Rationality theorems for Hecke operators on GLn

JOURNAL OF NUMBER THEORY

卷:102

Article

Rhodes, JA ; Shemanske, TR

关键词: Hecke operators;    rationality;    generating series;   

DOI  :  10.1016/S0022-314X(03)00081-7

来源: Elsevier

【 摘 要 】

We define n families of Hecke operators T-k(n)(p') for GL(n) whose generating series SigmaT(k)(n)(p')u' are rational functions of the form q(k)(u)(-1) where qk is a polynomial of degree ((n)(k)), 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 Pi(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 GL(3) and GL(4). (C) 2003 Elsevier Inc. All rights reserved.

【 授权许可】

Free