An arithmetic property of intertwining operators for p-adic groups

TL;DR

This paper proves an arithmetic property of normalized standard intertwining operators for p-adic groups, a key step in establishing algebraicity of special values of automorphic L-functions using Eisenstein cohomology.

Contribution

It establishes a general arithmetic property of intertwining operators within the Langlands-Shahidi framework, crucial for automorphic L-function studies.

Findings

Proves the arithmetic property of normalized intertwining operators for p-adic groups.

Addresses a key local step in automorphic L-function algebraicity proofs.

Provides a foundation for future research on automorphic L-functions and their special values.

Abstract

If one proposes to use the theory of Eisenstein cohomology to prove algebraicity results for the special values of automorphic L-functions as in my work with Harder for Rankin-Selberg L-functions, or its generalizations as in my work with Bhagwat for L-functions for orthogonal groups and independently with Krishnamurthy on Asai L-functions, then in a key step, one needs to prove that the normalised standard intertwining operator between induced representations for p-adic groups has a certain arithmetic property. The principal aim of this article is to address this particular local problem in the generality of the Langlands-Shahidi machinery. The main result of this article is invoked in some of the works mentioned above, and I expect that it will be useful in future investigations on the arithmetic properties of automorphic L-functions.