Toda Versus Pfaff Lattice and Related Polynomials

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TODA VERSUS PFAFF LATTICE AND RELATED POLYNOMIALS

M. ADLER and P. VAN MOERBEKE

Abstract We study the Pfaff lattice, introduced by us in the context of a Lie algebra splitting of gl(infinity) into sp(infinity) and lower-triangular matrices. We establish a set of bilinear identities, which we show to be equivalent to the Pfaff Lattice. In the semi-infinite case, the tau-functions are Pfaffians; interesting examples are the matrix integrals over symmetric matrices (symmetric matrix integrals) and matrix integrals over self- dual quaternionic Hermitian matrices (symplectic matrix integrals). There is a striking parallel of the Pfaff lattice with the Toda lattice, and more so, there is a map from one to the other. In particular, we exhibit two maps, dual to each other, (i) from the the Hermitian matrix integrals to the symmetric matrix integrals, and (ii) from the Hermitian matrix integrals to the symplectic matrix integrals. The map is given by the skew-Borel decomposition of a skew-symmetric operator. We give explicit examples for the classical weights.

Contents 0. Introduction ...... 2 1. Splitting theorems, as applied to the Toda and Pfaff lattices ...... 10 2. Wave functions and their bilinear equations for the Pfaff lattice ...... 15 3. Existence of the Pfaff τ-function ...... 21 4. Semi-infinite matrices m , (skew-)orthogonal polynomials, and matrix in- ∞ tegrals ...... 30 k 4.1. ∂m /∂tk m , orthogonal polynomials, and Hermitian matrix ∞ = ∞ integrals (α 0) ...... 30 = k k 4.2. ∂m /∂tk m m ⊤ , skew-orthogonal polynomials, and ∞ = ∞ + ∞ symmetric and symplectic matrix integrals (α 1) ...... 32 =± DUKE MATHEMATICAL JOURNAL Vol. 112, No. 1, c 2002 Received 9 September 1999. Revision received 22 December 2000. 2000 Mathematics Subject Classification. Primary 37K10, 70H06; Secondary 70F45, 82C23. Adler’s work supported by National Science Foundation grant number DMS-98-4-50790. Van Moerbeke’s work supported by National Science Foundation grant number DMS-98-4-50790, a North At- lantic Treaty Organization grant, a Fonds National de la Recherche Scientifique grant, and a Francqui Foun- dation grant.

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5. A map from the Toda to the Pfaff lattice ...... 37 6. Example 1: From Hermitian to symmetric matrix integrals ...... 41 7. Example 2: From Hermitian to symplectic matrix integrals ...... 46 Appendix A. Free parameter in the skew-Borel decomposition ...... 51 Appendix B. Simultaneous (skew-)symmetrization of L and N ...... 54 Appendix C. Proof of Lemma 3.4 ...... 55 References ...... 57

0. Introduction Consider a weight on R, depending on t (t1, t2,...) C , = ∈ ∞ i i ρ (z) g(z) ∞ ti z V (z) ∞ ti z ′ ρt (z) dz e 1 ρ(z) dz e− + 1 dz, with V ′(z) , = = − ρ(z) = = f (z) (0.1) with rational g and f ’s, and with ρ(z) decaying fast enough at . ∞ The Toda lattice, its τ-functions and Hermitian matrix integrals (revisited) This weight leads to a t-dependent moment matrix

k ℓ mn(t) µk ℓ(t) z + ρt (z) dz , = + 0 k,ℓ n 1 = R 0 k,ℓ n 1 ≤ ≤ − ≤ ≤ − with the semi-inﬁnite moment matrix m , satisfying the commuting equations ∞

∂m k k ∞ m m . (0.2) ∂tk = ∞ = ∞ is the customary shift matrix, with zeroes everywhere, except for 1’s just above the diagonal, that is, (v)n vn 1. Consider the Borel decomposition into a lower- and = + an upper-triangular matrix 1 1 m S− S⊤− , (0.3) ∞ = and the following t-dependent matrix integrals (n 0): ≥ i Tr( V (X) ∞ ti X ) τn(t) e − + 1 dX det mn and τ0 1, (0.4) := = = Hn where dX is Haar measure on the ensemble Hn (n n)–Hermitian matrices . As ={ × } is well known (e.g., see E. Witten [18] or M. Adler and P. van Moerbeke [6]), integral (0.4) is a solution to the following two systems.

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TODA VERSUS PFAFF LATTICE 3

(i) The KP-hierarchy∗ 1 ∂2 sk 4(∂) τn τn 0, for k, n 0, 1, 2,.... (0.5) + ˜ − 2 ∂t1∂tk 3 ◦ = = + (ii) The Toda lattice, that is, the tridiagonal matrix 1/2 ∂ τ1 τ0τ2 ∂t log τ 2 0 1 0 τ1  1/2 1/2  τ0τ2 ∂ log τ2 τ1τ3 1  τ 2 ∂t1 τ1 τ 2  L(t) SS−  1 2  , (0.6) := =  1/2   .   0 τ1τ3 ∂ log τ3 ..  τ 2 ∂t1 τ2   2   . .   .. ..      satisﬁes the following commuting Toda equations

∂ L 1 n (L )sk, L , ∂tn = 2 where (A)sk denotes the skew-part of the matrix A for the Lie algebra splitting into skew and lower-triangular matrices. Moreover, the following t-dependent polynomials in z, are defined by the S-matrix obtained from the Borel decomposition (0.3); it is also given, on the one hand, in terms of the functions τn(t), and, on the other hand, by a classic determinantal formula (for a C,define a (a, a2/2, a3/3,...) C ) ∈ [ ]:= ∈ ∞ n 1 i n τn(t z− ) pn(t z) Sni(t)z z −[ ] ; := = √τnτn 1 i 0 + = 1 z   1 . det . . = τ τ  mn(t) .  √ n n 1   +    n   µn,0(t)...µn,n 1(t) z   −    The pn(t z)’s are orthonormal with respect to the (symmetric) inner-product , sy, ; i j Break/display in footnote defined by z , z sy µij, which is a restatement of the Borel decomposition (0.3) = OK? (see [6]). The vector p(t z) (pn(t z))n 0 is an eigenvector of the matrix L(t) in ; = ; ≥ (0.6): L(t)p(t z) zp(t z). ; = ;

t zi i The sℓ’s are the elementary Schur polynomials e 1∞ i si (t)z and sℓ(∂) sℓ(∂/∂t1,(1/2)(∂/∂t2), ∗ := i 0 ˜ := ...).Given a polynomial p(t , t ,...),deﬁne the customary Hirota≥ symbol 1 2 p(∂t ) f g p(∂/∂y1,∂/∂y2,...)f (t y)g(t y) . ◦ := + − y 0 =

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4 ADLER and VAN MOERBEKE

The Pfaff lattice and its τ-functions For use throughout this paper, deﬁne the skew-symmetric matrix

. ..  01  10  0   −   01    2 J  10  , with J I, (0.7) =   =−  −   01       0 10   −   .   ..      and the involution on the space D gl of inﬁnite matrices, := ∞

J D D a J (a) Ja⊤ J. (0.8) : −→ : −→ := Also, consider the splitting of D k n into two Lie subalgebras k and n, with the = + corresponding projections denoted πk and πn, where k is the Lie algebra of lower- triangular matrices with some special feature (see (1.17)) and where

n a D such that Ja⊤ J a sp( ). := { ∈ = }= ∞ Given a skew-symmetric semi-inﬁnite matrix m , consider the commuting differen- ∞ tial equations ∂m i i ∞ m m ⊤ (0.9) ∂ti = ∞ + ∞ ; they maintain the skew-symmetry of m . The Borel decomposition of m into ∞ ∞ lower-triangular times upper-triangular matrices requires the insertion of the skew- symmetric matrix J: 1 1 m (t) Q− (t)JQ− ⊤(t). (0.10) ∞ = Dressing up the shift with the lower-triangular matrix Q(t) leads to the commuting equations (0.11) below.

THEOREM 0.1 The Pfaff lattice equations

∂ L i i πk L , L πn L , L (0.11) ∂ti =[− ]=[ ]

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TODA VERSUS PFAFF LATTICE 5

maintain the locus of semi-inﬁnite matrices of the form (ai 0): = 01 d1 a1 O  −  d1 1 L   . (0.12) =  d2 a2   −   d2   ∗   ..   .      The solutions L to (0.11) of the form (0.12) are given by

1 L(t) Q(t)Q− (t), = where Q is a lower-triangular matrix, whose entries are given by the coefﬁcients of 1 1 the polynomials, obtained by the ﬁnite Taylor expansion in z of τ2n(t z ) below − −[ − ] (h2n τ2n 2(t)/τ2n(t)): := + 2n 1 j 2n 1/2 τ2n(t z− ) q2n(t z) Q2n, j (t)z z h2−n −[ ] , ; := = τ2n(t) j 0 = 2n 1 1 + j 2n 1/2 (z ∂/∂t1)τ2n(t z− ) q2n 1(t z) Q2n 1, j (t)z z h2−n + −[ ] , (0.13) + ; := + = τ2n(t) j 0 =

with τ0,τ2,τ4, a sequence of functions of t1, t2,..., characterized by the following bilinear identities for all n, m 0, (τ0 1), ≥ =

i dz 1 1 ∞(ti t )z 2n 2m 2 τ2n t z− τ2m 2 t′ z− e 1 − i′ z − − z −[ ] + +[ ] 2πi =∞ i dz ∞(t ti )z 2n 2m τ2n 2 t z τ2m t′ z e 1 i′− − z − 0. (0.14) + + +[ ] −[ ] 2πi = z 0 = Remark Theorem 0.1 is robust and remains valid for the bi-inﬁnite matrix L. In that case, the summations in the expressions q2n and q2n 1 run from j , instead of running + =−∞ from j 0. =

The τ-functions are given by Pfafﬁans pf m2n(t) and satisfy, as a consequence of the bilinear relations (0.14), the Pfafﬁan KP-hierarchy for k, n 0, 1, 2,..., = 1 ∂2 sk 4(∂) τ2n τ2n sk(∂) τ2n 2 τ2n 2. (0.15) + ˜ − 2 ∂t1∂tk 3 ◦ = ˜ + ◦ − +

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2 3 The t-dependent polynomials qn(t z) (Q(t)(1, z, z , z ,...) )n in z, obtained in ; = ⊤ (0.13) are “skew-orthonormal” with respect to the skew inner-product , sk,deﬁned i j by y , z sk µij(t), namely, =

qi , q j sk 0 i, j< J, ≤ ∞ = and are eigenvectors for the matrix L:

L(t)q(t z) zq(t z). (0.16) ; = ;

Explicit representations of L, in terms of the τ2n’s, are as follows:

L L 00 ˆ 00 ˆ 01 L L L 0  ˆ 10 ˆ 11 ˆ 12  1 1/2 1/2 L QQ− h− L21 L22 L23 h , = =  ∗ ˆ ˆ ˆ   L32 L33 ...   ∗∗ˆ ˆ   .   .      with the entries Lij and the entries of h, being (2 2)-matrices ˆ ×

h diag(h0 I2, h2 I2, h4 I2,...), h2n τ2n 2/τ2n, = = +

and ( ∂/∂t1), · =

(log τ2n)· 1 − 00 Lnn , Ln,n 1 , ˆ   ˆ + 10 := s2(∂)τ2n s2( ∂)τ2n 2 := ˜ −˜ + (log τ2n 2)· − τ2n − τ2n 2 +   +  (log τ2n 2)·· Ln 1,n ∗ + . (0.17) ˆ + := ∗∗

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Also, the t-dependent polynomials qn(t, z) in z have Pfafﬁan expressions, in contrast with the determinantal expressions in the Hermitian case, mentioned earlier:

1 z 1   . q2n(t z) pf m2n 1(t) . , ; = √τ2nτ2n 2    + 2n  +  z     1 ... z2n 0   − −    q2n 1(t z) + ; 1 µ0,2n 1 + z µ1,2n 1  +  1 . . pf  m2n(t) . .  . = √τ2nτ2n 2  z2n 1 µ  +  − 2n 1,2n 1   −2n 1+   1 z ... 0 z +   − − 2n 1 −   µ0,2n 1 µ1,2n 1 ... z + 0   − + − +   (0.18)

Theorem 0.1 and the subsequent statements are established in Sections 2 and 3. (0.18) lines too wide; this We show how a general skew-symmetric infinite matrix flowing according to (0.11) break OK? and its skew-Borel decomposition (0.10), lead to wave vectors , satisfying bilinear relations and differential equations. Section 3 deals with the existence, in the general setting, of a so-called Pfaffian τ-function, satisfying bilinear equations and the so- called Pfaff-KP hierarchy. In [4], these results were obtained by embedding the system in 2-Toda theory, while in this paper, they are obtained in an intrinsic fashion. For k 0, the Pfaff-KP equation (0.15) has already appeared in the context of = the charged BKP hierarchy, studied by V. Kac and J. van de Leur [13]; the precise relationship between the charged BKP hierarchy of Kac and van de Leur and the Pfaff Lattice, introduced here, deserves further investigation. (See the recent paper by van de Leur [16]).

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Examples: Symmetric and “symplectic” matrix integrals An important example is given by the skew-symmetric matrix m (µij)i, j 0 of ∞ = ≥ moments deﬁned by (α 1) (see [4]) ∗ =± i i i j i j ti (y z ) α y , z sk y z e + 2D δ(y z)ρ(y)ρ(z) dydz (0.19) := R2 − i i (1) i j 1∞ ti (y z ) µij (t) y z e + ε(y z)ρ(y)ρ(z) dydz, = R2 −  for α 1,  =− =    i (2) i j 1∞ 2ti y 2 µij (t) y , y (y)e ρ(y) dy, for α 1.  = R{ } ˜ =+    (1) (2) The associated moment matrices m2n and m2n satisfy the differential equations (0.9) and lead to “symmetric” matrix integrals

1 i (1) Tr( V (X) ∞ ti X ) (1) τ (t) e − + 1 dX pf(m ), 2n := (2n) = 2n ! S2n and “symplectic” matrix integrals

1 i (1) 2Tr( V (X) ∞ ti X ) (2) τ (t) e − + 1 dX pf(m ), 2n := n = 2n ! T2n both expressed in terms of the Pfafﬁan of the upper-left-hand principal minors of the “moment” matrix m(i), where ∞ (1) for i 1, dX denotes Haar measure on the space S2n of symmetric matrices, = and, (2) for i 2, dX denotes Haar measure on the (2n 2n)-matrix realization T2n = × of the space of self-dual (n n)–Hermitian matrices, with quaternionic entries. × A remarkable map from Toda to Pfaff lattice Remembering the notation (0.1), we act with the z-operator,

f d k d f g k (1/2) tk z ′ 1/2 tk z nt fρt e− f (z) + (z) e (0.20) := ρt dz = dz − 2 on the t-dependent orthonormal polynomials pn(t, z) in z; in [6], we showed that the matrix N deﬁned by

f ′ g nt p(t, z) f (L)M + (L) p(t, z) N p(t, z) (0.21) = − 2 =: We have ε(x) 1, for x 0,ε(x) 1, for x < 0, and f, g f g fg . ∗ = ≥ =− { }= ′ − ′

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is skew-symmetric. The t-dependent matrix N is expressed in terms of L and a new matrix M,deﬁned by

k d k (1/2) tk z (1/2) tk z zp Lp and e− e p Mp. (0.22) = dz = 1 Consider now the skew-Borel decomposition of N (2t) and its inverse∗ N (2t)− ,in † terms of lower-triangular matrices O( )(t) and O( )(t), respectively: + − 1 1 1 N (2t)± O(− )(t)JO(⊤−) (t). (0.23) =− ± ± Then, the lower-triangular matrices O( )(t) map orthonormal into skew-orthonormal ± polynomials, and the tridiagonal L-matrix into an L-matrix:‡ ˜ ( ) pn(t z) qn± (t z) O( )(t)p(t z) , ; −→ ; = ± ; n 1 L(t) L(t) O( )(t)L(2t)O( )(t)− (0.24) −→ ˜ = ± ± (Toda lattice) (Pfaff lattice).

It also maps the weight into a new weight

V (z) V (z) (1/2)(V (z) log f (z)) ρ(z) e− ρ (z) e− ˜ e− ∓ , = −→ ˜ ± = := and the corresponding string of τ-functions into a new string of Pfafﬁan τ-functions i (remember Vt (z) V (z) ∞ ti z ): = − 1 ( ) Tr 2( V (X)) τ + (t) e ˜t dX (β 4), Tr( Vt (X)) 2n T − τk(t) e − dX := 2n = ( ) Tr( Vt (X)) = Hk −→ τ − (t) e − ˜ dX (β 1). 2n := !S2n = ! For the classical orthogonal polynomials pn(z), we have shown in [6] that N (0) is not only skew-symmetric but also tridiagonal; that is,

b0 a0 0 c0 a0 b1 a1 c0 0 c1    −  L . , N . . = a b .. − = c 0 ..  1 2   1   . .   − .   .. ..   ..         (0.25)

∗See Appendix B. †The upper-signs (resp., lower-signs) correspond to each other throughout this section. ‡ We have p(t z) (p0(t z), p1(t z),...) . ; := ; ; ⊤

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In Sections 6 and 7, we show that the maps O( ) and O( ), as in (0.24), only involve − + three steps, in the following sense:

( ) c2n q − (0 z) p2n(0 z), 2n ; = a ; " 2n ( ) a2n q − (0 z) 2n 1 ; = c + " 2n 2n c2n c2n 1 p2n 1(0, z) bi p2n(0 z) c2n p2n 1(0 z) × − − − + a2n ; + + ; 0 (β 1), (0.26) =

a2n 2 ( ) ( ) p2n(0 z) c2n 1 − q2+n 2(0 z) √a2nc2n q2+n (0 z) ; =− − c2n 2 − ; + ; " − a2n 2 ( ) p2n 1(0 z) c2n − q2+n 2(0 z), + ; =− c2n 2 − ; " − 2n c2n ( ) c2n ( ) bi q2+n (0 z) q2+n 1(0 z)(β 4). (0.27) − a2n ; + a2n + ; = 0 " " The abstract map O( ) for t 0 appears already in the work of E. Brezin´ and − = H. Neuberger [9]. This has been applied recently by [1] to a problem in the theory of random matrices.

1. Splitting theorems, as applied to the Toda and Pfaff lattices In this section, we show how each of the equations

∂m i ∂m i i ∞ m and ∞ m m ⊤ (1.1) ∂ti = ∞ ∂ti = ∞ + ∞ lead to commuting Hamiltonian vector ﬁelds related to a Lie algebra splitting. First recall the splitting theorem due to Adler, B. Kostant, and W. Symes in [5], and later recall the R-version due to A. Reyman and M. Semenov-Tian-Shansky [15]. The R- version allows for more general initial conditions.

PROPOSITION 1.1 Let g k n be a (vector space) direct sum of a Lie algebra g in terms of Lie = + subalgebras k and n, with g paired with itself via a nondegenerate ad-invariant inner product , ; this in turn induces a decomposition g k n and isomorphisms ∗ = ⊥ + ⊥ g g , k n , n k . Let πk and πn be projections onto k and n, respectively. ≃ ∗ ⊥ ≃ ∗ ⊥ ≃ ∗

∗ Adg X Y X, Adg 1 Y , g G, and thus z, x , y x, z, y . ; = − ∈ [ ] = −[ ]

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Let G , Gk, and Gn be the groups associated with the Lie algebras g, k, and n. Let I (g) be the Ad Ad-invariant functions on g g. ∗ ≃ ∗ ≃ (i) Then, given an element

ε g ε, k k⊥ and ε, n n⊥, ∈ :[ ]⊂ [ ]⊂ the functions ϕ(ε ξ ′) with ϕ I (g) and ξ ′ k⊥, (1.2) + |k⊥, ∈ ∈ respectively, Poisson commute for the respective Kostant-Kirillov symplectic struc- tures of n k ; the associated Hamiltonian ﬂows are expressed in terms of the Lax ∗ ≃ ⊥ pairs∗

ξ πk ϕ(ξ),ξ πn ϕ(ξ),ξ , for ξ ε ξ ′,ξ′ k⊥. (1.3) ˙ =[− ∇ ]=[ ∇ ] ≡ + ∈ (ii) The splitting also leads to a second Lie algebra gR, derived from g, such that g gR, namely, ∗R ≃ 1 1 gR x, y R Rx, y x, Ry πkx,πk y πnx,πn y , (1.4) :[ ] = 2[ ]+2[ ]=[ ]−[ ]

with R πk πn. The functions = − ϕ(ξ) g , with ϕ I (g) and ξ gR, | R ∈ ∈ respectively, Poisson commute for the respective Kostant-Kirillov symplectic struc- tures of g gR, with the same associated (Hamiltonian) Lax pairs ∗R ≃ ξ πk ϕ(ξ),ξ πn ϕ(ξ),ξ , for ξ gR. (1.5) ˙ =[− ∇ ]=[ ∇ ] ∈ Each of the equations (1.3) and (1.5) has the same solution expressible in two different ways:† ξ(t) AdK (t) ξ0 Ad 1 ξ0, (1.6) = = S− (t) with‡ t ϕ(ξ0) t ϕ(ξ0) K (t) π e ∇ and S(t) π e ∇ . = Gk = Gn i Example 1 (The standard Toda lattice and the equations ∂m/∂ti m for the = Hankel¨ matrix m ) ∞ Since, in particular, the matrix m is symmetric, the Borel decomposition into lower- ∞ times upper-triangular matrix must be done with the same lower-triangular matrix S:

1 1 m S− S⊤− . (1.7) ∞ =

∗ ϕ is deﬁned as the element in g∗ such that dϕ(ξ) ϕ,dξ , ξ g. †∇ 1 = ∇ 1 ∈ We naively write Ad ( ) ξ K (t)ξ K (t) ,Ad 1 ξ S (t)ξ S(t). K t 0 0 − S− 0 − 0 ‡ = = Consider the group factorization A πG A πG A. = k n

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In turn, the matrix S deﬁnes a wave vector , and operators∗ L and M, the same as the ones deﬁned in (0.22),

i (1/2) ∞ ti z 1 (t, z) e 1 Sχ, L SS− , := := 1 ∞ i 1 1 M S ∂ iti − S− , (1.8) := + 2 1 satisfying the following well-known equations:† ∂ L z, M , with L, M 1, = = ∂z [ ]= ∂ S 1 n ∂ 1 n (L )bo S, (L )sk, ∂tn =−2 ∂tn = 2 ∂ L 1 n ∂ M 1 n (L )sk, L (L )sk, M . (1.9) ∂tn = 2 ∂tn = 2 # $ # $ The wave vector can then be expressed in terms of a sequence of τ-functions τn(t) det mn(t), but it also has a simple expression in terms of orthonormal poly- = nomials, with respect to the moment matrix m : ∞ 1 i τn(t z ) (t, z) e(1/2) ti z zn −[ − ] n 0 = τn(t)τn 1(t) + ≥ i (1/2) ti z e pn( t, z) n 0. (1.10) = ≥ The vector ﬁelds (1.9) on L are commuting Hamiltonian vector ﬁelds, in view of the Adler-Kostant-Symes (AKS) splitting theorem (version (i)),

i 1 ∂ L tr L + i πk Hi , L πn Hi , L , Hi , Hi L , (1.11) ∂ti =[− ∇ ]=[ ∇ ] = i 1 ∇ = + with L ⊤a b a, a and b diagonal matrices, (1.12) = + + for the splitting of the Lie algebra of semi-inﬁnite matrices

D gl k n skew-symmetric lower-triangular = ∞ = + := { }+{ } k⊥ n⊥ symmetric strictly upper-triangular , = + := { }+{ } (1.13)

0 2 ∗In the formulas below χ(z) (z , z, z ,...)⊤, and ∂ is the matrix such that (d/dz)χ(z) ∂χ(z). † = = The notation ()sk and ()bo refers to the skew-part and the lower-triangular (Borel) part, respectively, that is, projection onto k and n, respectively.

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TODA VERSUS PFAFF LATTICE 13

with the form (1.12) of L being preserved in time. Note that the solution (1.6) to the differential equation (1.5) in the AKS theorem is nothing but the factorization of m ∞ followed by the dressing up of .

i i Example 2 (The Pfaff lattice and the equations ∂m/∂ti m m ) = + ⊤ Throughout this paper the Lie algebra D gl of semi-inﬁnite matrices is viewed = as composed of (2 2)-blocks. It admits the natural∞ decomposition into subalgebras: ×

D D D0 D D D− D+ D , (1.14) = − ⊕ ⊕ + = − ⊕ 0 ⊕ 0 ⊕ +

where D0 has (2 2)-blocks along the diagonal with zeros everywhere else and where × D (resp., D ) is the subalgebra of upper-triangular (resp., lower-triangular) matrices + − with (2 2)-zero matrices along D0 and zero below (resp., above). As pointed out in × (1.14), D0 can further be decomposed into two Lie subalgebras:

D− all (2 2)-blocks D0 are proportional to Id , 0 ={ × ∈ } D+ all (2 2)-blocks D0 have trace zero . (1.15) 0 ={ × ∈ } Remember from (0.7) and (0.8) in the introduction, the matrix J and the associated Lie algebra involution J . The splitting into two Lie subalgebras∗

D k n, (1.16) = + with

k D D− = − + 0 . .. 0  Q2n,2n 0  0 Q  2n,2n  algebra of   , =  Q 0   2n 2,2n 2   + +   0 Q2n 2,2n 2   + +   ∗ .   ..      n a D, such thatJ a a b J b, b D sp( ), (1.17) ={ ∈ = }={ + ∈ }= ∞ † with corresponding Lie groups. Gk and Gn Sp( ), play a crucial role here. Let = ∞ πk and πn be the projections onto k and n. Notice that n sp( ) and Gn Sp( ) = ∞ = ∞

∗Note n is the ﬁxed point set of J . † Gk is the group of invertible elements in k, that is, invertible lower-triangular matrices, with nonzero (2 2)-blocks proportional to Id along the diagonal. ×

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14 ADLER and VAN MOERBEKE

stand for the inﬁnite-rank afﬁne symplectic algebra and group n (e.g., see [12]). Any element a D decomposes uniquely into its projections onto k and n, as follows: ∈

a πka πna = + 1 1 a J a a0 J a0 a J a a0 J a0 . = − − + + 2 − + + + + + 2 + % & % & (1.18)

The following splitting, with

k D D− and n n, + = + + 0 + = is also used in Section 2; the projections take on the following form:

a πk a πn a = + + + 1 1 a J a a0 J a0 a J a a0 J a0 . = + − − + 2 − + − + − + 2 + ' ( ' (1.19)(

Note that J intertwines πk and πk : +

J πk πk J . (1.20) = +

For a skew-symmetric semi-infinite matrix m , the skew-Borel decomposition ∞ 1 1 m Q− JQ− ⊤, with Q Gk, (1.21) ∞ := ∈ is unique, as was shown in [2]. Here we may assume m to be bi-infinite, as long as ∞ factorization (1.21) is unique, upon imposing a suitable normalization. Then we use Q to dress up : 1 L QQ− . = i i Then letting m run according to the equations ∂m/∂ti m m⊤ , we show in ∞ = + the next proposition and corollary that L evolves according to a system of commuting equations, which by virtue of the AKS theorem are Hamiltonian vector fields (for details, see [2]).

PROPOSITION 1.2 For the matrices

1 1 1 m Q− JQ− ⊤ and L QQ− , with Q Gk, ∞ := := ∈ the following three statements are equivalent:

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TODA VERSUS PFAFF LATTICE 15

∂ Q 1 i (i) Q− πk L , ∂ti =− i ∂ Q 1 (ii) L Q− n, + ∂ti ∈ ∂m i i (iii) ∞ m m ⊤ . ∂ti = ∞ + ∞ Whenever the vector ﬁelds on Q or m satisfy (i), (ii), or (iii), then the matrix L QQ 1 is a solution of the AKS-Lax pair = − ∂ L i i πk L , L πn L , L . ∂ti =[− ]=[ ]

Proof Written out and using (1.18), Proposition 1.2 amounts to showing the equivalence of the three formulas: ∂ Q 1 i i 1 i i (I) Q− (L ) J(L )⊤ J (L )0 J((L )0)⊤ J 0, ∂ti + − − + + 2 − = i ∂ Q 1 i ∂ Q 1 ⊤ (II) L Q− J L Q− J 0, + ∂ti − + ∂ti = i i ∂m (III) m m ⊤ ∞ 0. ∞ + ∞ − ∂ti = The point is to show that 1 (I) 0,(I) (II) J (II )⊤ J,(I)0 (II)0 , + = − = − =− + = 2 1 1 Q− (II)JQ− ⊤ (III). (1.22) = The details of this proof are found in [2].

2. Wave functions and their bilinear equations for the Pfaff lattice Consider the commuting vector ﬁelds

i i ∂m /∂ti m m ⊤ (2.1) ∞ = ∞ + ∞ on the skew-symmetric matrix m (t) and the skew-Borel decomposition ∞ 1 1 m (t) Q− (t)JQ⊤− (t), Q(t) Gk (2.2) ∞ = ∈ ; remember from (1.17) that Q(t) Gk means that Q(t) is lower-triangular, with along ∈ the “diagonal” (2 2)-matrices c2n I , with c2n 0. × = In this section, we give the properties of the wave vectors and their bilinear relations. In this and the next section, the matrices are assumed to be bi-inﬁnite; the semi-inﬁnite case is dealt with by specialization. Upon setting

1 Q1 Q(t) and Q2 JQ⊤− (t), (2.3) = =

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16 ADLER and VAN MOERBEKE

the matrix Q(t) deﬁnes wave operators

i i 1∞ ti 1∞ ti ⊤ 1 W1(t) Q1(t)e , W2(t) Q2(t)e− JW1− ⊤(t), (2.4) := := = L-matrices

1 1 L L1 Q1Q− , L2 J (L1) Q2⊤ Q− , (2.5) := := 1 := − = 2 and wave and dual wave vectors

1 1 1 1(t, z) W1(t)χ(z)∗(t, z) W − (t)⊤χ(z− ) J2(t, z− ), = 1 = 1 =− 1 1 1 2(t, z) W2(t)χ(z)∗(t, z) W − (t)⊤χ(z− ) J1(t, z− ). (2.6) = 2 = 2 = From the deﬁnition, it follows that the wave functions 1 have the following asymptotics:

k tk z 2n 1 1,2n(t, z) e z c2n(t)ψ1,2n(t, z), ψ1,2n 1 O(z− ), := k = + tk z 2n 1 2 1,2n 1(t, z) e z + c2n(t)ψ1,2n 1(t, z), ψ1,2n 1 1 O(z− ), + + + = = + k tk z− 2n 1 1 2,2n(t, z) e− z + c2−n (t)ψ2,2n(t, z), ψ2,2n 1 O(z), = k = + tk z 2n 1 2 2,2n 1(t, z) e− − z ( c− (t))ψ2,2n 1(t, z), ψ2,2n 1 1 O(z ), + = − 2n + + = + (2.7) where the ci are the elements of the diagonal part of Q.

THEOREM 2.1 The following statements are equivalent: i i (i) m satisfies ∂m /∂ti m m ⊤ , ∞ ∞ = ∞ + ∞ (ii) Q1 satisfies the hierarchy of equations (with Li defined in (2.5))

∂ Q1 i (πk L1)Q1, (2.8) ∂ti =− 1 (iii) Q2 JQ⊤− satisﬁes = 1 ∂ Q2 i i J (πk L1) Q2 (πk L2)Q2, ∂ti =− = + t zi (iv) 1 e 1∞ i Q1(t)χ(z) satisﬁes = ∂1 i (πn L1)1, (2.9) ∂ti =

i 1∞ ti z− 1 (v) 2 e− JQ⊤−1 (t)χ(z) satisﬁes = ∂2 i i i (L2 πk L2)2 (πn L2), 2, ∂ti =− − + =− +

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(vi) 1 and 2 satisfy the bilinear identity for all n, m Z, ∈ 1 dz 1 dz 1,n(t, z)2,m(t′, z− ) 2,n(t, z)1,m(t′, z− ) 0. 2πiz + 0 2πiz = ∞ (2.10) If any one of these six conditions is satisﬁed, then

∂ L1 i ∂ L2 i πk L1, L1 , πk L2, L2 , (2.11) ∂ti =[− ] ∂ti =[ + ] and 1 L11 z1, L22 z− 2, = = ⋆ ⋆ ⋆ 1 ⋆ L⊤ z , L⊤ z− . (2.12) 1 1 = 1 2 2 = 2

For later use, we also consider the “monic” wave functions, with the factors c2n(t) removed; that is, 1 1(t, z) Q− 1 and 2(t, z) Q02 (2.13) ˆ := 0 ˆ := and the matrix L1, normalized so as to have 1’s above the main diagonal, with Q 1 ˆ ˆ := Q0− Q, 1 1 1 1 1 L1 Q− L1 Q0 (Q− Q)(Q− Q)− QQ− , ˆ = 0 = 0 0 = ˆ ˆ 1 1 L2 Q0 L2 Q− Q0J (L1)Q− J (L1) (2.14) ˆ = 0 =− 0 =− ˆ 1 Then, in terms of the elements qij of the matrix Q Q− Q, one easily computes ˆ ˆ := 0 by conjugation that L1 has the following block structure: ˆ Display too wide; is this 1 1 1 1 L1 Q− L1 Q0 (Q− Q)(Q− Q)− break OK? ˆ = 0 = 0 0 . . ... L L 00  ˆ 00 ˆ 01  L L L 0  ˆ 10 ˆ 11 ˆ 12    , =  L21 L22 L23   ∗ ˆ ˆ ˆ   L32 L33 ...   ∗∗ˆ ˆ   .   .    with  

q2i,2i 1 1 00 Lii ˆ − , Li,i 1 , ˆ := q2i 1,2i 1 q2i 2,2i q2i 2,2i 1 ˆ + := 10 ˆ + − −ˆ + −ˆ + + 2 q2i 2,2i 1 q2i 3,2i 1 q2i 2,2i Li 1,i ∗−ˆ+ + −ˆ + + +ˆ + . (2.15) ˆ + := )∗∗* These deﬁnitions lead to a new statement that is equivalent to Theorem 2.1.

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18 ADLER and VAN MOERBEKE

THEOREM 2.2 L , Q, , satisfy the following equations: ˆ i ˆ ˆ 1 ˆ 2 ∂ Q ˆ n 2 n 2 (L1) Q0− J((L1) )⊤ JQ0 Q, (2.16) ∂tn =− ˆ − − ˆ + ˆ and 1 L11 z1, L22 z− 2, (2.17) ˆ ˆ = ˆ ˆ ˆ = ˆ with

∂ n n 2 n 2 1(t, z) (L1) (L1)0 Q0− J ((L1) )Q0 1(t, z), ∂tn ˆ = ˆ + + ˆ + ˆ + ˆ ∂ n n 2 n 2 2(t, z) J (L1) (L1)0 Q0− J ((L1) )Q0 2(t, z) ∂tn ˆ = ˆ + + ˆ + ˆ + ˆ n n 2 n 2 (L2) (L2)0 Q0J ((L2) )Q− 2(t, z). =− ˆ − + ˆ + ˆ − 0 ˆ The proof of Theorem 2.1 hinges on the following proposition.

PROPOSITION 2.3 The following three statements are equivalent: i i (i) ∂m /∂ti m m ⊤ , ∞ = ∞ + ∞ (ii) the matrices W1(t) and W2(t) (defined in (2.4)) satisfy 1 1 W1(t)W1(t′)− W2(t)W2(t′)− , (2.18) = (iii) the i (t, z) Wi (t)χ(z) satisfy the bilinear identity = 1 dz 1 dz 1,n(t, z)2,m(t′, z− ) 2,n(t, z)1,m(t′, z− ) 0. 2πiz + 0 2πiz = ∞ Proof The solution to (2.1) is given by k k m (t) e tk m (0)e tk ⊤ . ∞ ∞ = Therefore skew-Borel decomposing m (t) and m (0),wefind ∞ ∞ i i 1 1 ti 1 1 ti Q− (0)JQ⊤− (0) e− Q− (t)JQ⊤− (t)e− ⊤ , (2.19) = and so, from the definition of W1 and W2, 1 1 1 W − (0)W2(0) Q− (0)JQ⊤− (0) 1 = t i 1 t i 1 Q(t)e 1∞ i − J Q(t)e i ⊤− (using (2.19)) = 1 1 W1(t)− JW1(t)⊤− = 1 W1(t)− W2(t), (2.20) =

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implying the independence in t of the right-hand side of (2.20). Therefore, we have

1 1 W1(t)− W2(t) W1(t′)− W2(t′), for all t, t′ C∞, = ∈ and so 1 1 W1(t)W − (t′) W2(t)W − (t′), 1 = 2 thus yielding (ii). Reversing the steps yields the differential equation (i). Finally, the proof of the bilinear identity (iii) proceeds as follows. Using the well- known formula (see [3, Prop. 4.1]),

1 dz W1(t)W1(t′)− 1(t, z) 1∗(t′, z) , 2 2 = 2 ⊗ 2πiz ∞0 2 statement (ii) becomes dz dz 1(t, z) 1∗(t′, z) 2(t, z) 2∗(t′, z) , ⊗ 2πiz = 0 ⊗ 2πiz ∞ whose (m, n)th component is dz dz 1,n(t, z)1∗,m(t′, z) 2,n(t, z)2∗,m(t′, z) 0. 2πiz − 0 2πiz = ∞ 1 1 Next we use the relations (t, z) J2(t, z ) and (t, z) J1(t, z ) to 1∗ =− − 2∗ = − yield

1 dz 1 dz 1(t, z) J2(t′, z− ) 2(t, z) J1(t′, z− ) 0, ⊗ 2πiz + 0 ⊗ 2πiz = ∞ which again leads to (iii). That (iii) implies (ii) is obtained by reversing the arguments.

Proof of Theorem 2.1 The proof of statement (ii) for Q1, namely,

∂ Q1 i (πk L1)Q1, ∂ti =− follows at once from Proposition 1.2. 1 The proof of (iii) for Q2 JQ1⊤− is based on the identity J πka πk J a. = = +

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20 ADLER and VAN MOERBEKE

Indeed, we compute

∂ Q2 1 1 ∂ Q1⊤ 1 1 Q2− JQ1⊤− Q1⊤− Q2− ∂ti =− ∂ti 1 i 1 JQ⊤− Q⊤(πk L )⊤ Q⊤− Q⊤ J =− 1 1 1 1 1 i J(πk L )⊤ J =− 1 i J (πk L ) =− 1 i πk J L1 =− + i πk J ( J L2) (using (2.5)) =− + − i i πk J ( 1) (J L2) =− + − i i 1 i πk J ( 1) ( 1) − J L2 =− + − − i πk L2. = + Statements (iv) and (v) for 1,2 are straightforwardly equivalent to (ii) and (iii), respectively. According to Propositions 1.2 and 2.3 combined, the bilinear identity (2.10) in (vi) is equivalent to statement (i), (ii), or (iii). The hierarchy concerning the Li ’s follows at once from (ii) and (iii), thus ending the proof of Theorem 2.1.

Proof of Theorem 2.2 To prove (2.16), remember from Theorem 2.1 that

∂ Q 1 n n n 1 n n Q− πk L (L ) J(L )⊤ J (L )0 J((L )0)⊤ J ∂tn =− =− − − + − 2 − ; hence, taking the ()0-part of this expression yields

∂ log Q0 ∂ Q 1 n 1 n 1 n Q− πk(L )0 (L )0 J(L )0⊤ J. ∂tn = ∂tn 0 =− =−2 + 2 1 Using the fact that Q0, Q0− , Q0 Gk D0 commute among themselves and com- ˙ ∈ ∩ 1 mute with J and the fact that D0D , D D0 D , we compute for Q Q0− Q, L1 1 ± ± ⊂ ± ˆ = ˆ Q− L1 Q0 (see (2.14)) = 0 ∂ Q ˆ 1 1 1 1 1 1 Q− Q0− Q0 Q0− QQ− Q0 Q0− QQ− Q0 ∂tn ˆ =− ˙ + ˙ 1 1 1 Q− Q0 Q− QQ− Q0 =− 0 ˙ + 0 ˙ 1 1 1 Q− ( Q0 Q− QQ− )Q0 = 0 − ˙ 0 + ˙ 1 n n Q0− (L1) J(L1 )⊤ J Q0 = − − + + 1 n 1 1 n 1 (Q− L1 Q0) Q− J Q0(Q− L1 Q0) Q− ⊤ JQ0 =− 0 − + 0 0 + 0 n 2 n 2 (L1) Q− J((L1) )⊤ JQ0. =− ˆ − + 0 ˆ +

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TODA VERSUS PFAFF LATTICE 21

Using this result and L11(t, z) z1(t, z),weﬁnd ˆ ˆ = ˆ ∂1(t, z) ∂ i ˆ e ti z Qχ(z) ˆ ∂tn = ∂tn i i n 1∞ ti z 1∞ ti z n 2 n 2 z e Qχ(z) e (L1) Q− J((L1) )⊤ JQ0 Qχ(z) = ˆ + − ˆ − + 0 ˆ + ˆ n n 2 n 2 L1 (L1) Q− J((L1) )⊤ JQ0 1(t, z) = ˆ − ˆ − + 0 ˆ + ˆ n n 2 n 2 (L1) (L1)0 Q− (J (L1) )Q0 1(t, z). (2.21) = ˆ + + ˆ + 0 ˆ + ˆ 1 But, we also have that 1 Q0− 1(t, z) and 2 Q02(t, z) satisfy, using W2 1 ˆ = ˆ = = JW1− ⊤, ∂ (t, z) ˆ 1 1 1 1 1 (Q0− W1)·χ(z) (Q0− W1)·(Q0− W1)− 1(t, z), (2.22) ∂tn = = ˆ

∂2(t, z) 1 ˆ (Q0W2)·χ(z) (Q0W2)·(Q0W2)− (Q02) ∂tn = = 1 1 (Q0W2 Q0W2)W − Q− (Q02) = ˙ + ˙ 2 0 1 1 1 (Q0 Q− Q0W2W − Q− )Q02 = ˙ 0 + ˙ 2 0 1 1 1 Q0 Q− Q0J (W1W − )Q− Q02 = ˙ 0 + ˙ 1 0 1 1 1 Q0 Q− Q0 J(W1W − )⊤ JQ− Q02 = ˙ 0 + ˙ 1 0 1 1 1 J Q0 Q− J(Q− W1W − Q0)⊤ JQ02 = − ˙ 0 + 0 ˙ 1 1 1 1 J( Q− Q0 Q− W1W − Q0)⊤ JQ02 = − 0 ˙ + 0 ˙ 1 1 1 1 J (Q− W1)·(Q− W1)− (Q02). (2.23) = 0 0 Comparing (2.21), (2.22), and (2.23), and invoking (2.14),

J (Ln) Ln, − ˆ 1 = ˆ 2 and so, in particular,

n n n n J ((L1) ) (L2) and J ((L1)0) (L2)0, − ˆ ± = ˆ ∓ − ˆ = ˆ ∂ (t, z) ˆ 2 n n 2 n 2 (L2) (L2)0 Q0(J (L2) )Q0− 2(t, z), ∂tn =− ˆ − + ˆ + − ˆ which establishes Theorem 2.2.

3. Existence of the Pfaff τ-function The point of this section is to show that the solution of the Pfaff lattice can be expressed in terms of a sequence of functions τ, which are not τ-functions in the usual sense but which enjoy a different set of bilinear identities and partial differential equations.

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22 ADLER and VAN MOERBEKE

PROPOSITION 3.1 There exist functions τ2n(t) such that

1 τ2n(t z− ) τ2n 2(t z ) ψ1,2n(t, z) −[ ] and ψ2,2n(t, z) + +[ ] . (3.1) = τ2n(t) = τ2n 2(t) + The proof of Proposition 3.1 is postponed until later. For future use, we deﬁne the diagonal matrix

τ2n 2 h diag(...,h 2, h 2, h0, h0, h2, h2,...) D0−, with h2n + . (3.2) = − − ∈ = τ2n

THEOREM 3.2 1 and 2 have the following τ-function representation:

1 i τ (t z ) ti z 2n 1/2 2n − 1,2n(t, z) e z h−2n −[ ] , = τ2n(t) 1 i (z ∂/∂t )τ (t z ) ti z 2n 1/2 1 2n − 1,2n 1(t, z) e z h− + −[ ] , + 2n = τ2n(t) i τ (t z ) ti z− 2n 1 1/2 2n 2 2,2n(t, z) e− z + h−2n + +[ ] , = τ2n(t) 1 i (z ∂/∂t )τ (t z ) ti z 2n 1 1/2 − 1 2n 2 2,2n 1(t, z) e− − z + h− − + +[ ] , (3.3) + 2n = τ2n(t)

with the τ2n(t) satisfying the following bilinear identity for all n, m Z: ∈

i dz 1 1 (ti t )z 2n 2m 2 τ2n t z− τ2m 2 t′ z− e − i′ z − − z −[ ] + +[ ] 2πi =∞ i dz (t ti )z 2n 2m τ2n 2 t z τ2m t′ z e i′− − z − 0. (3.4) + + +[ ] −[ ] 2πi = z 0 = Conversely, this bilinear relation characterizes the τ-function for the Pfaff lattice.

Remark Then L has the following representation in terms of the Pfafﬁan τ-functions: . . ... L L 00  ˆ 00 ˆ 01  L L L 0 1/2 1/2  ˆ 10 ˆ 11 ˆ 12  h Lh−   , =  L21 L22 L23   ∗ ˆ ˆ ˆ   L32 L33 ...   ∗∗ˆ ˆ   .   .     

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with ( ∂/∂t1), · =

(log τ2n)· 1 − 00 Lnn , Ln,n 1 , ˆ   ˆ + 10 := s2(∂)τ2n s2( ∂)τ2n 2 := ˜ −˜ + (log τ2n 2)· − τ2n − τ2n 2 +   +  (log τ2n 2)·· Ln 1,n ∗ + . (3.5) ˆ + := ∗∗ The following bilinear relations follow from (3.4) and are due to [4].

COROLLARY 3.3 The functions τ2n(t) satisfy the following “differential Fay identity”:∗

τ2n(t u ), τ2n(t v ) −[ ] −[ ] 1 1 + (u− v− ) τ2n(t, u )τ2n(t v ) τ2n(t)τ2n(t u v ) + − −[ ] −[ ] − −[ ]−[ ] uv(u v)τ2n 2(t u v )τ2n 2(t), (3.6) = − − −[ ]−[ ] + and Hirota-type bilinear equations, always involving nearest neighbours:

1 ∂2 pk 4(∂) τ2n τ2n pk(∂) τ2n 2 τ2n 2, k, n 0, 1, 2,.... + ˜ − 2 ∂t1∂tk 3 ◦ = ˜ + ◦ − = + (3.7)

LEMMA 3.4 Consider an arbitrary function ϕ(t, z) depending on t C ,z C, having the ∈ ∞ ∈ asymptotics ϕ(t, z) 1 O(1/z) for z and satisfying the functional relation = + ր∞ 1 1 ϕ(t z2− , z1) ϕ(t z1− , z2) −[ ] −[ ] , t C∞, z C. ϕ(t, z1) = ϕ(t, z2) ∈ ∈ Then there exists a function τ(t) such that

τ(t z 1 ) ϕ(t, z) −[ − ] . = τ(t)

Proof See Appendix C.

We deﬁne f, g f g fg , where ∂/∂t1. ∗ { }:= ′ − ′ ′ =

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24 ADLER and VAN MOERBEKE

LEMMA 3.5 The following holds for the Pfafﬁan wave functions 1 and 2, as in (2.7), 1 1 ψ1,2n(t z− , z1) ψ1,2n(t z− , z2) −[ 2 ] −[ 1 ] (3.8) ψ1,2n(t, z1) = ψ1,2n(t, z2) and 1 1 ψ2,2n 2 t z− , z− ψ1,2n(t, z) 1. (3.9) − −[ ] = Proof Setting (2.7) in the bilinear equation (2.12), with n 2n, m 2n 2, yields → → − c (t) i dz 2n (ti t )z 1 e − i′ ψ1,2n(t, z)ψ2,2n 2(t′, z− ) c2n 2(t) − 2πi − ∞ 2 c (t) i z dz 2n 2 (t ti )z 1 − e i′− − ψ2,2n(t, z)ψ1,2n 2(t′, z− ) 0. + c (t) − 2πi = 2n 0 Setting 1 1 t t′ z− z− − =[ 1 ]+[ 2 ] i in the above and using e 1∞ x /i 1/(1 x) yields = − 1 c2n ψ1,2n(t, z)ψ2,2n 2(t′, z− ) dz − c2n 2 (1 z/z1)(1 z/z2) 2πi − ∞ − − c2n 2 2 1 1 1 dz − z 1 1 ψ2,2n(t, z)ψ1,2n 2(t′, z− ) 0, =− c2n 0 − zz1 − zz2 − 2πi = the latter being equal to zero, because the integrand on the right-hand side is holomorphic. The integral on the left-hand side can be viewed as an integral along a contour encompassing and the points z1 and z2, thus leading to ∞ 1 1 1 ψ1,2n(t, z1)ψ2,2n 2 t z− z− , z− − −[ 1 ]−[ 2 ] 1 1 1 1 ψ1,2n(t, z2)ψ2,2n 2 t z− z− , z− (3.10) = − −[ 1 ]−[ 2 ] 2 with

1 1 1 1 1 ψ1,2n(t, z) 1 O(z− ), ψ2,2n 2 t z− z− , z− 1 O(z− ). = + − −[ 1 ]−[ 2 ] = + Therefore, letting z2 , one ﬁnds ր∞ 1 1 ψ1,2n(t, z1)ψ2,2n 2 t z− , z− 1, (3.11) − −[ 1 ] 1 = 1 yielding (3.9), and so, upon shifting t t z− , → −[ 2 ] 1 1 1 1 ψ2,2n 2 t z− z− , z− − −[ 1 ]−[ 2 ] 1 = 1 ; ψ1,2n(t z2− , z1) −[ ]

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TODA VERSUS PFAFF LATTICE 25

similarly,

1 1 1 1 ψ2,2n 2 t z− z− , z− . (3.12) − −[ 1 ]−[ 2 ] 2 = 1 ψ1,2n(t z1− , z2) −[ ] Setting the two expressions (3.12) in (3.10) yields

1 1 ψ1,2n(t z− , z1) ψ1,2n(t z− , z2) −[ 2 ] −[ 1 ] . ψ1,2n(t, z1) = ψ1,2n(t, z2)

Proof of Proposition 3.1 From Lemmas 3.4 and 3.5, there exists, for each 2n, a function τ2n such that the ﬁrst relation of (3.1) is satisﬁed; that is,

1 τ2n(t z− ) ψ1,2n(t, z) −[ ] , = τ2n(t) and so from (3.9) 1 τ (t) ψ 1 , 1 2n , 2,2n 2 t z− z− 1 − −[ ] = ψ1,2n(t, z) = τ2n(t z− ) −[ ] thus leading to τ2n(t z ) ψ2,2n 2(t, z) +[ ] , − = τ2n(t) which is the second relation of (3.1).

Proof of Theorem 3.2 1 At ﬁrst, remembering that Q Q− Q, observe that ˆ = 0 i ti z 1 e (Q)χ(z) Q− 1(t, z) ˆ 2n = 0 2n i ti z 2n e z ψ1,2n(t, z) = 1 i τ2n(t z ) e ti z z2n −[ − ] = τ2n(t) i ∞ sk( ∂)τ2n(t) e ti z z2n 1 −˜ , = + τ2n(t) n 1 =

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26 ADLER and VAN MOERBEKE

showing that a few subdiagonals of the matrix Q are given by ˆ . ..  10  01   Q   ˆ =  q q 10   2n,2n 2 2n,2n 1   ˆ − ˆ −   q2n 1,2n 2 q2n 1,2n 1 01   ˆ + − ˆ + −   .   ..      with ∂ s2( ∂)τ2n q2n,2n 1 log τ2n, q2n,2n 2 −˜ . (3.13) ˆ − =−∂t1 ˆ − = τ2n Remembering that (2.7), normalized, becomes

k tk z 2n 1 1,2n(t, z) e z ψ1,2n(t, z), ψ1,2n 1 O(z− ), ˆ = k = + (3.14) tk z 2n 1 2 1,2n 1(t, z) e z + ψ1,2n 1(t, z), ψ1,2n 1 1 O(z− ), ˆ + + + = = +

k tk z 2n 1 2,2n(t, z) e− − z + ψ2,2n(t, z), ψ2,2n 1 O(z), ˆ = k = + tk z 2n 2 2,2n 1(t, z) e− − z ψ2,2n 1(t, z), ψ2,2n 1 1 O(z ), ˆ + =− + + = + (3.15)

we now show (3.3). Compute, using Theorem 2.2,

i ∂ ∂ ti z 2n e z z ψ1,2n(t, z) 1(t, z) ˆ 2n ∂t1 + = ∂t1 2 2 (L1) (L1)0 Q− J(L1 )⊤ JQ0 1(t, z) (3.16) = ˆ + + ˆ + 0 ˆ + ˆ 2n and

i ∂ 1 ∂ ti z 2n 1 e− − z + ψ2,2n(t, z) 2(t, z) ˆ 2n ∂t1 − z = ∂t1 2 2 J (L1) (L1)0 Q− J(L1 )⊤ JQ0 2(t, z) . (3.17) = ˆ + + ˆ + 0 ˆ + ˆ 2n

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TODA VERSUS PFAFF LATTICE 27

In this expression, the matrix equals, according to (2.14),

2 2 (L1) (L1)0 Q− J(L1 )⊤ JQ0 ˆ + + ˆ + 0 ˆ + . . ... q0, 1 1 00 00 ˆ − q q q 10 00  1, 1 20 21   ˆ − −ˆ −ˆ   00 q21 1 00  2 2 ˆ  , =  c0/c2 0 q31 q42 q43 10  ˆ −ˆ −ˆ   00 00 q43 1   2 2 ˆ   00c /c 0 q53 q64 q65 ...  2 4 ˆ −ˆ −ˆ   .   .      and, acting with J on this matrix,

2 2 J (L1) (L1)0 Q− J(L1 )⊤ JQ0 ˆ + + ˆ + 0 ˆ + . . ... q21 1 00 00 ˆ q q q c2/c2 0 00  1, 1 20 0, 1 0 2   ˆ − −ˆ −ˆ −   00 q43 1 00  ˆ 2 2  , =  10q31 q42 q21 c2/c4 0   ˆ −ˆ −ˆ   00 00 q65 1   ˆ   00 10q53 q64 q43 ...  ˆ −ˆ −ˆ   .   .      using the fact that ab db J − . cd= c a − Therefore the 2nth rows of both matrices, respectively, have the form

(0,...,0, q2n,2n 1(t), 1, 0, 0,...), ˆ − 2↑n

(0,...,0, q2n 2,2n 1(t), 1, 0, 0,...), ˆ + + 2↑n

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28 ADLER and VAN MOERBEKE

and thus from (3.16) and (3.17), and expansions (3.14) and (3.15), we have

∂ 2n 2n 2n 1 z z ψ1,2n(t, z) q2n,2n 1(t)z ψ1,2n z + ψ1,2n 1, ∂t1 + =ˆ − + + ∂ 1 2n 1 2n 1 2n z− z + ψ2,2n(t, z) q2n 2,2n 1(t)z + ψ2,2n z ψ2,2n 1. (3.18) ∂t1 − =ˆ + + + + So, using the expression (3.13) for q2n,2n 1 and the ﬁrst expression of (3.1), ˆ − 2n 1 z + ψ1,2n 1(t, z) + ∂ 2n 2n z z ψ1,2n(t, z) q2n,2n 1(t)z ψ1,2n(t, z) = + ∂t1 −ˆ − ∂ 2n ∂ 2n z z ψ1,2n(t, z) log τ2n(t) z ψ1,2n(t, z) = + ∂t1 + ∂t1 ∂ τ (t z 1 ) ∂ τ (t z 1 ) 2n 2n − τ ( ) 2n 2n − z z −[ ] 2n t z 2−[ ] = + ∂t1 τ2n(t) + ∂t1 τ2n(t) 1 (z ∂/∂t1)τ2n(t z ) z2n + −[ − ] , (3.19) = τ2n(t) and similarly, using the second relation (3.18),

1 2n 2n 1 ( z− ∂/∂t1)τ2n 2(t z ) z ψ2,2n 1(t, z) z + − + + +[ ] . (3.20) + = τ2n 2(t) + This establishes (3.3) modulo the denominators. Therefore, we also have

1 1 ∂ ∂τ2n 1 2 ψ1,2n 1(t, z) z τ2n(t) z− s2( ∂)τ2n z− + = z τ2n(t) + ∂t1 − ∂t1 + −˜ +··· 1 ∂2 ( ∂) τ 2 ( 3) 1 2 s2 2n z− O z− = + τ2n(t) − ∂t + −˜ + ; 1 thus, referring to the matrix Q just preceding (3.13), ˆ 1 ∂2 s (∂)τ , ( ∂) τ 2 ˜ 2n . q2n 1,2n 0 q2n 1,2n 1 s2 2 2n − (3.21) ˆ + = ˆ + − = τ2n −˜ − ∂t = τ2n 1 To show (3.4), setting n 2n and m 2n in bilinear relation (2.10) and → → substituting, using (2.7) and the expressions for ψ1,2n(t, z) and ψ2,2n(t, z) in the proof of Proposition 3.1,

1 k τ (t z ) tk z 2n 2n − 1,2n(t, z) e z c2n(t) −[ ] = τ2n(t) and k τ2n 2(t′ z ) tk′ z− 2n 1 1 2,2n(t′, z) e− z + c2−n (t′) + +[ ] = τ2n 2(t′) +

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TODA VERSUS PFAFF LATTICE 29

into

1 dz 1 dz 1,2n(t, z)2,2n(t′, z− ) 2,2n(t, z)1,2n(t′, z− ) 0 2πiz + 0 2πiz = ∞ yields

1 1 c2n(t) (t t )zk τ2n(t z− )τ2n 2(t′ z− ) dz k k′ + e − −[ ] +[ ] 2 c2n(t′) τ2n(t)τ2n 2(t′) 2πiz ∞ + c (t ) k τ (t z )τ (t z ) dz 2n ′ (t tk )z 2n 2 2n ′ e k′ − − + +[ ] −[ ] . + c2n(t) 0 τ2n 2(t)τ2n(t′) 2πi + Setting t t α amounts to replacing the exponential: ′ = +[ ] k k 1 (tk t )z (t tk )z e − k′ 1 αz, e k′ − − , = − = 1 α/z − so that the ﬁrst integral has a simple pole at z and the second integral has one at =∞ z α. Evaluating the integrals yields = 2 c (t)(τ2n 2(t)/τ2n(t)) α 2n + α 2 0 − c (t′)τ2n 2(t′)/τ2n(t′) + = ; 2n + that is, i τ (t) (α /i)(∂/∂ti ) 2 2n 2 (e 1)cn(t) + 0 − τ2n(t) = yields the following relation, which involves a constant cn, independent of time,

2 τ2n(t) 1 c2n(t) cn cn h2−n (t). (3.22) = τ2n 2(t) = · + Rescaling τ2n τ2n/(c1c2 cn 1), in effect, sets cn 1, and then (3.22), (2.7), → ··· − = (3.1), (3.19), and (3.20) yield (3.3); substituting (3.3) into (2.10) yields (3.4). 1/2 Finally, identity (3.22) actually says Q0 h . To derive the form (3.5) of the = − matrix L, set (3.13) and (3.21) in the elements just below the main diagonal of matrix (2.15), to yield (· ∂/∂t1) = Eqn. too wide for line; 2 (s ( ∂) (∂2/∂t2))τ this break OK? 2 τ2n 2 ˜ 1 2n q2n,2n 1 q2n 1,2n 1 q2n,2n 2 ˙ − − −ˆ − −ˆ + − +ˆ − =− τ2n − τ2n s2( ∂)τ2n −˜ + τ2n τ2n τ2n 2 ¨ ˙ = τ2n − τ2n (log τ2n)·· =

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30 ADLER and VAN MOERBEKE

and 2 2 (s2( ∂) ∂ /∂t1 )τ2n s2( ∂)τ2n 2 q2n 1,2n 1 q2n 2,2n −˜ − −˜ + ˆ + − −ˆ + = τ2n − τ2n 2 + s2(∂)τ2n s2( ∂)τ2n 2 ˜ −˜ + , =− τ2n − τ2n 2 + concluding the proof of Theorem 3.2, upon substituting the two relations (3.13) and 1/2 (3.14) and also Q0 h into (2.15). = − 4. Semi-inﬁnite matrices m , (skew-)orthogonal polynomials, and matrix inte- ∞ grals In this section, consider the following inner-product for α 0, 1: ∗ = ∓ i i ti (y z ) α f, g t f (y)g(z)e + 2D δ(y z)ρ(y)ρ(z) dydz = R2 − ˜ ˜ i f (y)g(y)e 2ti y 2ρ(y)2dy, for α 0, R ˜ =  i i ∞ ti (y z )  f (y)g(z)e 1 + ε(y z)ρ(y)ρ(z) dydz, for α 1, =  R2 − ˜ ˜ =−   i f, g (y)e 1∞ 2ti y ρ(y)2 dy, for α 1.  R{ } ˜ =+   (4.1)  Each type of inner-product is discussed in Sections 4.1 and 4.2.

k 4.1. ∂m /∂tk m , orthogonal polynomials, and Hermitian matrix integrals ∞ = ∞ (α 0) = The inner-product above, with α 0, corresponds to Hermitian matrix integrals; this = theory is sketched here for the sake of completeness and analogy; it mainly summa- rizes [6]. Consider a t-dependent weight Creates bad break in i i ti z V (z) ti z text; OK to display? ρt (dz) e ,ρ(dz) e− + dz := = on R, as in (0.1) and the induced t-dependent measure i Tr( V (X) ti X ) e − + dX, (4.2) on the ensemble Hn of Hermitian matrices, with Haar measure dX; the latter can be decomposed into a spectral part (radial part) and an angular part: n 2 dX dXii (d Xij d Xij) (z) dz1 dzn dU, (4.3) := ℜ ℑ = ··· 1 1 i< j n - ≤-≤

We have ε(x) sign x, having the property ε 2δ(x). Also, consider the Wronskian f, g (∂ f /∂y)g ∗ = ′ = { }:= f (∂g/∂y). −

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TODA VERSUS PFAFF LATTICE 31

where (z) (zi z j ) is the Vandermonde determinant. Here we form = 1 i< j n − the following matrix≤ integral≤ . n i i Tr( V (X) ti X ) 2 ∞ ti z e − + dX cn (z) e 1 k ρ(dzk). (4.4) = Rn Hn k 1 -= The weight ρt (dz) defines a (symmetric) t-dependent inner-product of the type (4.1) for α 0: = sy i f, g f (z)g(z)e 1∞ ti z ρ(dz), t = with moments k i j sy i j tk z µij(t) z , z t z + e ρ(dz) := = R satisfying ∂µ k ij i j ℓ tk z z + + e ρ(dz) µi ℓ, j (t). ∂t = R = + ℓ Therefore the semi-infinite moment matrix m (t) (µij(t))i, j 0 satisfies ∞ = ≥ ∂m i i ∞ m m ⊤ . (4.5) ∂ti = ∞ = ∞ The point now is that the following integral can be expressed as a determinant of moments, namely, n i Tr( V (X) ∞ ti X ) 2 e − + 1 dX (z) ρt (dzk) = Rn Hn k 1 = - n ℓ 1 k 1 det(zσ(−k)zσ(−k))1 ℓ,k n ρt (dzk) = Rn ≤ ≤ σ Sn k 1 ∈ -= n ℓ k 2 det(zσ(+k)− )1 ℓ,k n ρt (dzσ(k)) = Rn ≤ ≤ σ Sn k 1 ∈ -= ℓ k 2 det zσ(+k)− ρt (dzσ(k)) = R 1 ℓ,k n σ Sn ≤ ≤ ∈ ℓ k 2 n det z + − ρt (dz) = ! R 1 ℓ,k n ≤ ≤ n det(µij)0 i, j n 1 n τn(t) = ! ≤ ≤ − = ! is a τ-function for the KP-equation; also, in view of (4.5) and the upper-lower Borel decomposition (0.3) of m , the integrals form a vector of τ-functions for the Toda ∞ lattice. The polynomials pn(t z) defined by the Borel decomposition m (t) 1 1 ; ∞ = S− S⊤− and p(t z) Sχ(z) are orthonormal with regard to the inner-product i j sy ; = z , z µij(t). t =

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32 ADLER and VAN MOERBEKE

k k 4.2. ∂m /∂tk m m ⊤ , skew-orthogonal polynomials, and symmetric ∞ = ∞ + ∞ and symplectic matrix integrals (α 1) =± Consider a skew-symmetric semi-inﬁnite matrix

m (t) µij(t) i, j 0, with mn(t) µij(t) 0 i, j n 1, ∞ = ≥ = ≤ ≤ − satisfying k n ∂m /∂tk m m ⊤ . (4.6) ∞ = ∞ + ∞ Then we have shown in Sections 2 and 3 that, upon skew-Borel decomposing m , ∞ these equations ultimately imply the existence of functions τ(t) satisfying bilinear equations (3.4). Remember also that

τ2n 2(t) h(t) diag(h0, h0, h2, h2,...) D0−, with h2n(t) + . = ∈ = τ2n(t)

Here, we need the Pfafﬁan pf(A) of a skew-symmetric matrix A (aij)0 i, j n 1 = ≤ ≤ − for∗ even n: 1 n/2 pf(A) dx0 dxn 1 aijdxi dxj r ∧···∧ − = (n/2) ∧ 0 i< j n 1 ! ≤ ≤ − 1 / ε(σ)ai0,i1 ai2,i3 ain 2,in 1 dx0 dxn 1, (4.7) n 2( / ) − − − = 2 n 2 σ ··· ∧···∧ ! so that pf(A)2 det A. We now state the following theorem due to Adler, E. Horozov, = and van Moerbeke [2], in complete analogy with the discussion of the Hermitian case.

THEOREM 4.1 Consider a semi-inﬁnite skew-symmetric matrix m , evolving according to (4.6); ∞ setting pf(m2n 2(t)) τ2n(t) pf m2n(t) and h2n + (4.8) = = pf(m2n(t)) ; then, modulo the exponential, the wave vector 1 (deﬁned by (3.3)) is a sequence of polynomials, i ti z 1,k(t, z) e qk(t, z), (4.9) = where the qk’s are skew-orthonormal polynomials of the form (0.13) and (0.18), satisfying sk i j sk ( qi , q j )0 i, j< J, with y , z µij. (4.10) ≤ ∞ = :=

∗In the formula below (i0, i1,...,in 2, in 1) σ(0, 1,...,n 1), where σ is a permutation and ε(σ) its − − = − parity.

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TODA VERSUS PFAFF LATTICE 33

The matrix Q deﬁned by q(z) Qχ(z) is the unique solution (modulo signs) to the = skew-Borel decomposition of m : ∞ 1 1 m (t) Q− JQ⊤− , with Q k. (4.11) ∞ = ∈ The matrix L QQ 1, also deﬁned by = − zq(t, z) Lq(t, z), = and the diagonal matrix h satisfy the equations

∂ L i 1 ∂h i πk L , L and h− 2πk(L )0. (4.12) ∂ti =[− ] ∂ti =

Sketch of proof At first note that looking for skew-orthogonal polynomials is tantamount to the skew- Borel decomposition of m , so that (4.10) and (4.11) are equivalent. The skew- ∞ orthogonality of the polynomials (0.18) follows from expanding the Pfaffians explic- itly in terms of z-columns, upon using the expression for the Pfaffian in terms of a column

k ( 1) aki pf(0,...,k,...,ℓ 1) pf(0,...,ℓ 1, i). − ˆ − = − 0 k ℓ 1 ≤≤ − For details, see [2]. On the other hand, Theorem 3.2 gives (t, z) and hence Q in terms of τn(t) pf m2n(t) of (4.8). By the uniqueness of decomposition (4.11), the = two ways of arriving at Q, (0.18), and (3.3) must coincide.

Important remark The polynomials (0.18) provide an explicit algorithm to perform the skew-Borel decomposition of the skew-symmetric matrix m . Namely, the coefﬁcients of the poly- ∞ nomials qi provide the entries of the matrix Q. This fact is used later in the examples.

Symmetric matrix integrals (α 1) =− Here we focus on integrals over the space S2n of symmetric matrices of the type

i Tr( V (X) ∞ ti X ) e − + 1 dX, (4.13) S2n

where dX denotes Haar measure for X U diag(z1,...,zn)U , UU I , = ⊤ ⊤ =

dX dXij (z) dz1 dzn dU. (4.14) := =| | ··· 1 i j n ≤-≤ ≤

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34 ADLER and VAN MOERBEKE

As appears below, integral (4.13) leads to a skew-inner-product of the type (4.1) with i i ti z V (z) ti z α 1, with weight ρt (z) e ρ(z) e− + : =− := =

f (x), g(y) f (x)g(y)ε(x y)ρt (x)ρt (y) dx dy, (4.15) := R2 − / 0 leading to skew-symmetric moments∗

i j µij(t) x y ε(x y)ρt (x)ρt (y) dx dy = R2 − i j j i (x y x y )ρt (x)ρt (y) dx dy = x y − ≥ Fj (x)Gi (x) Fi (x)G j (x) dx, (4.16) = R − where ( d/dx) ′ = x k k i tk y i tk x Fi (x) y e ρ(y) dy and Gi (x) F′(x) x e ρ(x). := := i = −∞ By simple inspection, the moments µkℓ(t) satisfy

∂µ n n kℓ k i ℓ k ℓ i tn(x y ) (x + y x y + )ε(x y)e + ρ(x)ρ(y) dx dy ∂t = R2 + − i µk i,ℓ µk,ℓ i , = + + + and so m satisﬁes (4.6). ∞ According to M. Mehta [14], the symmetric matrix integral can now be expressed in terms of the Pfafﬁan, as follows, taking into account a constant c2n, coming from

We have ε(x) 1, for x 0, and ε(x) 1, for x < 0. ∗ = ≥ =−

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TODA VERSUS PFAFF LATTICE 35

integrating the orthogonal group:

2n 1 i 1 k Tr( V (X) ti X ) dX tk z e − + 2n(z) e i ρ(zi ) dzi (2n) = (2n) R2n | | S2n(E) i 1 ! ! -= 2n i det z j 1ρt (z j 1) 0 i, j 2n 1 dzi = + +

i i k ℓ ∞ ti (y z ) pf y z ε(y z)e 1 + ρ(y)ρ(z) dydz = R2 − 0 k,ℓ 2n 1 ≤ ≤ − pf µij(t) 0 i, j 2n 1 τ2n(t), (4.17) = ≤ ≤ − = which is a Pfafﬁan τ-function. Eqn. lines too wide for page; these breaks OK? Symplectic matrix integrals: (α 1) =+ Here we concentrate on integrals of the type

i 2Tr( V (X) ∞ ti X ) e − + 1 dX, (4.18) T2n

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36 ADLER and VAN MOERBEKE

where dX denotes Haar measure∗ N (0) (0) (1) (1) dX dXk dX dX¯ dX dX¯ = kℓ kℓ kℓ kℓ -1 k-<ℓ on the space T2N of self-dual (N N)–Hermitian matrices, with real quaternionic × entries; the latter can be realized as the space of (2N 2N)-matrices with entries × X (i) C, ℓk ∈ (0) (1) Xkℓ Xkℓ T2N X (Xkℓ)1 k,ℓ N , Xkℓ   with Xℓk Xk⊤ℓ . = = ≤ ≤ = (1) (0) = ¯  X X  − ¯ kℓ ¯ kℓ    Another skew-symmetric moment matrix m satisfying (4.6) is given by inner- ∞α α tα y V (y) tα y product (4.1) for α 1, with ρt (y) ρ(y)e e− + , = = = i j 2 µij(t) y , y ρt (y) IE (y) dy = R{ } i j y ρt (y), y ρt (y) IE (y) dy = R + , Gi (y)Fj (y) Fi (y)G j (y) dy, (4.19) = R − upon setting ( d/dx) ′ = j j Fj (x) x ρt (x) and G j (x) F′ (x) x ρt (x) ′. = := j = That m satisﬁes (4.6) follows at once from the ﬁrst expression (4.18): ∞ k ℓ 2 k ℓ 1 2 µkℓ(t) y , y ρt (y) dy (k ℓ)y + − ρt (y) dy = { } = − ∂µkℓ k ℓ i 2 2 y , y y ρt (y) dy ∂t = { } i k i ℓ 1 k i ℓ 1 2 (k i ℓ)y + + − (k ℓ i)y + + − ρt (y) dy = + − + − − µk i,ℓ µk,ℓ i , = + + + thus leading to (4.6). Using the relation

4 i i i i i i (xi x j ) det x (x )′ x (x )′ x (x )′ , − = 1 1 2 2 ··· n n 0 i 2n 1 1 i, j n ≤ ≤ − ≤-≤

X means the usual complex conjugate. The condition on the (2 2)-matrices Xkℓ implies that Xkk Xk I , ∗ ¯ × = with Xk R and I the identity. ∈

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TODA VERSUS PFAFF LATTICE 37

one computes, using again de Bruijn’s lemma, n 1 i 1 2Tr( V (X) ti X ) 4 2 e − + dX (xi x j ) ρt (xi ) dxi (n) = n Rn − T2n 1 i, j n i 1 ! ! ≤-≤ -= n 1 2 ρt (xk) dxk = n Rn k 1 ! -= i i i i i i det x1 (x1)′ x2 (x2)′ xn (xn)′ 0 i 2n 1 × ··· ≤ ≤ − n 1 dyi det Fi (y1) Gi (y1)...Fi (yn) Gi (yn) 0 i 2n 1 = n Rn 1 ≤ ≤ − ! - 1/2 det Gi (y)Fj (y) Fi (y)G j (y) dy = R − 0 i, j 2n 1 ≤ ≤ − pf µij(t) 0 i, j 2n 1 τ2n(t), (4.20) = ≤ ≤ − = which is a Pfafﬁan τ-function as well.

5. A map from the Toda to the Pfaff lattice t zk Remember from (0.1) the notation ρt (z) ρ(z)e k and ρ /ρ g/f . Assuming, = ′ =− in addition, that f (z)ρ(z) vanishes at the endpoints of the interval under considera- tion (which could be finite, infinite, or semi-infinite), one checks that the t-dependent operator in z,

f d nt fρt :=ρt dz k d f g k e(1/2) tk z f (z) ′ + (z) e1/2 tk z = dz − 2 d f ′ gt ∞ k 1 f (z) + (z), with gt (z) g(z) f (z) ktk z − , (5.1) = dz − 2 = − 1 maintains H 1, z, z2,... and is skew-symmetric with respect to the t-dependent + ={sy } inner-product , ,deﬁned by the weight ρt (z) dz, t sy sy nt ϕ,ψ (nt ϕ)(z)ψ(z)ρt (z) dz ϕ(nt ψ)ρt dz ϕ,nt ψ . t = =− =− t E E The orthonormality of the t-dependent polynomials pn(t, z) in z imply sy pn(t, z), pm(t, z) δmn. t = The matrices L and M are deﬁned/ by 0

k d k (1/2) tk z 1/2 tk z zp Lp and e− e p Mp. = dz =

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38 ADLER and VAN MOERBEKE

The skewness of nt implies the skew-symmetry of the matrix

f ′ g N (t) f (L)M + (L) such that nt p(t, z) N p(t, z) (5.2) = − 2 = ;

so N (t) can be viewed as the operator nt , expressed in the polynomial basis (p0(t, z), p1(t, z),...). In the next theorem, we consider functions F of two (noncommutative) variables z and nt so that the (pseudo-)differential operator ut F(z, nt ) in z and the matrix := U F(L, N ) related by := ∗

F(z, nt )p(t, z) F(L, N )p(t, z), = are skew-symmetric as well. Examples of F’s are†

1 ℓ 2k 1 † F(z, nt ) nt , n− , or z , n + , := t { t } corresponding to

1 2k 1 ℓ † F(L, N ) N , N − , or N + , L . = { }

THEOREM 5.1 Any Hankel¨ matrix m evolving according to the vector ﬁelds ∞ ∂m (t) k ∞ m ∂tk = ∞

leads to matrices L and M, evolving according to the Toda lattice equations ∂ L/∂tn n n (1/2) (L )sk, L and ∂ M/∂tn (1/2) (L )sk, M (see (1.9)). Consider a function = [ ] = [ ] F of two variables such that the operator ut F(z, nt ) is skew-symmetric with sy := respect to , and so the matrix t

U (t) F L(t), N (t) , deﬁned by ut p(t, z) U p(t, z), = = is skew-symmetric. This induces a natural lower-triangular matrix O(t), mapping the Toda lattice into the Pfaff lattice (for notation πbo,πk, Gk, etc., see (1.9), (1.18), (1.17)):

pn(t, z) (S(t)χ(z))n orthonormal with respect to = i j sy 1 1 m (t) z , z t 0 i, j S− S⊤− , Toda lattice  ∞ = ≤ ≤∞ =  1 ∂ L 1 j  L(t) SS− satisﬁes πbo L , L , j 1, 2,... = ∂t j = − 2 =   ∗It is to be understood that F(L, N ) reverses the order of z, u in F(z, u). †We deﬁne A, B † AB BA. { } := +

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TODA VERSUS PFAFF LATTICE 39

      1 1  U (2t) O− (2t)JO⊤− (2t),  − =   map O(2t) such that  O(2t) is lower-triangular,      O(2t)S(2t) Gk  ∈             qn(t, z) O(2t)p(2t, z) n, skew-orthonormal with regard to 5 = 1 1 1 1 m (t) S− (2t)U (2t)S⊤− (2t) Q− (t)JQ⊤− (t)  ˜ ∞ := − i j sk =  z , z t  = 0 i, j  zi , u z j sy≤ ≤∞ , Pfaff lattice  2t 2t 0 i, j  = ≤ ≤∞  L(t) O(2t)L(2t)O(2t) 1 satisﬁes ˜ := −  ∂ L j  ˜ πk L , L , j 1,....  ∂t j =[− ˜ ˜ ] =    Too wide for page; this Proof break OK? Since U (t) is skew-symmetric, it admits a skew-Borel decomposition

1 1 U (t) O− (t)JO⊤− (t), with lower-triangular O(t). (5.3) − = But the new matrix, deﬁned by

1 1 m (t) S− (2t)U (2t)S⊤− (2t), (5.4) ˜ ∞ := − is skew-symmetric and thus admits a unique skew-Borel decomposition

1 1 m (t) Q− (t)J Q(t)⊤− , with Q(t) Gk. (5.5) ˜ ∞ = ˜ ˜ ˜ ∈ Comparing (5.3), (5.4), and (5.5) leads to a unique choice of matrix O(t), skew-Borel decomposing U (2t), as in (5.3), such that −

O(2t)S(2t) Q(t) Gk. (5.6) = ˜ ∈ Using, as a consequence of (5.2) and (1.9),

∂U k (2t) πsy L (2t), U (2t) ∂tk = # $

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40 ADLER and VAN MOERBEKE

and ∂ S k (2t) πbo L (2t) S(2t), ∂tk =− we compute

∂m 1 ∂ S 1 1 1 ∂ 1 ˜ ∞ (t) S− (2t)S− U (2t)S⊤− (2t) S− (2t) U (2t) S⊤− (2t) ∂tk = ∂tk − ∂tk 1 1 ∂ S⊤ 1 S− (2t)U (2t)S⊤− (2t)S⊤− + ∂tk 1 k 1 1 k 1 S− πbo L (2t) U S⊤− S− πsy L , U S⊤− =− − [ ] 1 k 1 S− U (πbo L )⊤ S⊤− − 1 k k 1 1 k k 1 S− (πbo L πsy L )U S⊤− S− U (πbo L )⊤ πsy L S⊤− =− + − − 1 k 1 1 k 1 S− L U S⊤− S− U L⊤ S⊤− (using (5.7)) =− − k 1 1 1 1 k 1 k k 1 S− U S⊤− S− U S⊤− ⊤ S⊤ S⊤− (using L S S− ) =− − = k k m (t) m (t)⊤ (by (5.4)). = ˜ ∞ +˜∞ For an arbitrary matrix A,wehave

A A⊤ A (Abo)⊤ Asy. (5.7) = ⇐⇒ = −

Indeed, remembering that∗ Abo 2A A0 and Asy A A , one checks = − + = + − −

(Abo)⊤ Asy A 2(A )⊤ A0 (A A ) A A A0 2(A (A )⊤), − − = − + − +− − − −− +− =− +− − so that the left-hand side vanishes, if the right-hand side does; the latter means A is symmetric. We now deﬁne L(t) by conjugation of L(2t) by O(2t): ˜ 1 1 1 1 L(t) O(2t)L(2t)O(2t)− O(2t)S(2t)S− (2t)O(2t)− Q(t)Q− (t) ˜ := = = ˜ ˜ ; thus, by Proposition 1.2, L(t) satisﬁes the Pfaff Lax equation. Therefore the sequence ˜ of polynomials

q(t, z) O(2t)p(2t, z) O(2t)S(2t)χ(z) Q(t)χ(z) := = = ˜ is skew-orthonormal sk qi (t, z), q j (t, z) Jij = / 0 ∗ A means the usual strictly upper-(lower-)triangular part, and A0 means the diagonal part in the common ± sense.

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TODA VERSUS PFAFF LATTICE 41

with regard to the skew inner-product speciﬁed by the matrix m : ˜ ∞ i j sk z , z µij(t). t =˜ In the last step, we show that ϕ,ψ sk ϕ,uψ sy. Since t = 2t 1 1 U (2t) O− (2t)JO⊤− (2t) U ⊤(2t), (5.8) =− =− we compute

sy sy qi (t, z), (u2t q) j (t, z) 2t (Op)i (2t), (uOp) j (2t) 2t = sy / 0 /(Op)i (2t), (Oup) j (2t)02t = sy /(Op)i (2t), (OU p) j (2t0) 2t = sy / O(2t) pk(2t), pℓ(2t) k,ℓ0 0(OU )⊤(2t) ij = ≥ O(2t)/I (OU )⊤(2t) 0 = ij O(2t)U ⊤(2t)O⊤(2t) = ij O(2t)U (2t)O⊤(2t) =− ij Jij (using (5.8)). (5.9) = Therefore, deﬁning a new skew-inner-product , sk′

sy ϕ,ψ sk′ ϕ,uψ , := 2t we have shown sk sk qi , q j ′ qi , q j Jij, t = t = and so by completeness of the basis qi ,wehave

, sk′ , sk, t = t thus ending the proof of Theorem 4.1.

6. Example 1: From Hermitian to symmetric matrix integrals Striking examples are given by using the map O(t) obtained from skew-Borel de- 1 1 composing N − (t) and N (t) (see (5.2)). This section deals with N − (t), whereas Section 7 deals with N (t).

PROPOSITION 6.1 The special transformation

1 1 f ′ g − U (t) N − (t) f (L)M + (L) (t) = = − 2

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42 ADLER and VAN MOERBEKE

maps the Toda lattice τ-functions with initial weight ρ e V , V g/f (Hermi- = − ′ =− tian matrix integral) to the Pfaff lattice τ-functions (symmetric matrix integral), with initial weight

ρ (z) 1/2 i 2t (1/2)(V (z) log f (z) 2 ∞ ti z ) ρt (z) e− + − 1 ˜ := f (z) = i i V (z) ∞ ti z ti z e− ˜ + 1 ρ(z)e . =: =˜ To be precise:

pn(t, z) orthonormal polynomials in z for the inner-product i sy ti z  ϕ,ψ t ϕ(z)ψ(z)e ρ(z) dz,  = Toda lattice  i j sy  µij(t) z , z t and mn (µij)0 i, j n 1,  = = ≤ ≤ −  1 i Tr( V (X) ∞ ti X ) τn(t) det mn e − + 1 dX  = = n Hn  !  

     1(2t) O 1(2t)JO 1(2t),  N − − ⊤−  − =  map O(2t) such that  O(2t) is lower-triangular,      O(2t)S(2t) Gk  ∈            qn(t, z) O(2t)p(2t, z) skew-orthonormal polynomials 5 = n in z for the skew-inner-product (weightρ),  ˜  sk 1 sy  ϕ,ψ t ϕ,n2−t ψ 2t  :=  1 i i  ti (x y )  v ϕ(x)ψ(y)ε(x y)e +  = 2 R2 −   ρ ρ Pfaff lattice  (x) (y) dx dy,  × f f  " " i j sk µij(t) x , y t and mn (µij)0 i, j n 1,  ˜ = ˜ = ˜ ≤ ≤ −   1 Tr( V (X) t Xi )  ˜ 1∞ i  τ2n(t) pf(m2n) n e − + dX,  ˜ = ˜ = 2 (2n) S  ! 2n  1  with V (z) V (z) log f (z) .  ˜  = 2 +  

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TODA VERSUS PFAFF LATTICE 43

In the ﬁrst integral deﬁning τn(t), dX denotes Haar measure on Hermitian matrices Lines in above display (see Section 4.1), whereas the second integral τ2n(t) involves Haar measure on sym- too wide for page; this ˜ break OK? metric matrices (see Section 4.2).

Proof At ﬁrst, check that d 1 1 − ϕ(x) ε(x y)ϕ(y) dy. (6.1) dx = 2 − Indeed, d d 1 1 ∂ − ϕ(x) ε(x y)ϕ(y) dy dx dx = 2 ∂x − ∂ δ(x y)ϕ(y) dy (using ε(x) 2δ(x)) = − ∂x = ϕ(x). = Consider now the operator

1 − 1 f d 1 1 ut nt− fρt , so that ut p nt− p N − p, = = )ρt dz * = = according to (5.2). Let it act on a function ϕ(x):

1 1 1 d − ρt (x) nt− ϕ(x) ϕ(x) = √ f (x)ρt (x) dx f (x) 1 ε(x y) ρt (y) − ϕ(y) dy (using (6.1)). = R √ f (x)ρ (x) 2 f (y) t One computes

sk sy ϕ,ψ ϕ,u2t ψ t = 2t 1 sy ϕ,n− ψ = 2t 2t 1 ρ (x) ρ (y) 2t ε(x y) 2t ϕ(x)ψ(y) dx dy = 2 R2 f (x) − f (y) 1 k k ∞ tk (x y ) ρ(x)ρ(y)e 1 + ε(x y)ϕ(x)ψ(y) dx dy. = 2 R2 ˜ ˜ − So, ﬁnally setting V (x) (1/2)(V (x) log f (x)) yields by (4.17) that ˜ = + 1 i Tr( V (X) ∞ ti X ) τ2n(t) pf(m2n) e − ˜ + 1 dX. ˜ = ˜ = (2n) ! S2n

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44 ADLER and VAN MOERBEKE

The map O for the classical orthogonal polynomials at t 0 = Then, the matrix O, mapping orthonormal pk into skew-orthonormal polynomials qk, is given by a lower-triangular three-step relation:

c2n q2n(0, z) p2n(0, z), = a " 2n a2n q2n 1(0, z) + = c " 2n 2n c2n c2n 1 p2n 1(0, z) bi p2n(0, z) c2n p2n 1(0, z) , − − − + a2n + + 0 (6.2)

where the ai and bi are the entries in the tridiagonal matrix deﬁning the orthonormal polynomials, and the ci ’s are the entries of the skew-symmetric matrix N . In [6], we showed that, in the classical cases below, N is tridiagonal, at the same time as L (see Appendix B):

b0 a0 0 c0 a0 b1 a1 c0 0 c1    −  L . , N . , = a b .. − = c 0 ..  1 2   1   . .   − .   .. ..   ..          (6.3) with the following precise entries:

z2 Hermite: ρ(z) e− , an 1 n/2, bn 0, cn an = − = = = ; z α Laguerre: ρ(z) e− z I 0, )(z ), an 1 n(n α), = [ ∞ − = + bn 2n α 1, cn a n/2 = + + = ; α ρ Jacobi: ρ(z) (1 z) (1 z) I 1,1 (z) = − + [− ] ; 4n(n α β)(n α)(n β) 1/2 an 1 + + + + , − = (2n α β)2(2n α β 1)(2n α β 1) + + + + + + + − α2 β2 bn − , = (2n α β)(2n α β 2) + + + + + α β cn an + n 1 . = 2 + + If the skew-symmetric matrix N has the tridiagonal form above, then one checks that

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TODA VERSUS PFAFF LATTICE 45

its inverse has the following form:

1 c1 c1c3 c1c3c5 0 0 − 0 − 0 − − c0 c0c2 c0c2c4 c0c2c4c6 1 0000000  c0  1 c3 c3c5 000 0 − 0 −  − c2 c2c4 c2c4c6   c1 0 1 00 00   c0c2 c2  1 c5 1  0 0000 0 −  N −  c4 c4c6  − =  c c c 1 −   1 3 0 3 0 00 0   c0c2c4 c2c4 c4   0 000000 1   − c6   c1c3c5 c3c5 c5 1   0 0 0 0   c0c2c4c6 c2c4c6 c4c6 c6   ..   .     (6.4) In order to find the matrix O, we must perform the skew-Borel decomposition of the matrix U : − 1 1 1 U N − O− JO⊤− . − =− = The recipe for doing so is given in Theorem 4.1 (see also the important remark following that theorem). It suffices to form the Pfaffians (0.18) by appropriately border- ing the matrix N 1, as in (0.18), with rows and columns of powers of z, yielding − − skew-orthonormal polynomials; we choose to call them r’s, instead of the q’s of The- orem 4.1, with Oχ(z) r(z). They turn out to be the following simple polynomials, = with 1/τ2n c0c2c4 c2n 2: ˜ = ··· − 2n 1 c2n z 1 2n r2n(z) c2n z , = c0c2 c2n = √c2n τ2nτ2n 2 ··· ˜ ˜ + 6 2n 1 2n 1 1 c2n z + c2n 1z − 1 2n 1 2n 1 r2n 1(z) − − (c2n z + c2n 1z − ). + = c0c2 c2n = √c2n − − τ2nτ2n 2 ··· ˜ ˜ + 6 Then, also from Appendix A, in order to get O O in the correct form, we compute → ˆ the skew-orthonormal polynomials rk, with Oχ(z) r(z): ˆ ˆ =ˆ 1 c2n 2n r2n(z) r2n(z) z , ˆ = √a = a 2n " 2n 2n 0 bi r2n 1(z) r2n(z) √a2nr2n 1(z) ˆ + = a + + √ 2n 2n a2n 2n 1 c2n 2n 2n 1 c2n 1z − bi z c2n z + . (6.5) = c2n − − + a2n + " 0 From the coefficients of the polynomial rk, one reads off the transformation ma- ˆ trix from orthonormal to skew-orthonormal polynomials; it is given by the matrix

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46 ADLER and VAN MOERBEKE

O such that Oχ(z) r(z). Therefore q(t, z) O(2t)p(2t, z) yields, after setting ˆ ˆ =ˆ = ˆ t 0, We added times symbol = to (6.6). OK? c2n q2n(0, z) p2n(0, z), = a " 2n a2n q2n 1(0, z) + = c " 2n 2n c2n c2n 1 p2n 1(0, z) bi p2n(0, z) c2n p2n 1(0, z) , × − − − + a2n + + 0 (6.6)

conﬁrming (6.2).

7. Example 2: From Hermitian to symplectic matrix integrals

PROPOSITION 7.1 The matrix transformation

f ′ g N f (L)M + (L) = − 2 maps the Toda lattice τ-functions with t-dependent weight

i V (z) ∞ ti z ρt (z) e− + 1 , V ′ g/f = = (Hermitian matrix integral) to the Pfaff lattice τ-functions (symplectic matrix integral), with t-dependent weight Is this break OK? i 1/2 (1/2)(V (z) log f (z) 2 ∞ ti z ) ρt (z) ρ2t (z) f (z) e− − − 1 ˜ := = i i V (z) ti z ti z e− ˜ + ρ(z)e . =: =˜ To be precise:

pn(t, z) orthonormal polynomials in z for the inner-product sy i ϕ,ψ ϕ(z)ψ(z)e ti z ρ(z) dz,  t Toda lattice i j sy =  µij(t) z , z t and mn (µij)0 i, j n 1,  = = ! ≤ ≤ −  1 i  Tr( V (X) ti X ) τn(t) det mn(t) e − + dX = = n  ! Hn  

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TODA VERSUS PFAFF LATTICE 47

      1 1  N (2t) O− (2t)JO⊤− (2t),  − =   map O(2t) such that  O(2t) is lower-triangular,      O(2t)S(2t) Gk  ∈             qn(t, z) O(2t)p(2t, z) skew-orthonormal polynomials 5 = n in z for the skew-inner-product (weight ρ ),  t sk sy ˜  ϕ,ψ t ϕ,n2t ψ  := 2t  1 i  ϕ(z), ψ(z) e2 ti z ρ(z) f (z) dz,  Pfaff lattice  =−2 R2 { }  i j sk  µij(t) z , z t and mn det(µij)0 i, j n 1,  ˜ = ˜ = ˜ ≤ ≤ − 1 i 2Tr( V (X) ti X ) τ2n(t) pf(m2n(t)) e − ˜ + dX,  ˜ = ˜ = ( 2)nn  T2n  − !  1  with V (z) V (z) log f (z) .  ˜ = 2 −   Proof  Representing d/dx as an integral operator d ∂ ϕ(x) δ(x y)ϕ′(y) dy δ(x y)ϕ(y) dy δ′(x y)ϕ(y) dy, dx = R − =− R ∂y − = R − compute

f d ut nt fρt , so that nt p(t, z) N p(t, z); = = ρt dz = remember N from (5.2). Let it act on a function ϕ(x):

f d ut ϕ(x) fρt ϕ(x) = ρ dx t f (x) δ′(x y) f (y)ρt (y)ϕ(y) dy. = R ρt (x) −

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48 ADLER and VAN MOERBEKE

Then

sk sy sy ϕ,ψ ϕ,u2t ψ ϕ,n2t ψ t = 2t = 2t f (x) ρ2t (x)ϕ(x) δ′(x y) f (y)ρ2t (y)ψ(y) dx dy = R2 ρ2t (x) − f (x)ρ2t (x)ϕ(x)δ′(x y) f (y)ρ2t (y)ψ(y) dx dy = R2 − ∂ f (x)ρ2t (x)ϕ(x) δ(x y) f (y)ρ2t (y)ψ(y) dx dy =− R2 ∂x − ∂ f (x)ρ2t (x)ϕ(x) f (x)ρ2t (x)ψ(x) dx =− R ∂x 1 ∂ f (x)ρ2t (x)ϕ(x) f (x)ρ2t (x)ψ(x) dx =−2 R ∂x 1 ∂ f (x)ρ2t (x)ϕ(x) f (x)ρ2t (x)ψ(x) dx + 2 R ∂x 1 f (x)ρ2t (x)ϕ(x), f (x)ρ2t (x)ψ(x) dx =−2 R 1 + i , 2 2 1∞ ti x ϕ(x), ψ(x) ρ0 (x)e dx, =−2 R ˜ + , V (x) using the notation in the statement of this proposition. Setting ρ(x) e ˜ , with ˜ = − V (x) (1/2)(V (x) log f ), ˜ = − 1 i xi , x j sk xi , x j ρ2(x)e2 1∞ ti x dx =−2 R{ }˜ 1 i i j 2(V (x) ti x ) x , x e− ˜ − dx, =−2 R{ } and so 1 i 2Tr( V (x) ti x ) τ2n(t) pf m2n(t) e − ˜ + dx. = ˜ = ( 2)nn T2n − !

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TODA VERSUS PFAFF LATTICE 49

The map O 1 for the classical orthogonal polynomials at t 0 − = Then, the matrix O, mapping orthonormal pk into skew-orthonormal polynomials qk, is given by a lower-triangular three-step relation:

a2n 2 p2n(0, z) c2n 1 − q2n 2(0, z) √a2nc2n q2n(0, z), =− − c2n 2 − + " − a2n 2 p2n 1(0, z) c2n − q2n 2(0, z) + =− c2n 2 − " − 2n c2n c2n bi q2n(0, z) q2n 1(0, z), − a2n + a2n + 0 " " (7.1)

where the ai and bi are the entries in the tridiagonal matrix defining the orthonormal polynomials, and the ci are the entries in the skew-symmetric matrix. In this case, we need to perform the following skew-Borel decomposition at t = 0: 1 1 U N O− JO⊤− , − =− = where N is the matrix (6.3). Here again, in order to find O, we use the recipe given in Theorem 4.1, namely, writing down the corresponding skew-orthogonal polynomials (0.18), but where the µij are the entries of U N : consider the Pfaffians of the − =− bordered matrices (0.18); they have the leading term

n 1 − τ2n c2 j . ˜ = -0 Then one computes

n n i 1 i 1 1 2n 2i − − − r2n z − c2 j c2n 2 j 1 , = − − τ2nτ2n 2 i 0 0 0 ˜ ˜ + = - - 6 n 1 n n i 1 i 1 1 2n 1 − 2n 2i − − − r2n 1 z + c2 j z − c2 j c2n 2 j 1 , + = + − − τ2nτ2n 2 0 i 1 0 0 ˜ ˜ + - = - - 6 (7.2)

with

τ2nτ2n 2 c0c2 c2n 2√c2n, τ0τ2 √c0. ˜ ˜ + = ··· − ˜ ˜ = 6 Setting 6

D diag( τ0τ2, τ0τ2, τ2τ4, τ2τ4,...), := ˜ ˜ ˜ ˜ ˜ ˜ ˜ ˜ 6 6 6 6

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50 ADLER and VAN MOERBEKE

the matrix O is the set of coefﬁcients of the polynomials above, that is,

10000000 01000000   c1 0c0 00000    c2 00c0 000 0  1   O D−  c1 c3 0c0 c3 0c0 c2 00 0  =    c1 c4 0c0 c4 00c0 c2 00   c1 c3 c5 0c0 c3 c5 0c0 c2 c5 0c0 c2 c4 0    c c c 0cc c 0cc c 00cc c   1 3 6 0 3 6 0 2 6 0 2 4   .   ..   1  D− R. (7.3) =: As before, in order to get the skew-symmetric polynomials in the right form, from the orthogonal ones, one needs to multiply to the left with the matrix E,deﬁned in (A.2):

1 O EO ED− R, (7.4) ˆ = = and so, 1 1 1 O− R− DE− (7.5) ˆ = ; it turns out the matrix O is complicated, but its inverse is simple. Namely, compute We made all c’s roman ˆ here and in (7.3). OK? 10 0 0 0 0 0 0  01 0 0 0 0 0 0  c1 0 1 00 00 0 c0 c0 − c   2 00 1 0000  − c0 c0   00 c3 0 1 00 0  1  − c0 c2 c0 c2  R−  c4 1  , =  00 c c 00c c 00   − 0 2 0 2   00 0 0 c5 0 1 0   − c0 c2 c4 c0 c2 c4   00 0 0 c6 00 1   c0 c2 c4 c0 c2 c4   −       ..   .    

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TODA VERSUS PFAFF LATTICE 51

and α0 0 1 β0  − α0 0   α2 0   1  1  β2  E−  − α2  , (7.6) =    α4 0   1   β4   − α4     ..   0 .      with α2n and β2n as in (A.5). Carrying out the multiplication (7.5) leads to the matrix O 1, with a few nonzero bands, yielding the map (7.1), by the recipe of Proposition ˆ − 6.1 inverted.

Appendix A. Free parameter in the skew-Borel decomposition 1 1 If the Borel decomposition of H O JO is given by a matrix O Gk, with − = − ⊤− ∈ the diagonal part of O being

σ0 0 0 σ  0 0  σ 0  2   0 σ2  (O)0   , (A.1) =    σ 0   4     0 σ4     .   0 ..      then the new matrix Is “O=:EO,” a separate equation? 1/α0 0 β α  0 0 0  1/α 0  2   β α  O  2 2  O EO, (A.2) ˆ :=   =:  1/α 0   4     β4 α4     .   0 ..     

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52 ADLER and VAN MOERBEKE

with free parameters α2n,β2n, is a solution of the Borel decomposition H − = O 1 J O 1, as well. The diagonal part of O consists of (2 2)-blocks ˆ − ˆ ⊤− ˆ × 1/α 0 σ 0 σ /α 0 2n 2n 2n 2n . β α 0 σ = β σ α σ 2n 2n 2n 2n 2n 2n 2n Imposing the condition that

k i qi (z) Oij p j (z), with pk(z) pki z , = ˆ = 0 j i i 0 ≤≤ = 2n has the required form, that is, the same leading term for q2n and q2n 1 and no z -term + in q2n 1, + 2n q2n(z) q2n,2n z , = +··· 2n 1 2n 1 q2n 1(z) q2n,2n z + q2n,2n 1z − (A.3) + = + − +··· implies

σ2n p2n,2n σ2nα2n p2n 1,2n 1, α2n = + + σ2nβ2n p2n,2n σ2nα2n p2n 1,2n 0 + + =

yielding, upon using the explicit form of the coefﬁcients pkℓ of the polynomials pk, associated with three-step relations (see Lemma A.1),

2 p2n,2n α2n a2n, = p2n 1,2n 1 = + + 2n β2n p2n 1,2n 0 bi + . (A.4) α =− p = a 2n 2n,2n 2n Hence 1 2n α2n √a2n and β2n bi . (A.5) = = √a2n 0 So, if r(z) Oχ(z), =

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TODA VERSUS PFAFF LATTICE 53

then (A.2) yields

1/α0 0 β α  0 0 0  1/α 0  2   β α  r(z) Oχ(z)  2 2  r(z) Er(z), ˆ := ˆ =   =  1/α 0   4     β4 α4     .   0 ..      and thus Is “r(z) = Er(z),” a 1 separate equation? r2n(z) r2n(z), (A.6) ˆ = √a2n 2n 0 bi r2n 1(z) r2n √a2nr2n 1(z). (A.7) ˆ + = a + + √ 2n

LEMMA A.1 n i A sequence of polynomials pn(z) pni z of degree n satisfying three-step = i 0 recursion relation = ∗

zpn an 1 pn 1 bn pn an pn 1, n 0, 1,..., (A.8) = − − + + + = has the form n pn,n n 1 n pn 1(z) z + bi z . + = an − +··· 0 Proof n 1 n Equating the z + and z coefﬁcients of (A.8) divided by pn,n yields

pn 1,n 1 1 + + pn,n = an and pn,n 1 pn 1,n − an + bn. pn,n = pn,n + Combining both equations leads to

pn 1,n pn,n 1 an + an 1 − bn, pn,n − − pn 1,n 1 =− − −

∗We set a 1 0. − =

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54 ADLER and VAN MOERBEKE

yielding n pn 1,n an + bi (using a 1 0). pn,n =− − = 0

Appendix B. Simultaneous (skew-)symmetrization of L and N

CLAIM For the classical polynomials, the matrices L and N can be simultaneously symmetrized and skew-symmetrized.

Sketch of proof This statement has been established by us in [6]. Given the monic orthogonal polynomials pn with respect to the weight ρ, with ρ /ρ g/f , we have that the operators ˜ ′ =− z and f d d f g n fρ f ′ − = ρ dz = dz + 2 acting on the polynomials pn’s have the following form: ˜ 2 z pn an 1 pn 1 bn pn pn 1, ˜ = − ˜ − + ˜ +˜+ npn ... γn pn 1, (B.1) ˜ = − ˜ + in view of the fact that for the classical orthogonal polynomials,∗ Hermite: n d z, = dz − Laguerre: n z d 1 (z α 1),  = dz − 2 − −  Jacobi: n (1 z2) d 1 ((α β 2)z (α β)).  = − dz − 2 + + + − For the orthonormal polynomials, the matrices L and N are symmetric and skew-  − symmetric, respectively. Therefore the right-hand side of these expressions must have the form:

2 z pn an 1 pn 1 bn pn pn 1, ˜ = − ˜ − + ˜ +˜+ 2 npn an 1γn 1 pn 1 γn pn 1. ˜ = − − ˜ − − ˜ + Therefore, upon rescaling the pn’s, to make them orthonormal, we have ˜ zpn (Lp)n an 1 pn 1 bn pn an pn 1, = = − − + + + npn (N p)n an 1γn 1 pn 1 anγn pn 1, = = − − ˜ − − ˜ +

2 They have the respective weights ρ e z ,ρ e z zα,ρ (1 z)α(1 z)β . ∗ = − = − = − +

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TODA VERSUS PFAFF LATTICE 55

from which it follows that

0 c0 c0 0 c1  −  N . , with cn anγn , − = c 0 .. =  1   − .   ..      where γn is the leading term in expression (B.1). −

Appendix C. Proof of Lemma 3.4 For future use, consider the ﬁrst-order differential operators

j ∞ z− ∂ ∂ ∞ j 1 ∂ η(t, z) and B(z) z− − (C.1) = j ∂t j =−∂z + ∂t j j 1 j 1 = = having the property

η(z) 1 B(z)e− f (t) B(z) f t z− 0. (C.2) = −[ ] = LEMMA C.1 Consider an arbitrary function ϕ(t, z) depending on t C ,z C, having the ∈ ∞ ∈ asymptotics ϕ(t, z) 1 O(1/z) for z and satisfying the functional relation = + ր∞ 1 1 ϕ(t z2− , z1) ϕ(t z1− , z2) −[ ] −[ ] , t C∞, z C. (C.3) ϕ(t, z1) = ϕ(t, z2) ∈ ∈ Then there exists a function τ(t) such that

τ(t z 1 ) ϕ(t, z) −[ − ] . (C.4) = τ(t)

Proof Applying B1 B(z1) to the logarithm of (C.3) and using (C.1) and (C.2) yields := η(z2) (e− 1)B1 log ϕ(t z1) B1 log ϕ(t, z2) − ; =− ∞ j 1 ∂ z1− − log ϕ(t, z2), =− ∂t j j 1 = which, upon setting

j f j (t) Resz1 z B1 log ϕ(t, z1), = =∞ 1

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56 ADLER and VAN MOERBEKE

yields termwise in z1, ∂ η(z2) (e− 1) f j (t) log ϕ(t, z2). (C.5) − =−∂t j

Acting with ∂/∂ti on the latter expression and with ∂/∂t j on the same expression with j replaced by i, and subtracting,∗ one ﬁnds ∂ f ∂ f η(z2) i j (e− 1) 0, − ∂t j − ∂ti = yielding ∂ f ∂ f i j 0 ∂t j − ∂ti = ;

the constant vanishes because ∂ fi /∂t j never contains constant terms. Therefore there exists a function log τ(t1, t2,...)such that

∂ j log τ f j (t) Resz z B log ϕ, −∂t j = = =∞ and hence, using (C.5),

∂ η(z) ∂ log ϕ(t, z) (e− 1) log τ, ∂t j = − ∂t j or, what is the same,

∂ η log ϕ (e− 1) log τ 0, ∂t j − − = from which it follows that

η ∞ bi i log ϕ (e− 1) log τ z− − − =− i 1 1 is, at worst, a holomorphic series in z− with constant coefﬁcients, which we call bi /i. Hence − i τ(t z 1 e 1∞(bi /i)z− ϕ(t, z) −[ − ] − = τ(t) i 1 ∞ bi (ti z /i) τ(t z− )e 1 − − −[ ] = τ(t)e 1∞ bi ti ; η(z) It is obvious that ∂/∂ti , e 0. ∗ [ − ]=

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TODA VERSUS PFAFF LATTICE 57

that is, τ(t z 1 ) ϕ(t, z) ˜ −[ − ] , = τ(t) ˜ where τ τ(t)e 1∞ bi ti . ˜ = Thus Lemma 3.4 is proved.

References

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Proc. Lecture Notes 14, Amer. Math. Soc., Providence, 1998, 159–202. MR 99a:17026 [14] M. L. MEHTA, Random Matrices, 2d ed., Academic Press, Boston, 1991. MR 92f:82002 [15] A. G. REYMAN and M. A. SEMENOV-TIAN-SHANSKY, Reduction of Hamiltonian systems, afﬁne Lie algebras and Lax equations, Invent. Math. 54 (1979), 81–100. MR 81b:58021 [16] J. VAN DE LEUR, Matrix integrals and the geometry of spinors, J. Nonlinear Math. Phys. 8 (2001), 288–310. MR CMP 1 839 189 [17] P. VAN MOERBEKE, “Integrable lattices: Random matrices and permutations” in Random Matrix Models and Their Applications, ed. P. Bleher and A. Its, Math. Sci. Res. Inst. Publ. 40, Cambridge Univ. Press, Cambridge, 2001, 321–406. MR CMP 1 842 794 [18] E. WITTEN, “Two-dimensional gravity and intersection theory on moduli space” in Surveys in Differential Geometry (Cambrdige, Mass., 1990), Lehigh Univ., Bethlehem, Pa., 1991, 243–310. MR 93e:32028

Adler Department of Mathematics, Brandeis University, Waltham, Massachusetts 02454, USA; [email protected]

van Moerbeke Department of Mathematics, Universite´ de Louvain, 1348 Louvain-la-Neuve, Belgium and Department of Mathematics, Brandeis University, Waltham, Massachusetts 02454, USA; [email protected] and @math.brandeis.edu

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