A formal power series over a noncommutative Hecke ring and the rationality of the Hecke series for $GSp_4$

TL;DR

This paper extends previous work on noncommutative Hecke rings associated with p-adic Lie algebras, demonstrating a rationality property of Hecke series for GSp_4, analogous to classical results for GL_2.

Contribution

It introduces a formal power series over a noncommutative Hecke ring linked to a 5-dimensional Heisenberg Lie algebra and proves its rationality, generalizing prior results for lower dimensions.

Findings

Established a rationality identity for the Hecke series of GSp_4.

Extended the noncommutative Hecke ring framework to higher-dimensional Lie algebras.

Demonstrated the similarity to classical Hecke series rationality results.

Abstract

The present paper studies Hecke rings derived by the automorphism groups of certain algebras $L_p$ over the ring of $p$-adic integers. Our previous work considered the case where $L_p$ is the Heisenberg Lie algebra (of dimension 3) over the ring of $p$-adic integers. Although this Hecke ring is noncommutative, we showed that a formal power series with coefficients in this Hecke ring satisfies an identity similar to the rationality of the Hecke series for $GL_2$ due to E.~Hecke. In the present paper, we establish an analogous result in the case of the Heisenberg Lie algebra of dimension 5 over the ring of $p$-adic integers. In this case, our identity is similar to the rationality of the Hecke series for $GSp_4$, due to G.~Shimura.