Research Interests | Research Papers | Notes / Expository Articles

ArXiv version 2017

International Journal of Number Theory. 13, No. 4, pp. 853--884 (2017)

ArXiv version 12 April 2012

ArXiv version 18 June 2010

pdf version Journal of Number Theory Volume 130, Issue 1, January 2010, Pages 101-115

pdf version 1 July 2008

To address the specific question of characterizing families of Hecke operators whose generating series have ``Euler'' products, we define $(n+1)$ families of polynomial Hecke operators $t_k^n(p^\ell)$ (in ${\mathbb{Q}}[x_0^{\pm1}, \dots,x_n^{\pm1}]$) for $Sp_n$ whose generating series $\sum t_k^n(p^\ell) v^\ell$ are rational functions of the form $q_k(v)^{-1}$, where $q_k$ is a polynomial in ${\mathbb{Q}}[x_0^{\pm1}, \dots,x_n^{\pm1}][v]$ of degree $2^k\binom{n}{k}$ ($2^n$ if $k = 0$). For $k=0$ and $k=1$ the form of the polynomial is essentially that of the local factors in the spinor and standard zeta functions. For $k>1$, these appear to be new expressions. Taking advantage of the generating series and our ability to explicitly invert the Satake isomorphism, we explicitly compute the classical operators with the analogous properties in the case of genus 2. It is of interest to note that these operators lie in the full, but not generally the integral, Hecke algebra.

pdf version 9 May 2006

We then give a natural representation of the local Hecke algebra over $K$ acting on the special vertices of the Bruhat-Tits building for $Sp_n(K)$. Finally, we give an application of the Hecke operators defined on the building by characterizing minimal walks on the building for $Sp_n$.

(revised PDF) 12 July 2005

Transactions AMS 359, (2007), 3409-3423.

We also offer some insight (disjoint from the representation theory) for why there should be a correspondence between the local Hecke algebra and a ring of polynomials invariant under an associated Weyl group.

pdf version 19 Dec 2003

pdf version 9 Aug 2002 (Acta Arith 112 (2004))

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 $GL_4$.

pdf version 31 July 2002 (JNT 102 278 -- 297)

pdf version 11 Apr 2002