Rationality theorems for Hecke operators on GLn

We define n families of Hecke operators Tkn( pℓ) for GLn whose generating series ∑Tkn( pℓ)uℓ are rational functions of the form qk(u)−1 where qk is a polynomial of degree , and whose form is that of the kth exterior product. This work can be viewed as a refinement of work of Andrianov (Math. USSR Sb. 12(3) (1970)), in which he defined Hecke operators the sum of whose generating series was a rational function with nontrivial numerator and whose denominator was essentially ∏k qk. By a careful analysis of the Satake map which defines an isomorphism between a local Hecke algebra and a ring of symmetric polynomials, we define n families of (polynomial) Hecke operators and characterize their generating series as rational functions. We then give an explicit means by which to locally invert the Satake isomorphism, and show how to translate these polynomial operators back to the classical double coset setting. The classical Hecke operators have generating series of exactly the same form as their polynomial counterparts, and hence are of number-theoretic interest. We give explicit examples for GL3 and GL4.