Math. Ann. (2001) Mathematische Annalen Digital Object Identifier (DOI) 10.1007/s002080100153 Pfaff τ-functions M. Adler · T. Shiota · P. van Moerbeke Received: 20 September 1999 / Published online: ♣ – © Springer-Verlag 2001 Consider the two-dimensional Toda lattice, with certain skew-symmetric initial condition, which is preserved along the locus s =−t of the space of time variables. Restricting the solution to s =−t, we obtain another hierarchy called Pfaff lattice, which has its own tau function, being equal to the square root of the restriction of 2D-Toda tau function. We study its bilinear and Fay identities, W and Virasoro symmetries, relation to symmetric and symplectic matrix integrals and quasiperiodic solutions. 0. Introduction Consider the set of equations ∂m∞ n ∂m∞ n = Λ m∞ , =−m∞(Λ ) ,n= 1, 2,..., (0.1) ∂tn ∂sn on infinite matrices m∞ = m∞(t, s) = (µi,j (t, s))0≤i,j<∞ , M. Adler∗ Department of Mathematics, Brandeis University, Waltham, MA 02454–9110, USA (e-mail: [email protected]) T. Shiota∗∗ Department of Mathematics, Kyoto University, Kyoto 606–8502, Japan (e-mail: [email protected]) P. van Moerbeke∗∗∗ Department of Mathematics, Universit´e de Louvain, 1348 Louvain-la-Neuve, Belgium Brandeis University, Waltham, MA 02454–9110, USA (e-mail: [email protected] and @math.brandeis.edu) ∗ The support of a National Science Foundation grant # DMS-98-4-50790 is gratefully ac- knowledged. ∗∗ The support of Japanese ministry of education’s grant-in-aid for scientific research, and the hospitality of the University of Louvain and Brandeis University are gratefully acknowledged. ∗∗∗ The support of a National Science Foundation grant # DMS-98-4-50790, a Nato, a FNRS and a Francqui Foundation grant is gratefully acknowledged. 2 M. Adler et al. where t = (t1,t2,...)and s = (s1,s2,...)are two sequences of scalar indepen- dent variables, Λ = (δi,j−1)0≤i,j<∞ is the shift matrix, and Λ its transpose. In 1 [2,4], it was shown that Borel decomposing m∞ into lower- and upper-triangular matrices S1 = S1(t, s) and S2 = S2(t, s): −1 ∞ m∞(t, s) = µij = S S , t s ∈ C , 0≤i,j<∞ 1 2 for “generic” , (0.2) L := S ΛS−1 leads to a two-Toda (two-dimensional Toda) system for 1 1 1 and L = S ΛS−1 2 2 2 , ∂L ∂L i =[(Ln) ,L ] , i =[(Ln) ,L ] ,i= , , 1 + i 2 − i 1 2 ∂tn ∂sn with A = A− + A+ being the decomposition into lower- and strictly upper- triangular matrices. The solution L1 and L2 can be expressed entirely in terms of a sequence of τ-functions τ = (τ0,τ1,...)given by τn(t, s) = det mn(t,s), mn(t, s) = (µi,j (t, s))0≤i,j<n , (0.3) for n ∈ Z≥0 := {0, 1,...}. As readily seen from formula (0.1), the 2-Toda flow then maintains the relation m∞(t, s) =−m∞(−s,−t) , and hence, by formula (0.3) n τn(t, s) = (−1) τn(−s,−t). (0.4) The main point of this paper is to study equation (0.1) with skew-symmetric initial condition m∞(0, 0) and the restriction of the system to s =−t. When s →−t, formula (0.4) shows that in the limit the odd τ-functions vanish, whereas the even τ-functions are determinants of skew-symmetric matrices. In particular, the factorization (0.2) fails; in fact in the limit the system leaves the main stratum to penetrate a deeper stratum in the Borel decomposition. In this paper we show this specialization s =−t leads to its own system, the Pfaff lattice on a successor L to the 2-Toda Lax pair (L1,L2), whereas in [7], we have shown this system is integrable by producing a Lax pair ∂L i i = −πkL ,L = πnL ,L , ∂ti ∞ 1 Here “t, s ∈ C ” is an informal way of saying that t and s are two sequences of independent ∞ scalar variables. Under suitable assumptions, m∞(t, s) exists for all t, s ∈ C and the decompo- ∞ sition holds for “generic” t, s ∈ C , but in general a function of those variables may be defined ∞ ∞ only in an open subset of C × C , or may even be a formal power series in t and s. Pfaff τ-functions 3 on semi-infinite matrices of the form 01 −d1 a1 O d1 1 L = . −d2 a2 d 2 1 . ∗ .. .. The projections πk and πn correspond to the Lie algebra splitting (in the formula below, lower-triangular with special diagonal means: the diagonal consists of 2 × 2 blocks, each of them proportional to the 2 × 2 identity) k ={ } gl(∞) = k ⊕ n lower-triangular matrices, with special diagonal n = sp(∞) ={a such that JaJ = a}, where 01 O − 10 01 − J := 10 with J 2 =−I. O 01 − 10 . .. The precise projections take on the following form2 a = (a) + (a) k n 1 = (a− − J(a+) J)+ (a0 − J(a0) J) 2 1 + (a+ + J(a+) J)+ (a + J(a ) J) . 2 0 0 The solution to the Pfaff lattice can be expressed in terms of “Pfaff τ-functions" τ(t)˜ as follows: L(t) = Q(t)ΛQ(t)−1, 2 a± refers to projection onto strictly upper (strictly lower) triangular matrices, with all 2 × 2 diagonal blocks equal zero. a0 refers to projection onto the “diagonal", consisting of 2 × 2 blocks. 4 M. Adler et al. where Q(t) is the lower-triangular matrix whose entries are given by the coeffi- cients of the polynomials in λ: 2n j q2n(t, λ) := Q2n,j (t)λ j=0 λ2n −1 = τ˜2n(t −[λ ]) τ˜2nτ˜2n+2 2n+1 j q2n+1(t, λ) := Q2n+1,j (t)λ j=0 λ2n ∂ = (λ + )τ˜ (t −[λ−1]) ∂t 2n τ˜2nτ˜2n+2 1 where τ˜2n(t) are Pfaffians: 1/2 1/2 τ˜2n(t) := Pf m2n(t, −t) = (det m2n(t, −t)) = τ2n(t, −t) , (0.5) for every even n ∈ Z≥0. The qi are skew-orthonormal polynomials with respect to a skew inner-product , , namely qi,qj =Jij in terms of the J -matrix defined earlier; see [7]. The “Pfaffian τ˜-function” is itself not a 2-Toda τ-function, but it ties up remarkably with the 2-Toda τ-function τ (see (0.3)) as follows3: τ2n(t, −t −[α]+[β]) =˜τ2n(t)τ˜2n(t +[α]−[β]) (0.6) τ2n+1(t, −t −[α]+[β]) = (β − α)τ˜2n(t −[β])τ˜2n+2(t +[α]). When β → α, we approach the deeper stratum in the Borel decomposition of m∞ in a very specific way. It shows that the odd τ-functions τ2n+1(t, −t −[α]+[β]) approach zero linearly as β → α, at the rate depending on α: lim τ n+ (t, −t −[α]+[β])/(β − α) =˜τ n(t −[α])τ˜ n+ (t +[α]). β→α 2 1 2 2 2 Equations (0.6) are crucial in establishing bilinear relations for Pfaffian τ˜-func- tions: for all t,t ∈ C∞ and m, n positive integers ∞ i −1 −1 i= (ti −ti )z 2n−2m−2 τ˜2n(t −[z ])τ˜2m+2(t +[z ])e 0 z dz z=∞ ∞ −i i= (ti −ti )z 2n−2m + τ˜2n+2(t +[z])τ˜2m(t −[z])e 0 z dz = 0 , (0.7) z=0 3 [α]:=(α, α2/2,α3/3,...). Pfaff τ-functions 5 This bilinear identity leads to different types of relations, involving nearest neighbors, like the “differential Fay identity”, {˜τ (t −[u]), τ˜ (t −[v])} 2n 2n −1 −1 + (u − v ) τ˜2n(t −[u])τ˜2n(t −[v]) −˜τ2n(t)τ˜2n(t −[u]−[v]) = uv(u − v)τ˜2n−2(t −[u]−[v])τ˜2n+2(t) , (0.8) and the Hirota bilinear equations4, ˜ 1 ˜ pk+ (D) − D Dk+ τ˜ n ·˜τ n = pk(D)τ˜ n+ ·˜τ n− . (0.9) 4 2 1 3 2 2 2 2 2 2 For k = 0, this equation can be viewed as an inductive expression of τ˜2n+2 in terms of τ˜2n−2 and derivatives of τ˜2n. In analogy with the 2-Toda or KP theory, we establish Fay identities for the Pfaff τ˜-functions. In this instance, they involve Pfaffians rather than determinants: (zj − zi)τ˜2n−2(t −[zi]−[zj ]) Pf τ˜ (t) 2n 1≤i,j≤2k τ˜ t − 2k [z ] 2n−2k i=1 i = ∆(z) , (0.10) τ˜2n(t) In particular for k = 2, (z1 − z0)(z2 − z3)τ˜2n(t −[z0]−[z1])τ˜2n(t −[z2]−[z3]) 1→2→3→1 =− (zi − zj ) τ˜2n+2(t)τ˜2n−2(t −[z0]−[z1]−[z2]−[z3]) 0≤i<j≤3 which has a useful interpretation in terms of the Pfaffians of Christoffel-Darboux kernels of the form Kn(µ, λ) n−1 ∞ i i i= ti (µ +λ ) = e 1 q2k(t, λ)q2k+1(t, µ) − q2k(t, µ)q2k+1(t, λ) , (0.11) k=0 4 ˜ ˜ ∂ = (∂/∂t1,(1/2)∂/∂t2,(1/3)∂/∂t3,...), D = (D1,(1/2)D2,(1/3)D3,...) is the corre- P(D)f˜ · g := P(∂/∂y ,( / )∂/∂y ,...)f(t + y)g(t − y)| sponding Hirota symbol: 1 1 2 2 y=0, and p ∞ p (t)zk := ( ∞ t zi) k are the elementary Schur functions: k=0 k exp i=1 i . 6 M. Adler et al. where the qm(t, λ) form the system of skew-orthogonal polynomials, mentioned above (see [7]). This is the analogue of the Christoffel-Darboux kernel for or- thogonal polynomials.