Spin Calogero Particles and Bispectral Solutions of the Matrix KP Hierarchy

SPIN CALOGER O P AR TICLES AND BISPECTRAL SOLUTIONS OF THE MA TRIX KP HIERARCHY MAAR TEN BERG VEL T, MICHAEL GEKHTMAN, AND ALEX KASMAN Abstract. P airs of n × n matrices wh ose c ommut ator differ from t he ide nt ity b y a matrix o f rank r a re used to construct bisp ectral di fferen tial op erators with r × r matrix coefficients satisfyi ng the Lax equations of the Matrix KP hierarch y . Moreov er, the b isp ectral in v olution on t hese operators has dynami- cal significance for the spin Calogero particles system whose phase space such pairs represent . In the case r = 1, this reproduces we ll-known results o f Wilson and others f rom the 1990’s relating (spinless) Calogero-Moser systems to the bisp ectralit y of (scalar) different ial op erators. This new class of pairs ( L, Λ) of bisp ectral matrix differential op erators is different than those previously studied in that L acts from the left, but Λ f rom the r igh t on a common r × r eigenmatrix. 1. Introduction 1.1. Bac kground. Let (1) CM n = { ( X , Z ) ∈ M n × n | ra nk ([ X, Z ] − I ) = 1 } be the se t of pairs o f complex n × n matrices whose commutator differs fro m the ident ity by a matrix of rank one. This space a rises na turally in the study of the int egrable Calo gero-Mos er-Sutherland particle system [vDV00, Pol06]. In particu- lar, the eige n v a lues o f the time dependent matrix X + itZ i − 1 mov e acco rding to the i th Ha miltonian of this in tegrable hierar c hy and even allows the con tin uation of the dynamics through collisio ns [K KS78, Wil9 8]. The KP hiera rch y is the collection of nonlinear par tia l differential e quations (2) ∂ ∂ t i L = [( L i ) + , L ] , i = 1 , 2 , 3 . . . . for a monic pseudo-differential op erator L o f order one whose co efficients are scalar functions dep ending on the time v ariables t i [SS83, SW85]. If the co efficien ts of L ar e further assumed to b e rational functions of t 1 which v anish a s t 1 → ∞ , t hen the solutions can be wr itten in terms o f the matrices in CM n and the po les move accor ding to th e dynamics of the Caloger o-Moser- Sutherland s ystem [AMM77, Kri79, Shi94, Wil98]. This w as in terpreted as a specia l case of a mor e general rela tionship betw een “r ank one conditions” and the KP hier arch y in [GK06]. Although it seems at firs t to be quite different in nature, having no obvious dynamical in terpretation, the bisp ectral problem [HK98] turns out to b e another asp ect of this relationship b et ween the KP hier arch y a nd the Caloger o-Moser- Sutherland particle system. As origina lly formulated in [DG86], the bisp ectral problem s e eks to find scalar co efficient ordinar y differential ope rators L and Λ in Date : October 30, 2018. 1 2 MAAR TEN BERGVEL T, MICHAEL GEKHTMAN, AND ALEX KASMAN the v ariables x and z resp ectively such t hat there is a commo n eigenfunction ψ ( x, z ) satisfying the eigenv alue equations (3) Lψ = p ( z ) ψ Λ ψ = π ( x ) ψ for non-consta n t functions p and π . As it turns out, if one additionally requires the op erato r L to c o mm ute with another ordinary differential op era tor of rela tiv ely prime order, the solutions to the bis p ectra l problem are ex actly the same as the rational solutions to the K P hierarch y ment ioned ab ov e [Wil93]. (Specifically , up to trivial renormaliz a tions, the bisp ectral o per a tors are the ordina r y differential op er- ators that comm ute with the pseudo-differe ntial op erato r L with the ident ification x = t 1 .) Moreov er, the bisp ectral prop erty for these op erators is a manifestation of the in volution on CM n given by ( X , Z ) 7→ ( Z ⊤ , X ⊤ ) which linear izes the dynamics of the particle system [K as95, Wil98]. In [Wil98] and the c o nference proceeding s [Wil00], Wilson sugges ts that the corres p ondence should generaliz e naturally to the cas e in which the “rank o ne co n- dition” (1) is repla ced with a “rank r condition” . Indeed, v arious a uthors ha ve demonstrated that a similar relationship exists betw een the matrice s who s e com- m utator differs f rom the identit y b y a matrix of rank r (1 < r ≤ n ), the “spin generaliza tion” o f the integrable particle s ystem, a nd matrix generaliza tio ns of the KP hierarch y . In particular, the spin genera lized sys tem [GH84] w as s hown to be related to the matrix KP equation in [KBBT95] and to the m ulti-comp o nen t KP hierarch y in [BZN07], a nd rather genera l r ank r conditions were shown to pro duce solutions to the matrix p otential KP hierarchy in [DMH07 ]. None of these, ho w- ever, has specifically addre ssed the question of whether and ho w the results r elating to bispe c tr alit y generalize to the ma tr ix ca se. 1.2. Outline. Section 1.3 will introduce a v ersion of the bis p ectra l problem in which matrix differential opera tors act on a common eigenmatrix from oppos ite sides. The main result of this pa per will b e to demonstrate that this formulation allows for the generalization o f the results on bisp ectra lity to the spin version of the particle system and the matrix K P hie r arch y . Section 2 introduces the g eneralization of (1) to the case of ar bitrary r ank a nd relates it to the dynamics of the spin Caloger o particle system. Specia l a tten tion is paid to the blo ck deco mpositio ns of the asso ciated o p era tors co rresp onding to the generalized eigenspace s of the matrix Z . A wa ve function and pseudo- differen tial op erator are constructed from a choice of n × n matrices s atisfying the rank r co ndition in Section 3. This r × r matrix pseudo-differential o per ator is sho wn to satisfy the La x equation of the KP hierarch y (2). A key compo nen t of the pro o f is the explicit co nstruction of an r n -dimensional space of finitely supp orted distr ibutio ns in the sp ectra l para meter which annihilate the wa v e function. Several obvious group actio ns on the space of matrices sa tisfying the rank r condition are inv estigated in Section 4 with emphasis on their effect on the corre- sp onding KP s olution. O f sp ecial interest is the bisp ectral inv olution which has the effect of exchanging v ariables and transp osing the wa v e function. The main theorem is the construction in Section 5 o f commutativ e rings of matrix differential op erato r s in x a nd z a nd the demons tr ation that t hey hav e the K P w a ve function as a common eigenfunction. SPIN CALOGER O AND BISPECTRAL MA TRIX KP 3 A final sec tion cont ains closing remar ks a nd lists problems for future research on this topic. 1.3. Notation and Matrix Bisp ectr alit y. W e will make use of the nota tion M L and M R to distinguish betw een the cases in whic h the o per ator M is acting from the left or the right, resp ectively . So, for instance, if M a nd P are b oth r × r ma tr ices, then [ M , P ] = ( M L − M R ) P. Similarly , if (4) L = N X i =0 M i ( x ) ∂ i x is an ordinary differen tial opera tor in x o f degree N with co efficients M i ( x ) that are r × r matrices and ψ ( x ) is an r × r matr ix function we define L L ( ψ ) = L ( ψ ) = N X i =0 M i ( x )  ∂ i ∂ x i ψ ( x )  , as usual. Ho wev er, the op erato r can also act fro m the r ight L R ( ψ ) = N X i =0  ∂ i ∂ x i ψ ( x )  M i ( x ) . Equiv alen tly , if w e denote by L ⊤ the differential operator with co efficients M ⊤ i that are the ordinary matrix tra ns pos e o f the co efficients of L , w e can say (5) L R ( ψ ) =  L ⊤ ( ψ ⊤ )  ⊤ . Definition 1.1. A bis p ectra l triple ( L, Λ , ψ ) c onsists of a differ ential op e r ator L in x as in ( 4), a differ ential op er ator Λ in the variable z also havi ng r × r matrix c o efficients, and an r × r matrix function ψ ( x, z ) of x and z satisfying the e quations (6) L L ( ψ ) = p ( z ) ψ and Λ R ( ψ ) = π ( x ) ψ , wher e p ( z ) and π ( x ) ar e non-c onstant, s c alar eigenvalues. This seems to b e a natural matrix generalizatio n o f the scalar bispectr a l problem for different ial op erator s considered in [DG86]. How ever, we note that this differs from the matrix generaliza tion previously considered by Z ubelli [Zub90] in w hich bo th op erato rs acted from the same side, and also from the “ bundle bisp ectrality” considered by Sakhnovic h-Zubelli [SZ01] where the op erators L and Λ were allow ed to dep end on b o th v ariables. (Here w e are interested only in the case that L is independent of z and Λ is independent of x .) In the rest of the pap er w e will use the notation I k for the k × k iden titit y matrix. Also we will abuse notation b y using the symbol I to deno te the identit y transformatio n on man y differ en t v ector spaces whenever its use sho uld mak e it clear which is intended. 4 MAAR TEN BERGVEL T, MICHAEL GEKHTMAN, AND ALEX KASMAN 2. Spin Calo ger o Ma trices Let sCM r n be the the set of 4-tuples o f ma tr ices ( X , Z, A, B ) s uc h that the n × n commutator [ X , Z ] differs from the identit y by the rank r matrix B A : (7) sCM r n = { ( X , Z , A, B ) | X , Z ∈ M n × n , A, B ⊤ ∈ M r × n , [ X , Z ] − I = B A 6 = 0 } . This space ar ises na tur ally in the description o f g eneric initia l conditions of the Spin Caloger o particles, a s we will s ee b elow. Mo r e imp o rtantly , the dynamics linear izes there (when one considers the phase space to b e sCM r n mo dulo the action of GL( n ) to be describ ed in the section on s y mmetries). Let q i (1 ≤ i ≤ n ) b e the distinct p ositions of n pa rticles on the complex plane, ˙ q i be their momen ta, a nd f ij = β j α i be their “spins” repr esent ed a s the pro ducts of n column r -vectors α i and n r ow r -vectors β j sub ject to the c o nstraint f ii = − 1. W e asso ciate to this data the ma trices ( X , Z, A, B ) ∈ sCM r n in the form X ij = q i δ ij Z ij = ˙ q i δ ij + (1 − δ ij ) f ij q i − q j A =  α 1 · · · α n  B =    β 1 . . . β n    . The dynamics o f the eigenv alues of X + itZ i − 1 are gov erned by the Hamiltonia n H = tr Z i . This is the spin Calo ger o system [GH84, Kr i7 9]. In the sp e cial case r = 1, this reduces to the mo re famous (spinless ) Calogero-Mo ser-Sutherland particle system [vDV 00]. 2.1. Blo c k Decomp osi tion. F or a fixed choice of ( X , Z , A, B ) ∈ sCM r n we get a decomp osition of V = C n int o generalize d eigenspace s of Z : V = M λ V λ V λ = { v ∈ C n | ( Z − λI ) k v = 0 for s ome k ≥ 0 } . The restriction of Z to V λ will b e denoted by Z λ = λI + N λ where I is the iden tit y o per ator on V λ and N λ is nilp otent. Ra tional expre s sions in z I − Z λ below will alwa ys be in terpreted by e xpansion in p ositive p ow ers of the nilpo ten t N λ . W e will utilize subscripts λ and µ which will run over the eigenv alues of Z to similarly denote the blo cks of other linear o pera tors ass ocia ted to this decomp ositio n of C n . Sp ecifically , A λ : V λ → C r will b e the restric tio n of the map A , B λ : C r → V λ will b e the map B followed by pro jection on to V λ , and for a linear op erato r M from C r to itself (such as X ) M λ,µ will b e the blo ck cor resp onding to the map fr o m V µ to V λ . The sCM condition (7) inv olves the commutator [ X , Z ] = ( Z R − Z L )( X ). Interest- ingly , although the op erator Z R − Z L = − ad( Z ) is not inv ertible, its “ off-diagonal” action is inv ertible which allows us to solve for X λµ when µ 6 = λ . Lemma 2. 1: Let λ 6 = µ be generalized eigenv alues of Z . Then X µλ =  ( Z λ ) R − ( Z µ ) L  − 1 B µ A λ = X k ≥ 0 (( N λ ) R − ( N µ ) L ) k ( λ − µ ) k +1 B µ A λ . SPIN CALOGER O AND BISPECTRAL MA TRIX KP 5 Pro of: Since ( Z R − Z L ) µλ = ( λ − µ ) I µλ + ( N λ ) R − ( N µ ) L differs from the a nonzero m ultiple of the identit y by a nilp otent matr ix , it is inv ertible. Sp ecifically , w e may inv ert it in gener al using ( Z R − Z L ) − 1 µλ = X k ≥ 0 (( N λ ) R − ( N µ ) L ) k ( λ − µ ) k +1 . Applying this when solving [ X , Z ] − I = B A for any o ff-diagonal blo ck of X yields the claimed formula. It will la ter b e necessary to ev aluate residues of ma trix functions written in terms of the blo cks Z λ . F or this purp ose the following “o b vious” lemma will be useful. F or conv enience we introduce notatio n for “divided deriv ativ es”: f [ k ] = 1 k ! d k f dz k . Lemma 2. 2: Let f ( z ) b e a ra tional function that is r egular at z = λ , then Res z = λ  f ( z ) ( z I − Z λ ) k +1  = f [ k ] ( Z λ ) . Pro of: Res z = λ  f ( z ) ( z I − Z λ ) k +1  = Res z = λ  f ( z ) ( − ∂ z ) k k ! 1 z I − Z λ  = Res z = λ  f [ k ] ( z ) 1 z I − Z λ  = Res z = λ   f [ k ] ( z ) X s ≥ 0 N s λ ( z − λ ) s +1   = Res z = λ   f [ k ] ( z ) X s ≥ 0 ( − ∂ z ) s /s ! N s λ ( z − λ )   = Res z = λ     X s ≥ 0 ∂ s z /s ! f [ k ] ( z ) N s λ   1 z − λ   = Res z = λ  f [ k ] ( z + N λ ) 1 z − λ  = f [ k ] ( Z λ ) . Of cours e , in the ab ov e lemma f [ k ] ( Z λ ) is a matrix and may not co mm ute with other matrices app earing. So, one needs a little care in applying the Lemma 2.2 . 3. Ma trix KP H ierar chy Let ν = ( X , Z , A, B ) ∈ s CM r n and asso ciate to it the wave function ψ ν depe nding on the sp ectral parameter z and the times ~ t = ( t 1 , t 2 , t 3 , . . . ): (8) ψ ν ( ~ t, z ) = γ ( ~ t, z )  I r + A ˜ X − 1 ( z I n − Z ) − 1 B  , where 1 ˜ X = ˜ X ( ~ t ) = ∞ X i =1 it i Z i − 1 − X , and γ ( ~ t, z ) = exp ∞ X i =1 t i z i ! . 1 Note that the dep endence on t i in − ˜ X is such that its eigenv alue dynamics are gov e rned by the i th spin Calogero Hamil tonian. 6 MAAR TEN BERGVEL T, MICHAEL GEKHTMAN, AND ALEX KASMAN If q ν ( z ) = det ( z I − Z ) is the characteristic poly nomial of Z then the w a v e function (8) can be multiplied by q ν and an expone ntial so a s to yield a poly no mial in z with co efficients that are rationa l in the times (9) K ( ~ t, z ) = γ − 1 ( ~ t, z ) ψ ν ( ~ t, z ) q ν ( z ) . Indeed, q ν ( z )( z I − Z ) − 1 is the class ical a djo int of z I n − Z is and hence p olynomial in z . Letting ∂ = ∂ ∂ x be the differential op erator in x = t 1 , we note tha t the or dinary differential op erator K ν = K ( ~ t, ∂ ) sa tis fies (10) ψ ν ( ~ t, z ) = 1 q ν ( z ) K ν γ ( ~ t, z ) . The main goal of this se ction is to prov e that the pseudo - differen tial op erator L ν = K ν ◦ ∂ ◦ K − 1 ν is a solution to the matrix KP hierarch y in that it satisfies the Lax e q uation (2). As in [Wil93] (see also [Kas95, SW85]), the pro of will inv olv e identifying finitely suppo rted distributions in z tha t annihilate the function ψ ν . 3.1. Conditions satisfied b y ψ ν . Consider a generalized eig e n v a lue λ of Z with m ultiplicit y ℓ and use the notation of Section 2.1 to denote by Z λ , X λµ , A λ , e tc. the blo cks of the op erato rs X , Z , A and B . Let v ∈ C r ℓ + ℓ hav e the dec o mpos ition v =          v 0 v 1 v 2 . . . v ℓ − 1 w          where v i ∈ C r and w ∈ C ℓ and define the distribution c v, λ taking r × r matrix functions of z to r component co nstan t vectors by the formula (11) c v, λ ( f ( z )) = Res z = λ f ( z ) · A λ ( z I − Z λ ) − 1 w + ℓ − 1 X i =0 ( z − λ ) i v i !! . In this section w e will show that there are r ℓ linear ly indep endent distributions of this form satisfying c v, λ ( ψ ν ( ~ t, z )) ≡ 0. Consequently , by r unning through a ll of the eigenv alues of Z we obtain in this manner an r n -dimensional spa ce of conditions satisfied by the w av e function. Indeed, if { λ i } ar e the generalized eigenv alues of the n × n matrix Z with multiplicities { ℓ i } , then n = P ℓ i . Consider the r × ( rℓ + ℓ ) matrix Γ λ =  B λ N λ B λ N 2 λ B λ · · · N ℓ − 1 λ B λ − X λλ  . Lemma 3.1: If v ∈ ker Γ λ then c v, λ  ψ ν ( ~ t, z )  = 0 for all v alues of the v ariables ~ t . Pro of: Note first that ψ ν ( ~ t, z ) has the blo ck decomp osition (12) ψ ν ( ~ t, z ) = γ ( z , t ) I − X κ,µ A κ ( ˜ X − 1 ) κµ ( z I − Z µ ) − 1 B µ ! where aga in the sum is taken over all (not necessa rily distinct) pa ir s of gener alized eigenv alues κ and µ of Z . SPIN CALOGER O AND BISP ECTRAL MA TRIX KP 7 Now, w e wish to use Lemma 2 .2 to expand the r esidue in (11) where f ( z ) is replaced b y (12). It will b e conv enient to in tro duce the abbreviatio n C µ = − X κ A κ ˜ X − 1 κµ , so that we hav e (13) ψ ν ( ~ t, z ) = γ ( ~ t, z ) I + X µ C µ ( z I − Z µ ) − 1 B µ ! and (14) A λ = − X µ C µ ˜ X µλ . The v ario us con tributions to the res idue are usefully org a nized a c cording to depe ndence on C µ . First o f a ll, there is the contribution indep endent of C µ . It is given b y (A) Res z = λ  γ ( ~ t, z ) A λ ( z I − Z λ ) − 1 w  = A λ γ ( ~ t, Z λ ) w. Making use of Lemma 2 .1 one finds tha t the contributions co n ta ining C µ for µ 6 = λ are (B) X µ 6 = λ Res z = λ  ( z I − Z µ ) − 1 B µ A λ ( z I − Z λ ) − 1 wγ ( ~ t, z )  = C µ ˜ X µλ γ ( ~ t, Z λ ) w. Next we turn to the terms inv olving C λ . The first one is C λ Res z = λ γ ( ~ t, z )( z I − Z λ ) − 1 B λ ℓ − 1 X i =0 ( z − λ ) i v i ! = C µ ℓ − 1 X i,j =0 Res z = λ  γ ( ~ t, z )( z I − Z λ ) i +1 N i λ B λ ( z − λ ) j v j  = C λ γ ( ~ t, Z λ ) ℓ − 1 X i =0 N i λ B λ v i . (C) The other term linear in C λ is C λ Res z = λ  γ ( ~ t, z )( z I − Z λ ) − 1 B λ A λ ( z I − Z λ ) − 1 w  = C λ Res z = λ  γ ( ~ t, z )( z I − Z λ ) − 1 ([ X λλ , Z λ ] − I )( z I − Z λ ) − 1 w  = − C λ Res z = λ  γ ( ~ t, z )( z I − Z λ ) − 2 w  + C λ Res z = λ  γ ( ~ t, z )( z I − Z λ ) − 1 ([ X λλ , Z λ ] − z )( z I − Z λ ) − 1 w  = − C λ γ ′ ( ~ t, Z λ ) w − C λ Res z = λ ( γ ( ~ t, z )( z I − Z λ ) − 1 X λλ w ) + C λ Res z = λ ( X λλ ( z I − Z λ ) − 1 γ ( ~ t, z ) w ) = − C λ γ ′ ( ~ t, Z λ ) w − C λ γ ( ~ t, Z λ ) X λλ w + C λ X λλ γ ( ~ t, Z λ ) w = C λ ˜ X λλ γ ( ~ t, Z λ ) w − C λ γ ( ~ t, Z λ ) X λλ w. (D) 8 MAAR TEN BERGVEL T, MICHAEL GEKHTMAN, AND ALEX KASMAN Since v ∈ ker Γ λ is equiv alent to the statement (15) ℓ − 1 X k =0 N k λ B λ v k − X λλ w = 0 , we see that ( C ) cancels against the seco nd term in ( D ). So, co m bining all f our terms gives ( A ) + ( B ) + ( C ) + ( D ) = A λ γ ( ~ t, Z λ ) w + X µ 6 = λ C µ ˜ X µλ γ ( ~ t, Z λ ) w + C λ ˜ X λλ γ ( ~ t, Z λ ) w. Applying (14) shows that this is eq ual to zero as required. Lemma 3. 2: The distributions c v, λ for v ∈ ker Γ λ form an r ℓ -dimensional space . Pro of: Note that the map Ω : v ∈ ker Γ λ 7→ c v, λ is itself a linea r map. What we need to prov e, therefor e is that dim ker Γ λ − dim k er Ω = r ℓ. A v ector v clearly does not lie in the kernel of Ω if v i 6 = 0 for any i . The dimension of the kernel o f Ω is therefore equal to the dimension o f the spa c e of vectors w with the pro per ty tha t X λλ w = 0 and A λ ( z I − Z λ ) − 1 w = 0 . In fact, we will show that the o nly such w is the zero vector (and hence that dim ker Ω = 0). Beginning with the fact that [( z I − Z λ ) , X λλ ] − B λ A λ = I . Multiplying by ( z I − Z λ ) − 1 on the right, applying bo th sides of the resulting equa- tion to w and then multiplying by ( z I − Z λ ) − 1 on the left gives us that X λλ ( z I − Z λ ) − 1 w = ( z I − Z λ ) − 2 w. Expanding b oth sides o f this equation in terms of p ow ers of ( z − λ ) and equating like p ow ers gives us that X λλ N k λ w = k N k − 1 λ w, for k > 0 . Since N λ is nilp otent, for a sufficiently larg e k the left-hand side is equal to zer o. But the eq uation then tells us that N k − 1 λ w is then also equal to zero, which ag a in means that the left hand side would be zer o for a smaller v a lue of k . Repea ting this pr o c ess until k = 1 we find that w = 0. A similar arg umen t shows that dim ker Γ λ = r ℓ . Considering ins tead the vectors w such that w ⊤ Γ λ = 0 implies that w ⊤ X = w ⊤ ( z I − Z λ ) − 1 B λ = 0 and the same pro cess r eveals that w = 0 so that Γ λ has rank ℓ . Consequent ly , its k er nel has dimension ( rℓ + ℓ ) − ℓ = rℓ . 3.2. The Kernel of K ν . The results o f the previous section o n the distributions annihilating ψ ν give us info r mation ab out the kernel o f the matr ix ordina ry differ- ent ial op erator K ν defined in (10): Corollary 3.3: Let c v, λ be a s in Lemma 3.1. Then the r comp onent vector v a lued function (16) φ v, λ ( ~ t ) = c v, λ  γ ( ~ t, z ) q ν ( z )  is in the kernel o f the o pera tor K ν . SPIN CALOGER O AND BISP ECTRAL MA TRIX KP 9 Pro of: Since c v, λ commutes with m ultiplica tion and differentiation in x = t 1 , we hav e K ν φ v, λ = c v, λ  K ν γ ( ~ t, z ) q − 1 ν ( z )  = c v, λ ( ψ ν ) = 0 , by (8) and Lemma 3.1. In fact, the en tire kernel of K ν is spanned by functions of this form, and a s a consequence they satisfy certa in useful linea r different ial equations. Corollary 3.4: If φ ( ~ t ) is a vector in the kernel of K ν then it is a linear com bination of the φ v, λ ( ~ t ), and so sa tisfies the equation ∂ k ∂ t k 1 φ ( ~ t ) = ∂ ∂ t k φ ( ~ t ) . Pro of: B y m aking use of all of the eigenv alues of Z , Coro llary 3.3 gives us r n linearly indep enden t vector functions in the kernel of the n th o rder or dinary differ - ent ial op e rator with r × r matrix co efficients. Since they are linea r ly indep endent (those corresp onding to the same eigen v alue are linearly independent by Lemma 3.2 and those cor resp onding to differ en t eigenv alues cannot be linearly dep endent due to the factor of e xλ ) this acco unts fo r the ent ire kernel of K ν . Note that γ ( ~ t, z ) trivially sa tisfies ∂ k ∂ t k 1 γ ( ~ t, z ) = z k γ ( ~ t, z ) = ∂ ∂ t k γ ( ~ t, z ) . Now, the pr o o f he r e is elementary b ecause differ e ntiation in t i commutes with the residue, m ultiplication by functions of z and matrix multiplication in the definition of φ v, λ and applies by linear it y to the entire kernel. 3.3. The Lax E quation. No w we come to main p oin t of this section. If the ν mov es ac c o rding the spin Calogero dynamics the wa ve function ψ ν depe nds on the time v ar iables ~ t , and this pr oduces a solution of the matrix KP hierarch y . More precisely: Theorem 3. 5: The pseudo- differen tial op erator L ν = K ν ◦ ∂ ◦ K − 1 ν satisfies ∂ ∂ t i L ν = [( L i ν ) + , L ν ] . Pro of: First, we note that the ( pseudo)-differential op erato r ( L i ν ) − ◦ K ν is a ctually a differ en t ial operato r since ( L i ν ) − ◦ K ν + ( L i ν ) + ◦ K ν = K ν ◦ ∂ i and therefore ( L i ν ) − ◦ K ν = − ( L i ν ) + ◦ K ν + K ν ◦ ∂ i . Now, let φ ( x ) be a vector function in the kernel of the o per ator K ν . Then applying ∂ ∂ t i to the equality K ν φ = 0 and using Co rollary 3 .4 w e find 0 = ∂ ∂ t i ◦ K ν ( φ ) = ( K ν ) t i ( φ ) + K ν ( φ t i ) = ( K ν ) t i ( φ ) + K ν ( ∂ i φ ) = ( K ν ) t i φ + L i ν ◦ K ν ( φ ) = ( K ν ) t i ( φ ) + ( L i ν ) + ◦ K ν ( φ ) + ( L i ν ) − ◦ K ν ( φ ) 10 MAAR TEN BERGVEL T, MICHAEL GEKHTMAN, AND ALEX KASMAN How ever, since φ is in the kernel of K ν we know that ( L i ν ) + ◦ K ν ( φ ) = 0. Then the last display ed equality gives us that the en tire kernel of K ν is in the kernel of the ordinary differential op erator ( K ν ) t i + ( L i ν ) − ◦ K ν . Since this o per a tor has or de r strictly les s than n , it can o nly have such a large k er nel if it is the zero o per ator and we conclude ( K ν ) t i = − ( L i ) − ◦ K ν . Using this we find tha t ( L ν ) t i = ( K ν ) t i ◦ ∂ ◦ K − 1 ν − K ν ◦ ∂ ◦ K − 1 ν ◦ ( K ν ) t i ◦ K − 1 = − ( L i ν ) − ◦ K ν ◦ ∂ ◦ K − 1 ν + K ν ◦ ∂ ◦ K − 1 ν ◦ ( L i ν ) − ◦ K ν ◦ K − 1 ν = [ L ν , ( L i ν ) − ] = [( L i ν ) + , L ν ] . 4. Symmetries The symmetry X 7→ X + cZ j of sCM r n induces the integrable dyna mics o f b oth the particle system and the w ave eq uations of the matrix KP hierarch y . Here are some other symmetries and how they affect the KP Lax op erator . 4.1. Action of GL ( n ) . F or any G ∈ GL( n ) we define S G acting on sCM r n by S G : sCM r n → sCM r n ( X, Z, A, B ) 7→ ( GX G − 1 , GZ G − 1 , AG − 1 , GB ) . The Lax op erator L ν is unaffected by this action: L S G ( ν ) = L ν . Since this symmetry do es no t affect the corres ponding dynamical ob jects from the previo us sectio ns, it makes sense to consider s CM r n mo dulo this group actio n as the phase spac e of the spin Calogero particle dynamics as well as the corresp onding matrix KP solutions. 4.2. Action of GL ( r ) . Clearly , if we hav e G ∈ GL( r ) then w e can conjugate solutions to the matrix KP hierarch y to get so lutions that are technically different, but not very different. This als o manifests itself as a g roup action on the lev el of sCM r n . Let G ∈ GL( r ) then if s G : sCM r n → sCM r n ( X, Z, A, B ) 7→ ( X, Z, GA, B G − 1 ) one finds that L s G ( ν ) = G L ν G − 1 . 4.3. Changing r . There is an easy way to take an r × r solution and turn it int o an R × R s olution for r < R . Let a b e an R × r and b an r × R matrix s uch that ba = I is the r × r iden tity matr ix. Defining U a,b : sCM r n → sCM R n by U a,b ( X, Z, A, B ) = ( X, Z, aA, B b ) . one then has L U a,b ( ν ) ( ~ t ) = a L ν ( ~ t ) b. SPIN CALOGER O AND BISP ECTRAL MA TRIX KP 11 4.4. The Bisp ectral In v oluti on. Finally , we ha ve an impor tant discrete symme- try whose effect on the KP s olution will be the sub ject o f the next section: ♭ : sCM r n → sCM r n ν = ( X , Z , A, B ) 7→ ν ♭ = ( Z ⊤ , X ⊤ , B ⊤ , A ⊤ ) The significa nc e o f this symmetry on the KP solution is most easily seen b y lo oking at the wav e function ψ ν as a function of x = t 1 and z only (setting all o f the other times eq ual to zer o). As in the ca se r = 1 , it inv olves a n exchange of x and z , but when r > 1 one must a lso take the transp ose o f the function: (17) ψ ν ( x, z ) = ψ ⊤ ν ♭ ( z , x ) . 5. Bispectrality Let ν ∈ sCM r n and q ν ( z ) = det( z I − Z ). In this se c tion, since the dynamics are not significant, we will co ns ider x = t 1 and t i = 0 for i > 1 . Th us, for instance, we will wr ite ψ ν ( x, z ) for ψ ν (( x, 0 , 0 , . . . ) , z ) and γ ( x , z ) = e xz . In the next sectio n we will asso ciate tw o commutativ e r ings of ordina r y differ en tia l op erators (one acting in x and one acting in z ) to the choice o f ν ∈ sCM r n and then in the following section we will demo nstrate that ψ ν ( x, z ) is a common eigenfunction for the op erator s in the rings. In particular , any op erator from e ach of the rings alo ng with ψ ν form a bisp e ctral triple as in Definition 1 .1. 5.1. Commutat iv e Rings of Matrix Differential Op erators. Asso cia te to ν ∈ sCM r n the ring R ν ⊂ C [ z ] defined b y the prop erty that the p olyno mials pre s erve the conditions annihilating ψ ν ( ~ t, z ) from Lemma 3.1: R ν = { p ∈ C [ z ] | c v, λ ( ψ ν ) = 0 ⇒ c v, λ ( p ( z ) ψ ν ) = 0 } . Lemma 5. 1: The ring R ν is non-empty . In particular, q 2 ν ( z ) C [ z ] ⊂ R ν . Pro of: Note that q ν ( z ) ψ ν ( ~ t, z ) = K ν γ ( ~ t, z ) is non-singula r in z . Then the claim follows from the fact that c v, λ ( q ν ( z ) f ( z )) = 0 for any no n-singular function f . By substituting the pseudo-differ en tia l o per ator L ν int o these p olynomia ls, we asso ciate a commutativ e ring of pseudo-differ e ntial op erators R ν = { p ( L ν ) | p ∈ R ν } to ν . How ever, a s the next lemma demonstrates, these a re in fact differential op erators. Lemma 5.2: If p ∈ R ν then L = p ( L ν ) is a differential op erator (as o ppos ed to a general pseudo-differential op erator ) satisfying the eig e n v alue equa tion Lψ ν ( x, z ) = p ( z ) ψ ν ( x, z ) . Pro of: Since the leading coefficient of K ν is a nonsingular matrix (in f act, it is the iden tity matrix b ecause of the form o f ψ ν ), it is sufficient to show that the kernel of K ν is c on tained in the kernel of K ν ◦ p ( ∂ ) b ecause then w e know that this ordina ry differential op erator facto r s as L ◦ K ν for so me ordinar y differential op erator L which meets all of the o ther criteria. 12 MAAR TEN BERGVEL T, MICHAEL GEKHTMAN, AND ALEX KASMAN So, now let φ ( x ) b e a function in the kernel of K ν . By Lemma 3.3 we know that φ ( x ) is a linear combination o f functions of the form (16). How ever, K ν ◦ p ( ∂ ) c v, λ ( γ ( x, z ) q − 1 ν ( z )) = c v, λ ( p ( z ) ψ ν ( x, z )) = 0 by (10) and the definition of R ν . W e will also asso ciate a commutativ e ring o f ordinar y different ial op era tors in z to ν . Applying the pr o c edure ab ov e to the p oin t ν ♭ = ( Z ⊤ , X ⊤ , B ⊤ , A ⊤ ) ∈ sCM r n we hav e a nother commutativ e r ing R ν ♭ of differential op erator s in x . W e con vert them to differen tial op erators in z by simply r eplacing x with z , ∂ x with ∂ z and transp osing the co efficients: R ♭ ν =  L ⊤ ( z , ∂ z ) | L ( x, ∂ x ) ∈ R ν ♭  . 5.2. Common Eigenfunction. Our main result is the o bserv ation that ψ ν ( x, z ) is a commo n eigenfunction for the differen tia l oper ators in the rings R ν and R ♭ ν satisfying eigenv alue equations of the for m (6): Theorem 5.3: Let p ∈ R ν and π ∈ R ν ♭ , then there exist ordinary differen tial op erators L ( x, ∂ x ) ∈ R ν and Λ( z , ∂ z ) ∈ R ♭ ν such that Lψ ν ( x, z ) = p ( z ) ψ ν ( x, z ) and Λ R ψ ν ( x, z ) = π ( x ) ψ ν ( x, z ) . Pro of: The first equation follows from L e mma 5.2. Similarly , it follows fro m Lemma 5.2 that there is a differ en tia l op era tor Q ( x, ∂ x ) with the prop erty that Qψ ν ♭ ( x, z ) = π ( z ) ψ ν ♭ ( x, z ) . Exchanging the roles of x and z in this equa tion, taking the transp ose (see ), a nd applying (5) and (17) res ults in the second equation of the claim. 6. Example F or the sake o f clar it y , we briefly illustr ate the main ideas with an exa mple. Consider ν = ( X, Z, A, B ) ∈ sCM 2 3 where X =   0 0 0 − 1 0 − 1 1 0 2   Z =   0 1 0 0 0 0 0 0 0   A =  0 1 0 0 0 1  B =   0 1 − 2 0 1 − 1   . Then ψ ν ( x, z ) = e xz I + − 2 z x 2 +3 z x +2 x − 2 ( x − 2) x 2 z 2 1 ( x − 2) x 2 z xz − 2 ( x − 2) xz 2 1 − x ( x − 2) xz !! . This can be written as ψ ν = K ν e xz /q ν ( z ) wher e q ν ( z ) = det( z I − Z ) = z 3 and K = 2( x − 1) ( x − 2) x 2 0 − 2 ( x − 2) x 0 ! ∂ + 3 − 2 x ( x − 2) x 1 ( x − 2) x 2 1 x − 2 1 − x ( x − 2) x ! ∂ 2 +  1 0 0 1  ∂ 3 is an ordinary differential o per ator. SPIN CALOGER O AND BISP ECTRAL MA TRIX KP 13 T o find the conditions satisfied b y ψ ν ( x, z ), we note that the kernel o f Γ 0 =   0 1 − 2 0 0 0 0 0 0 − 2 0 0 0 0 0 1 0 1 1 − 1 0 0 0 0 − 1 0 − 2   is made up of vectors of the form v =  c 1 c 2 c 2 2 c 4 c 5 c 6 3 c 1 + c 2 c 8 − c 1 − c 2  ⊤ . Hence, we conclude that c v, 0 ( ψ ν ( x, z )) = 0 wher e c v, 0 ( f ( z )) = Res z =0  f ( z )  c 5 z 2 + 1 2 c 2 z + c 1 + c 8 z c 6 z 2 + c 4 z + c 2 + − c 1 − c 2 z  . Moreov er, c v, 0 ( p ( z ) ψ ν ) = 0 whenever p ∈ z 4 C [ z ] = R ν . Consequently , we can find an ordinar y differential op erato r having any of these poly nomials as its eigen- v a lue. In par ticular, so lving K ν ◦ ∂ 4 = L ◦ K ν for L we find L =   − 8 ( 3 x 4 − 18 x 3 +50 x 2 − 63 x +30 ) ( x − 2) 4 x 4 4 ( 23 x 3 − 93 x 2 +138 x − 72 ) ( x − 2) 4 x 5 8 ( 2 x 3 − 5 x 2 +9 x − 6 ) ( x − 2) 4 x 3 − 4 ( 2 x 4 − 8 x 3 +27 x 2 − 42 x +24 ) ( x − 2) 4 x 4   +   4 ( 6 x 3 − 27 x 2 +49 x − 30 ) ( x − 2) 3 x 3 − 4 ( 13 x 2 − 35 x +26 ) ( x − 2) 3 x 4 − 4 ( 4 x 2 − 7 x +6 ) ( x − 2) 3 x 2 4 ( 2 x 3 − 6 x 2 +13 x − 10 ) ( x − 2) 3 x 3   ∂ +   − 8 ( x 2 − 3 x +3 ) ( x − 2) 2 x 2 4(3 x − 4) ( x − 2) 2 x 3 4 ( x − 2) 2 − 4 ( x 2 − 2 x +2 ) ( x − 2) 2 x 2   ∂ 2 + ∂ 4 which satisfies Lψ ν = z 4 ψ ν . Of cour se, w e can follow this same pro cedur e beg inning with another element of sCM 2 3 . In par ticular, if we b egin with ν ♭ = ( Z ⊤ , X ⊤ , B ⊤ , A ⊤ ) instead then the differential op era to r we pro duce will be Q =   − 4 ( 16 x 2 +65 x +90 ) x 6 80 x +216 x 6 8 x +12 x 4 − 4 ( 4 x 2 +21 x +36 ) x 6   +   4 ( 16 x 2 +65 x +90 ) x 5 − 8(10 x +27) x 5 − 4(2 x +3) x 3 4 ( 4 x 2 +21 x +36 ) x 5   ∂ + − 2 ( 12 x 2 +65 x +90 ) x 4 40 x +108 x 4 6 x 2 − 8 x 2 +42 x +72 x 4 ! ∂ 2 +  34 x +60 x 3 − 4(2 x +9) x 3 0 14 x +24 x 3  ∂ 3 +  4 − 12 x 2 6 x 2 0 4 − 6 x 2  ∂ 4 − 4 ∂ 5 + ∂ 6 . The function ψ ν ♭ is a n eigenfunction for this opera to r satisfying Qψ ν ♭ ( x, z ) = (4 z 4 − 4 z 5 + z 6 ) ψ ν ♭ ( x, z ). More interestingly , since ψ ν ♭ ( x, z ) = ψ ⊤ ν ( z , x ), if we trans p ose the matrix co effi- cients on Q , replace x with z and ∂ = ∂ x with ∂ z , we g et a differential op erator Λ 14 MAAR TEN BERGVEL T, MICHAEL GEKHTMAN, AND ALEX KASMAN in the v ar iable z . This oper ator applied to ψ ν ( x, z ) (the wa ve function co mputed earlier) fr om the right satisfies Λ R ψ ν ( x, z ) = (4 x 4 − 4 x 5 + x 6 ) ψ ν ( x, z ) , demonstrating bisp ectrality . 7. Concl usions and Comments The main results of the present pap er ca n b e viewed as another step in address - ing the “ bispectr al problem” of F.A. Gr ¨ un baum [HK98, DG86], seeking o per ators satisfying eig en v a lue equations of the form (3). In [DG05], the author s c onsidered the case in whic h one of the op erators is a second order differe nc e o per ator with matrix co efficients and the t wo op erator s act o n matrix eigenfunctions from dif- ferent directions. How ever, bisp ectrality for matrix differential op erato r s has only bee n studied with b oth op erators acting from the left [Zub90]. Here w e co ns ider the case (6) in whic h the op erator s ar e r × r matrix different ial oper ators acting from different directio ns. Since our construction conv eniently repro duces the r e- sults of Wilson’s seminal paper [Wil93] in the sp ecial ca s e r = 1, this par ticula r formulation of the bisp ectral pro blem app ears to b e the corre ct o ne for gener alizing those results to the ca se of matrix differ en tia l o per ators. How ever, the method of pro of and esp ecially the explicit form ulation of the “ conditions” s atisfied by the wa ve function ab ov e are nov el even for r = 1. In [Wil93], it was shown that the bisp ectra l o per a tors asso ciated to sCM 1 n are in fact the only bispectra l scalar o rdinary differ en tia l op erators which co mmute with op erators of relatively prime order up to obvious renormaliza tions a nd c hanges of v a riable 2 . By Lemma 5.1 it fo llo ws that the differential op erators pro duced b y the construction in this pap er a lso all hav e the pro per t y that they commute with other differential op erator s of relatively prime order. In addition, this pa per can be seen a s con tributing to the literature establishing a link b etw een bisp ectrality and dua lit y in classic a l and q ua n tum integrable systems. (See, for instance, [F GNR00, Ha i07, Kas95, Kas00, Kas01, Wil98].) Again, the main results o f the present pa per for the spin C a logero s ystem in the case r = 1 repro duce r esults pr eviously presented for the s pinless c ase in [K as95, Wil98]. Some que s tions ar ise naturally whic h we hav e not pursued. There are a dditional commuting Hamiltonians fo r the spin Calogero system [GH84] and co rresp onding isosp ectral deformations for the multi-component KP hier arch y [BtK95], but their relationship to bispectra lit y has not been explored here . W e ha ve not looked at the alg ebro-geo metr ic implications of the rings R ν . Certainly a s in the case r = 1 [Wil93, Wil93], these contain op erato r s of relatively prime order are isomorphic to the co o rdinate ring s of rational curves with only cuspidal sing ularities. How ever, whether there is a ny further algebro-geometr ic significa nce such as was found in [BW02] or whether every commutativ e ring of matrix or dinary differential op era- tors with these prop erties is necessa rily bisp ectral have not b een considered. These questions, a long with the obvious question o f what other matrix differential op era- tors s atisfy equations of the for m (6) will hop efully be addressed in future pap ers. Ac kno wledgme nts: The second author was partially supp orted by the NSF Gran t DMS-04004 84. The third author appreciates helpf ul discussio ns with T o m Iv ey , 2 These were called “rank one” op erators in that context , but w e wi ll av oid that terminology here to av oid confusion wi th the rank r whic h is something different. 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(Bergv elt) Dep a r tment of Ma thematic s, University of Illinois, Urbana, IL 61801, USA E-mail addr e ss : bergv@uiuc.edu (Gekh tman) Dep art ment of Ma thema tics, University of Notre D ame, Notre Dame, IN 46556, US A E-mail addr e ss : gekhtman.1@nd.e du (Kasman) Dep ar tment of Ma thema tics, College of Charleston, Ch arleston, SC 2942 4, USA E-mail addr e ss : kasmana@cofc.ed u