Matrix integrals, Toda symmetries, Virasoro constraints, and orthogonal polynomials
The paper is organized by the following points: 1. The Toda lattice revisited, 2. Symmetries of the Toda lattice and the \(w_2\)-algebra, 3. The action of the symmetries on \(\tau\)-functions, 4. Orthogonal polynomials, matrix integrals, skew-symmetric matrices, and Virasoro constraints, 5. Classical orthogonal polynomials, 6. Appendix: Virasoro constraints via the integrals. Theorem 1.2 states that the operators \(L\) and \(M\) defined by a new wave operator \(S\) satisfy the usual equations ``multiplication by \(z\)'' and ``differentiation by \(z\)'' with respect to a new wave function \(\psi\). Moreover the tridiagonal matrix \(L\) is proved to be symmetric, and the bracket of \(L\) with \(M\) equals 1. In point 2 symmetries as vector fields acting on the manifold of wave functions are proved to commute with the Toda vector fields and to induce vector fields on \(S\) and \(L\) (Theorem 2.2). The fundamental theorem 3.2 states a relationship between the action of symmetries on the wave functions and on the \(\tau\)-functions, which is the central part of the paper. In point 4 an orthonormal polynomial basis with respect to a suitable weight on \(H^+\) is considered and semi-infinite matrices \(L\) and \(M\) and a semi-infinite vector \(\psi\) are defined. Theorem 4.2 states that the wave vector constructed from basis of orthogonal polynomials is a fixed point with respect to a Lie algebra of symmetries defined by semi-infinite matrices \(L\) and \(Q\). The matrix \(Q\) is defined by \(L\) and \(M\) and turns out to be skew-symmetric, and the anticommutator matrices of \(Q\) with \(L^m\) form a (non-standard) representation of the Lie algebra of holomorphic vector fields \(\text{Diff} (S^1)^+\). This in turn implies Virasoro like constraints on the \(\tau\)-vector (Theorem 4.2). In point 5 the classical orthogonal polynomials are revisited from the point of view of the analysis discussed before. There is a conjecture that the existence of two operators, a symmetric and a skew-symmetric one, both represented by tridiagonal matrices may characterize the orthogonal polynomials on the line.
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