction. In fact, the resultis more general, and provides an isomorphism between the groups SL + ∗ (2 , M ( A )) and SL −∼ (2 , M ( A )),where ( A , ∗ ) is a unitary involutive ring and ∼ is another involution in A obtained from ∗ .Also, we study the structure of the associated unitary group and using this group we get an initialdecomposition of the representation.In section 6 we show compatibility of the method of Guti´errez, Pantoja and Soto-Andrade with theoryof dual pairs. For this, we prove that the representation of O q (2 n, n ) constructed using this method isequal to the restriction of the Weil representation to O q (2 n, n ) for the dual pair (Sp(2 , k ) , O q (2 n, n )).Also, we prove that the initial decomposition mentioned above is the same as decomposition with respectto the action of Sp(2 , k ) via the Weil representation. SL ε ∗ (2 , A ) . Let ( A, ∗ ) be a unitary ring with an involution ∗ . We can extend the involution ∗ in A to the ring M ( A )putting T ∗ = a bc d ! ∗ = a ∗ c ∗ b ∗ d ∗ ! . We consider ε = ± ∈ A and J ε = ε ! ∈ M ( A ) . Let us denote by H ε the associated ε -hermitian form defined by the matrix J ε . Definition 2.1.
The group SL ε ∗ (2 , A ) is the set of all automorphisms g of the A -module M = A × A such that H ε ◦ ( g × g ) = H ε . In matrix form:SL ε ∗ (2 , A ) = { T ∈ M ( A ) | T J ε T ∗ = J ε } Remark 2.2.
In [10] it is shown that a matrix a bc d ! ∈ M ( A ) is in SL ε ∗ (2 , A ) if and only if thefollowing equalities hold:1. ab ∗ = − εba ∗ ;2. cd ∗ = − εdc ∗ ;3. a ∗ c = − εc ∗ a ;4. b ∗ d = − εd ∗ b ;5. ad ∗ + εbc ∗ = a ∗ d + εc ∗ b = 1. e note that if ( A, ⋄ ) is the matrix ring M m ( F q ) with the transpose involution ⋄ , then SL − ⋄ (2 , A ) isthe symplectic group Sp(2 m, F q ) defined over F q . On the other hand SL ⋄ (2 , A ) gives the split orthogonalgroup O q ( m, m ).In what follows we put SL + ∗ (2 , A ) = SL ∗ (2 , A ), SL −∗ (2 , A ) = SL − ∗ (2 , A ) and ⋄ always will be thetranspose involution in a matrix ring.Let ( A , ∗ ) be a unitary ring with involution ∗ and A = M ( A ). Let us consider the matrix J = − ! ∈ A × , which satisfies J − = J ∗ = − J . Using the matrix J we can define a newinvolution in A , namely, a ∼ = Ja ∗ J − .Let us consider M ( A ) provided with the involutions ∗ and ∼ inherited from A. Theorem 2.3.
The groups SL + ∗ (2 , A ) and SL −∼ (2 , A ) are isomorphic. Proof 2.4.
Let U = J J ! ∈ GL ( A ) . A direct computation proves that T ∼ U = UT ∗ for all T ∈ M ( A ) . It is clear that
T J − T ∼ = J − if and only if T J − UT ∗ = J − U . Then we must showthat J − U and J + are equivalent. In fact, the (orthogonal) matrix P = J ! ∈ M ( A ) satisfies P J + P ∗ = J − U .Although the split orthogonal group is naturally a “SL + ”- group, in practical terms it is better tolook at it as a “SL − ”-group, because this fact will greatly facilitate technical aspects. Corollary 2.5.
The split orthogonal group O q (2 n, n ) is isomorphic to the group SL −∼ (2 , M n ( F q )),where the involution ∼ in M n ( F q )is given by a ∼ = J n a ⋄ J − n . ( J n = I n − I n ! ∈ M n ( F q )) . Proof 2.6.
Taking ( A , ∗ ) as the involutive ring ( M n ( F q ) , ⋄ ) and using the theorem (2.3) we get thatthe groups SL + ⋄ (2 , M n ( F q )) and SL −∼ (2 , M n ( F q )) are isomorphic. SL ε ∗ (2 , A ) Let A be a unitary ring with involution ∗ .We will write A sε, ∗ to denote the set of all ε - symmetricelements in A respect to the involution ∗ . Namely; A sε, ∗ = { a ∈ A | a ∗ = − εa } . In order to facilitate the notation, we put A s + , ∗ = A s , ∗ and A s − , ∗ = A s − , ∗ .Let us consider h t = t t ∗− ! ( t ∈ A × ) , w = ε ! , u s = s ! ( s ∈ A sε, ∗ ) Definition 3.1.
We will say that SL ε ∗ (2 , A ) has a Bruhat presentation if it is generated by the aboveelements with defining relations:1. h t h t ′ = h tt ′ , u s u s ′ = u s + s ′ ;2. w = h ε ;3. h t u s = u tst ∗ h t ;4. wh t = h t ∗− w ;5. wu t − wu − εt wu t − = h − εt , t ∈ A × ∩ A sε, ∗ . emark 3.2. Observing the last relation, we note that in order to have a Bruhat presentation forSL ε ∗ (2 , A ) is necessary that A × ∩ A sε, ∗ = ∅ . In [9] it is proved that if A is a simple artinian ring with involution ∗ that either is infinite or isomorphicto the full matrix ring over F q with q >
3, then the group SL −∗ (2 , A ) has a Bruhat presentation.Thus, the group SL −∼ (2 , M n ( F q )) mentioned in Corollary (2.5) has a Bruhat presentation if q > O q (2 n, n ) . In this section, our aim is to construct a Weil representation for the split orthogonal group seen asthe group SL −∼ (2 , M n ( F q )). One way to this goal is to construct the representation using the Bruhatpresentation. For this purpose we will use the result that follows ([3]).Let A be a ring with an involution ∗ . Let us suppose that the ring A is finite and that the group G = SL ε ∗ (2 , A ) has a Bruhat presentation. Let M be a finite right A -module and let us consider thefollowing data:1. A bi-additive function χ : M × M −→ C × and a character α ∈ b A × such that for all x, y ∈ M, t ∈ A × :(a) χ ( xt, y ) = α ( tt ∗ ) χ ( x, yt ∗ )(b) χ ( y, x ) = χ ( − εx, y )(c) χ ( x, y ) = 1 for all x ∈ M ⇒ y = 02. A function γ : A sε, ∗ × M −→ C × such that for all s, s ′ ∈ A sε, ∗ , x, z ∈ M , r ∈ A × , t ∈ A sε, ∗ ∩ A × :(a) γ ( s + s ′ , x ) = γ ( s, x ) γ ( s ′ , x )(b) γ ( s, xr ) = γ ( rsr ∗ , x )(c) γ ( t, x + z ) = γ ( t, x ) γ ( t, z ) χ ( x, zt )3. c ∈ C × such that c | M | = α ( ε ), and for all t ∈ A sε, ∗ ∩ A × the following equality holds: X y ∈ M γ ( t, y ) = α ( εt ) c Theorem 4.1. (Guti´errez, Pantoja and Soto-Andrade, [3])
Let M be a finite right A -module. Denote L ( M ) the vector space of all complex-valued functions on M , endowed with the inner product withrespect to the counting measure on M . Set:1. ρ ( h t )( f )( x ) = α ( t ) f ( xt ), f ∈ L ( M ) , t ∈ A × , x ∈ M ;2. ρ ( u s )( f )( x ) = γ ( s, x ) f ( x ), f ∈ L ( M ) , b ∈ A sε, ∗ , x ∈ M ;3. ρ ( w )( f )( x ) = c P y ∈ M χ ( − εx, y ) f ( y ), f ∈ L ( M ) , x ∈ M (where α denotes the complex conjugate of the character α ). These formulas define a unitary linearrepresentation ( L ( M ) , ρ ) of G , called the generalized Weil representation of G associated to the data( M, α, γ, χ ). Remark 4.2.
Let us note that this definition contains the classic Weil representation of SL ( K ), where K is a field (see [11], for instance). n what follows, we will focus on finding the necessary data to construct a generalized Weil represen-tation for O q (2 n, n ).From now on we put k = F q , A = M n ( k ) and we consider q odd greater than 3.We will apply Theorem (4.1) to the group SL −∼ (2 , A ) ∼ = O q (2 n, n ). To do this, we recall the fol-lowing fact. Let E be a finite dimensional vector space over a field K . In [5], the authors describe acorrespondence between the linear anti-automorphisms of End K ( E ) and the equivalence classes of nondegenerate bilinear forms on E modulo multiplication by a factor in K × . Under this correspondence, K -linear involutions on End K ( E ) correspond to non degenerate bilinear forms which are either symmet-ric or skew-symmetric. Let B be a non degenerate bilinear form. The aforementioned correspondenceassociates B with σ B , where σ B is a linear anti-automorphism in End K ( E ) defined by the followingequality; B ( f ( x ) , y ) = B ( x, σ B ( f ) y ) , f ∈ End K ( E ) , x, y ∈ E. (1)Now, let V a vector space of k -dimension 2 n . We fix a basis for V in order to put M n ( k ) ≃ End k ( V ) . Let < , > : V × V −→ k be the non degenerate symmetric bilinear form given by the standard dotproduct. We consider the non degenerate skew-symmetric bilinear form [ , ] : V × V −→ k , given by[ x, y ] = < x, yJ n > . According to the correspondence between involutions and non degenerate bilinear forms described above,the symmetric bilinear form < , > corresponds to the transpose involution ⋄ . Similarly, the skew-symmetric bilinear form [ , ] corresponds to the new involution ∼ . That is, for all x, y ∈ V and a ∈ A ; < xa, y > = < x, ya ⋄ > (2)[ xa, y ] = [ x, ya ∼ ] (3)Now, let ψ be a non trivial character of k + . Using the notation above let us consider:1. M the right A − module V with the following action:( x, y ) a = ( xa, ya ) a ∈ A, x, y ∈ V. χ : M × M −→ C × , χ (( x, y ) , ( v, z )) = ψ ([ x, z ] − [ y, v ]) . α the trivial character of A × . γ : A s − , ∼ × M −→ C × , γ ( u, ( x, y )) = ψ ([ xu, y ]) . Lemma 4.3.
For all u ∈ A × ∩ A s − , ∼ , the map Q u : V −→ k given by Q u (( x, y )) = [ xu, y ] is a non degenerate split quadratic form. Furthermore, for u, u ′ ∈ A × ∩ A s − , ∼ thequadratic forms Q u and Q u ′ are equivalent. Proof 4.4.
Let λ ∈ k, ( x, y ) , ( v, z ) ∈ V . Clearly Q u ( λ ( x, y )) = λ Q u (( x, y )). We will prove that B (( x, y ) , ( v, z )) = Q u (( x + v, y + z )) − Q u (( x, y )) − Q u (( v, z ))is a symmetric non degenerate bilinear form. We have; B (( x, y ) , ( v, z )) = [ xu + vu, y + z ] − [ xu, y ] − [ vu, z ] = [ xu, z ] + [ vu, y ] . ow; B (( x, y ) + ( r, t ) , ( v, z )) =[( x + r ) u, z ] + [ vu, ( y + t )]=[ xu, z ] + [ ru, z ] + [ vu, y ] + [ vu, t ]= B (( x, y ) , ( v, z )) + B (( r, t ) , ( v, z )) B ( λ ( x, y ) , ( v, z )) =[ λxu, z ] + [ vu, λy ]= λ [ xu, z ] + λ [ vu, y ]= λB (( x, y ) , ( v, z )) . Then, B is a symmetric bilinear form.Let us suppose that B (( x, y ) , ( v, z )) = 0 for all ( v, z ) ∈ V . If we choose v = 0, then [ xu, z ] = 0 forall z ∈ V . Since [ , ] is non degenerate and u is invertible, we get x = 0. Similarly y = 0. Therefore, B is non degenerate.Now, if u, u ′ ∈ A × ∩ A s − , ∼ then uJ ⋄ n and u ′ J ⋄ n are invertible skew symmetric matrices. In fact if u ∈ A s − , ∼ then u ∼ = J n u ⋄ J − n = u . Also J ⋄ n = − J n , so we get that ( uJ ⋄ n ) ⋄ = J n u ⋄ = uJ n = − uJ ⋄ n . Thus uJ ⋄ n and u ′ J ⋄ n represent a non degenerate skew symmetric bilinear form, therefore theyare equivalent. So, there exists j ∈ A × such that uJ ⋄ n = ju ′ J ⋄ n j ⋄ . Thus, Q u ′ (( xj, yj )) = [ xju ′ , yj ]= Q u (( x, y ))If we choose u = I n , the quadratic form Q u is represented by the matrix − J n J n ! . Thus, Q u is split. Theorem 4.5.
The data (
M, α, γ, χ ) describe a Generalized Weil Representation for G = O q (2 n, n ).Furthermore, this representation is independent of the choice of the character ψ . Proof 4.6.
We will check that χ satisfies the corresponding conditions.Let ( x, y ) , ( v, z ) ∈ M, a ∈ A. (a) χ (( x, y ) a, ( v, z )) = ψ ([ xa, z ] − [ ya, v ])= ψ ([ x, za ∼ ] − [ y, va ∼ ]) , by (3) = χ (( x, y ) , ( v, z ) a ∼ ) . (b) χ (( v, z ) , ( x, y )) = ψ ([ v, y ] − [ z, x ])= ψ ([ x, z ] − [ y, v ])= χ (( x, y ) , ( v, z )) . (c) Let us suppose that χ (( x, y ) , ( v, z )) = 1 for all ( v, z ) ∈ M . If v = 0, then ψ ([ x, z ]) = 1 for all z ∈ V . If x = 0, then [ x, · ] : V −→ k is a non trivial linear functional. Therefore it is surjective. Let λ ∈ k such that ψ ( λ ) = 1, and t = t ( λ ) ∈ V such that λ = [ x, t ], then we get the following contradiction:1 = ψ ([ x, t ]) = ψ ( λ ) . Therefore x = 0, and similarly y = 0. ow, we will prove that γ satisfies the corresponding properties. Let u, u ′ ∈ A s − , ∼ , a ∈ A × ,( x, y ) , ( v, z ) ∈ M ; (a) γ ( u + u ′ , ( x, y )) = ψ ([ xu + xu ′ , y ])= ψ ([ xu, y ]) ψ ([ xu ′ , y ])= γ ( u, ( x, y )) γ ( u ′ , ( x, y )) . (b) γ ( u, ( x, y ) a ) = ψ ([ xau, ya ])= ψ ([ xaua ∼ , y ])= γ ( aua ∼ , ( x, y )) . (c) γ ( u, ( x, y ) + ( v, z )) = ψ ([( x + v ) u, y + z ])= ψ ([ xu, y ]) ψ ([ xu, z ]) ψ ([ vu, y ]) ψ ([ vu, z ])= γ ( u, ( x, y )) γ ( u, ( v, z )) ψ ([ x, zu ] − [ y, vu ])= γ ( u, ( x, y )) γ ( u, ( v, z )) χ (( x, y ) , ( v, z ) u ) . Now, we must choose c ∈ C × satisfying c | M | = 1 and show that for u ∈ A × ∩ A s − , ∼ the followingequality holds: X ( x,y ) ∈ M γ ( u, ( x, y )) = X x,y ∈ V ψ ([ xu, y ]) = 1 c . According to the lemma 4.3, we know that P ( x,y ) ∈ M γ ( u, ( x, y )) is a Gauss sum associated to a splitquadratic form in a vector space of even dimension 4 n . This sum is calculated, for instance, in [11]. Infact, X ( x,y ) ∈ M γ ( u, ( x, y )) = q n , We choose c = 1 q n . Thus, X ( x,y ) ∈ M γ ( u, ( x, y )) = 1 c . Now, let ψ and ψ be two non trivial characters of k + . Let us prove that the corresponding repre-sentations are isomorphic.Let λ ∈ k × such that ψ ( r ) = ψ ( λr ) for all r ∈ k . Let ( L ( M ) , ρ ) and ( L ( M ) , ρ ) the Weilrepresentations obtained from ψ and ψ respectively. Then, the linear automorphismΨ : L ( M ) −→ L ( M ) given by (Ψ f )( x, y ) = f ( x, λy ) is a isomorphism between the representations( L ( M ) , ρ ) and ( L ( M ) , ρ ). Definition 5.1.
The group U( γ, χ ) is the group of all A -linear automorphisms β of M such that:1. γ ( u, β ( x, y )) = γ ( u, ( x, y )) for all u ∈ A sε, ∗ , ( x, y ) ∈ M. χ ( β ( x, y ) , β ( v, z )) = χ (( x, y ) , ( v, z )) for all ( x, y ) , ( v, z ) ∈ M. In what follows we will denote U( γ, χ ) simply by U. ollowing the idea of [3], if we know the structure of the group U and the set of its irreduciblerepresentations, we can find an initial decomposition of the Weil Representation in the sense that wedo not know if the components obtained are irreducible. In what follows, we make this decompositionexplicit.For β ∈ U and x ∈ M we put β.x = β ( x ). The group U acts naturally on L ( M ). That is to say theaction is given by: σ : U −→ Aut C ( L ( M )) ,σ β ( f )( x ) = f ( β − .x )In [3] it is shown that the natural action of U on L ( M ) commutes with the action of the WeilRepresentation.Let b U be the set of the irreducible representations of U. We consider the isotypic decomposition of L ( M ) with respect to U: L ( M ) ∼ = M ( V π ,π ) ∈ b U n π V π . Since n π = dim C ( Hom U ( V π , L ( M ))) = dim C ( Hom U ( L ( M ) , V π )), we can write this decomposition inthe following way: L ( M ) ∼ = M ( V π ,π ) ∈ b U ( Hom U ( L ( M ) , V π ) ⊗ C V π . )If we put m π = dim C ( V π ), we get; L ( M ) ∼ = M ( V π ,π ) ∈ b U m π Hom U ( L ( M ) , V π . )If ( V π , π ) ∈ b U and β ∈ U, we denote by π β the map π ( β ) : V π −→ V π . The space Hom U ( L ( M ) , V π ) isformed by linear functions Θ : L ( M ) −→ V π such that for any β ∈ UΘ ◦ σ β = π β ◦ Θ . (4)Let us consider the Delta functions { e x | x ∈ M } and the map θ : M −→ V π such that θ ( x ) = Θ( e x )for all x ∈ M . Since σ β ( e x ) = e β.x , condition (4) becomes: θ ( β.x ) = π β ◦ θ ( x ) . (5)Conversely, let θ : M −→ V π satisfying (5). We extend linerarly and we get a map Θ : L ( M ) −→ V π such that (4) holds.Thus, we can see the space Hom U ( L ( M ) , V π ) as the function space formed by maps θ : M −→ V π such that θ ( β.x ) = π β ◦ θ ( x ) for all β ∈ U , x ∈ M . The group G = SL ε ∗ (2 , A ) acts on this space viathe Weil representation, using the same formulas as defined in Theorem (4.1). Similarly, it is possibleto define the natural action of the group U in this space, because- like L ( M )- it is formed for functionswith domain M . et ρ denote the Weil action of G on L ( M ) and b ρ the Weil action of G on L ( V π ,π ) ∈ b U m π Hom U ( L ( M ) , V π ).Because of how we define the Weil representation, there exist scalars K g ( x, y ) ∈ C depending only on g ∈ G and x, y ∈ M such that for all f ∈ L ( M ), Λ ∈ L ( V π ,π ) ∈ b U m π Hom U ( L ( M ) , V π ) the followingstatements holds: ρ g ( f ) = X y ∈ M K g ( · , y ) f ( y ); (6) b ρ g (Λ) = X y ∈ M K g ( · , y )Λ( y ) . (7)In this way, we get: Lemma 5.2. ( L ( M ) , ρ ) and ( L ( V π ,π ) ∈ b U m π Hom U ( L ( M ) , V π ) , b ρ ) are isomorphic representations of G . Proof 5.3.
The linear isomorphism between L ( M ) and L ( V π ,π ) ∈ b U m π Hom U ( L ( M ) , V π )is an isomor-phism between representations.Finally, we have: Proposition 5.4.
The space
Hom U ( L ( M ) , V π ) is invariant under the Weil action of G . Proof 5.5.
Let g ∈ G , θ ∈ Hom U ( L ( M ) , V π ), β ∈ U, x ∈ M :( b ρ g θ )( β.x ) = σ β − ( b ρ g θ )( x ) , (definition of σ β )= b ρ g ( σ β − θ )( x )= b ρ g ( π β ◦ θ )( x ) , ( σ β − ( θ ) = π β ◦ θ )= π β ( b ρ g θ ( x ) . )The last equality holds because (7).Now, having made the decomposition above explicit, our purpose is to obtain an initial decompositionfor our particular case G = O q (2 n, n ) ∼ = SL −∼ (2 , M n ( k )). For this it is enough to know the structure ofthe group U and the set of irreducible representations. Remark 5.6.
We note that since in our case A = M n ( k ) and A s − , ∼ ∩ A × = ∅ , the first condition indefinition (5.1) implies the second one (see [3]). Theorem 5.7.
Let γ and χ be the functions defined above. Then,U( γ, χ ) ∼ = SL ( k ). Proof.
Let β ∈ U( γ, χ ). In particular β is k − linear, therefore we can suppose that β ∈ M n ( k ). We canwrite the action of A on M in matrix language as follows:( x y ) a a ! = ( xa ya ) x, y ∈ V, a ∈ A. Since β is A − linear we have that β ( x, y ) a = β ( xa, ya ). In matrix language; β a a ! = a a ! β. (8) et β , β , β , β ∈ A such that β = β β β β ! . Then, using (8) we get that each of these blocksmust be scalar. Thus, there are b , b , b , b ∈ k such that β = b I n b I n b I n b I n ! and hence γ ( u, β ( x, y )) = ψ ([( b x + b y ) u, b x + b y ]) . Let us note that the bilinear form ( x, y ) [ xu, y ] is skew symmetric for all u ∈ A s − , ∼ , hence [ xu, x ] = 0for all x ∈ V, u ∈ A sym . Thus for all x, y ∈ V, u ∈ A s − , ∼ γ ( u, β ( x, y )) = ψ (( b b − b b )[ xu, y ])= ψ ([ xu, y ])= γ ( u, ( x, y )) . Consequently ψ (( b b − b b − xu, y ]) = 1 for all x, y ∈ V, u ∈ A s − , ∼ . From this last equality it follows that b b − b b = 1. In fact, let u ∈ A s − , ∼ ∩ A × , x = 0 and let ussuppose b b − b b − = 0.The map F x,u : V −→ k given by F x,u ( z ) = [( b b − b b − xu, z ] is a non trivial linear functionaland therefore is surjective. Let λ ∈ k such that ψ ( λ ) = 1 and z = z ( λ ) ∈ V such that λ = [( b b − b b − xu, z ]. Then ψ ( λ ) = ψ ([( b b − b b − xu, z ]) = 1 . This contradicts our assumptionand therefore our result follows.Thus, for our case, we get an initial decomposition of the Weil Representation ( L ( M ) , ρ ). We expectto address the question about irreducibility elsewhere. In this section we will prove that the representation ( L ( M ) , ρ ) of O q (2 n, n ) constructed in section 4 isequal to the restriction of the Weil representation to O q (2 n, n ) for the dual pair (Sp(2 , k ) , O q (2 n, n )).Also, we will prove that the initial decomposition described above is the same as decomposition withrespect to the action of Sp(2 , k ) via the Weil representation.Let J n = I n − I n ! ∈ M n ( k ) and F = J n − J n ! ∈ M n ( k ) . The matrix F defines thefollowing non-degenerate split symmetric bilinear form in V = k n ( u, v ) = v t F u u, v ∈ V The group G of isometries of this form is isomorphic to the split orthogonal group O q (2 n, n ). As before,set a ∼ = J n a t J n a ∈ M n ( k ) . A direct calculation shows that the following matrices belong to the group G : h a = a
00 ( a ∼ ) − ! , w = I n − I n ! , u s = I n s I n ! ( a ∈ M n ( k ) × , s = s ∼ ∈ M n ( k )).Therefore G = SL −∼ (2 , M n ( k ). et V = k and W = Hom ( V , V ). The following formula defines a non-degenerate symplectic formon W . ≪ w , w ≫ = tr ( w F w t J ) ( w , w ∈ W )The group G acts on W by g ( w ) = w g − ( g ∈ G, w ∈ W ) . This action preserves the symplectic form ≪ · , · ≫ . In fact, since g ∈ G , ≪ w g − , w g − ≫ = tr ( w g − F ( g − ) t w t J ) = tr ( w F w t J ) = ≪ w , w ≫ . Let X = { ( x, | x ∈ M , n ( k ) } ; Y = { (0 , y ) | y ∈ M , n ( k ) } . Then W = X ⊕ Y is a complete polarization. We will consider the Schr¨odinger model of the Weilrepresentation of Sp( W ) attached to the above complete polarization realized on L ( X ) as in [1]. Let( L ( X ) , ω ) such representation.We identify X with M , n ( k ) in the canonical way X ∋ ( x, ! x ∈ M , n ( k ) . Remark 6.1.
Let us note that the module M in section 4 is canonically isomorphic to X . Consequentlythe spaces L ( M ) and L ( X ) are also isomorphic.Let ψ a non-trivial character of the additive group k + . For all x ∈ X , y ∈ Y it is clear that h a ( x ) = xh − a ∈ X and h a ( y ) = yh − a ∈ Y , then the matrix h a preserves X and Y . Also, det ( h a | X ) ∈ k × .Thus, proposition 34 in [1] shows that: ω ( h a ) f ( x ) = f ( xa ) ( f ∈ L ( X ))Thus, ω ( h a ) = ρ ( h a ) . Now, let us see the action of ω on u s . The matrix u s acts trivially on Y and on W/Y . Therefore,proposition 35 in [1] shows that: ω ( u s ) f ( x ) = ψ ( ≪ xc ( − u s ) , x ≫ ) f ( x ) , where c ( − u s ) = − s/
20 0 ! ∈ M n ( k ) is the Cayley transform for − u s .Let x = x x ! ∈ X , x , x ∈ k n . Then, ≪ xc ( − u s ) , x ≫ = x sJ n x t . In order to prove that ω ( u s ) = ρ ( u s ) we have to check that[ x s, x ] = ≪ xc ( − u s ) , x ≫ , where [ , ] is the symplectic form defined in section 4. In fact,[ x s, x ] = [ x , x s ] = − [ x s, x ] = x sJ n x t t is clear that the matrix w maps X bijectively onto Y and Y onto X , and w = −
1. Then, usingproposition 36 of [1] we get: ω ( w ) f ( x ) = 1 p | X | X x ′ ∈ X ψ ( ≪ w ( x ) , x ′ ≫ ) f ( x ′ )Thus, in order to prove that ω ( w ) = ρ ( w ) we have to check that χ ( x, x ′ ) = ψ ( ≪ xw − , x ′ ≫ ) . Let x = x x ! , x ′ = x ′ x ′ ! , x , x , x ′ , x ′ ∈ k n . So, ψ ( ≪ xw − , x ′ ≫ ) = ψ ( x J n ( x ′ ) t − x J n ( x ′ ) t ) . On the other hand, χ ( x, x ′ ) = ψ ([ x , x ′ ] + [ x ′ , x ])= ψ (( x x ) − J n J n ! ( x ′ ) t ( x ′ ) t ! )= ψ ( x J n ( x ′ ) t − x J n ( x ′ ) t )) . Thus, we have showed that the representation constructed in section 4 is equal to the restriction of theWeil representation to O q (2 n, n ) for the dual pair (Sp(2 , k ) , O q (2 n, n )).Furthermore, since an element g ∈ Sp(2 , k ) = SL ( k ) preserves X and Y and det ( g | X ) ∈ k × , usingproposition 34 in [1] we get that the group SL ( k ) acts on L ( X ) as follows: ω ( g ) f ( x ) = f ( g − x ) g ∈ SL ( k ) , f ∈ L ( X ) , x ∈ X. Therefore, the initial decomposition in section 5 is the same as the decomposition with respect to theaction of SL ( k ) via the Weil representation. Acknowledgments:
The author would like to thank Jos´e Pantoja, Jorge Soto, Luis Guti´errez andAnne-Marie Aubert for their never ending support and willingness. eferences [1] A.Aubert, T. Przebinda, A reverse engineering approach to Weil Representation. , To appear inCEJM (2012) .[2] L. Guti´errez,
A Generalized Weil Representation for SL ∗ (2 , A m ) , where A m = F q [ x ] / h x m i , Journalof Algebra, (2009), 42–53.[3] L. Guti´errez, J.Pantoja and J. Soto Andrade, On Generalized Weil Representations over InvolutiveRings , Contemporary Mathematics, (2011), 109–122.[4] L. Guti´errez, J. Pantoja and J. Soto Andrade,
Geometric Weil representations for star-analoguesof
SL(2 , k ), Contemporary Mathematics, (2011), 211–226.[5] M. Knus, A. Merkurjev, M. Rost and J. Tignol,
The Book of Involutions , AMS ColloquiumPublications, (1998).[6] P. Kutzko, The exceptional representations of GL2 , Composite Math, (1984), 3–14.[7] G. Lion and M. Vergne, The Weil Representation, Maslov Index and Theta Series , Progr. Math.,Birkhauser, Basel, (1980).[8] J. Pantoja and J. Soto-Andrade, A Bruhat decomposition of the group Sl ∗ (2 , A ), Journal of Algebra, (2003), 401–412.[9] J. Pantoja, A presentation of the group Sl ∗ (2 , A ) , A a simple artinian ring with involution ,Manuscripta math., (2006), 97–104.[10] J. Pantoja and J. Soto-Andrade, Bruhat presentations for ∗− classical groups , Communications inAlgebra, (2009), 4170–4191.[11] J. Soto-Andrade, Repr´esentations de certain groupes symplectiques finis , Bull. Soc. Math. FranceMem., (1978).[12] Weil A.,
Sur certains groupes d’op´erateurs unitaires , Acta Math., (1964), 143–211.Andrea Vera Gajardo.Universidad de Santiago de Chile.Avenida Libertador Bernardo O’Higgins 3363, Estaci´on Central, Santiago, Chile.email: [email protected](1964), 143–211.Andrea Vera Gajardo.Universidad de Santiago de Chile.Avenida Libertador Bernardo O’Higgins 3363, Estaci´on Central, Santiago, Chile.email: [email protected]