Gelfand-Zeitlin theory from the perspective of classical mechanics. I

Summary

Let M(n) be the algebra (both Lie and associative) of n × n matrices over ℂ. Then M(n) inherits a Poisson structure from its dual using the bilinear form (x, y) = −tr xy. The Gl(n) adjoint orbits are the symplectic leaves and the algebra, P(n), of polynomial functions on M(n) is a Poisson algebra. In particular, if f ∈ P(n), then there is a corresponding vector field ξ _f on M(n). If m ≤ n, then M(m) embeds as a Lie subalgebra of M(n) (upper left hand block) and P(m) embeds as a Poisson subalgebra of P(n). Then, as an analogue of the Gelfand-Zeitlin algebra in the enveloping algebra of M(n), let J(n) be the subalgebra of P(n) generated by P(m)^Gl(m) for m = 1, . . ., n. One observes that

$$ J\left( n \right) \cong P\left( 1 \right)^{Gl\left( 1 \right)} \otimes \cdots \otimes P\left( n \right)^{Gl\left( n \right)} $$

. We prove that J(n) is a maximal Poisson commutative subalgebra of P(n) and that for any p ∈ J(n) the holomorphic vector field ξ _p is integrable and generates a global one-parameter group σ _p(z) of holomorphic transformations of M(n). If d(n) = n(n + 1)/2, then J(n) is a polynomial ring ℂ[p ₁, . . ., p _d(n)] and the vector fields $\xi _{p_i }$ , i = 1, . . ., d(n − 1), span a commutative Lie algebra of dimension d(n − 1). Let A be a corresponding simply-connected Lie group so that A ≅ ℂ^d(n−1). Then A operates on M(n) by an action σ so that if a ∈ A, then

$$ \sigma \left( a \right) = \sigma _{p_1 } \left( {z_1 } \right) \cdots \sigma _{pd\left( {n - 1} \right)} \left( {z_{d\left( {n - 1} \right)} } \right) $$

where a is the product of exp z _i $\xi _{p_i }$ for i = 1, . . ., d(n − 1). We prove that the orbits of A are independent of the choice of the generators p _i. Furthermore, for any matrix the orbit A · x may be explicitly given in terms of the adjoint action of a n − 1 abelian groups determined by x. In addition we prove the following results about this rather remarkable group action.

• (1) Let x ∈ M(n). Then A · x is an orbit of maximal dimension (d(n − 1)) if and only if the differentials (dp _i)_x, i = 1, . . ., d(n), are linearly independent.

• (2) The orbits, O _x, of the adjoint action of Gl(n) on M(n) are A-stable, and if O _x is an orbit of maximal dimension (n(n − 1)), that is, if x is regular, then the A-orbits of dimension d(n − 1) in O _x are the leaves of a polarization of a Zariski open dense subset of the symplectic manifold O _x.

  The results of the paper are related to the theory of orthogonal polynomials. Motivated by the interlacing property of the zeros of neighboring orthogonal polynomials on ℝ, we introduce a certain Zariski open subset M _Ω(n) of M(n) and prove

• (3) M_Ω(n) has the structure of (ℂ^×)^d(n−1) bundle over a (d(n)-dimensional) variety of Hessenberg matrices. Moreover, the fibers are maximal A-orbits. The variety of Hessenberg matrices plays a major role in this paper.

In Part II of this two-part paper, we deal with a commutative analogue of the Gelfand-Kirillov theorem. The fibration in (3) leads to the construction of n ² + 1 functions (including a constant function) in an algebraic extension of the function field of M(n) which, under Poisson bracket, satisfies the commutation relations of the direct sum of a 2 d(n − 1) + 1 dimensional Heisenberg Lie algebra and an n-dimensional commutative Lie algebra.

With admiration, To a dear friend and brilliant colleague. B.K.

To Tony on his 60th birthday: We honor your past accomplishments and anticipate your future successes. N.W.

Research supported in part by NSF grant DMS-0209473 and in part by the KG & G Foundation.

Research supported in part by NSF grant DMS-0200305.