Holomorphy of normalized intertwining operators for certain induced representations I: a toy example

TL;DR

This paper proves the holomorphy of normalized local intertwining operators for certain induced representations of quasi-split classical groups, highlighting an intrinsic non-symmetry property that could extend to broader cases.

Contribution

It introduces a novel approach based on non-symmetry of normalization factors to establish holomorphy, offering potential for generalization beyond the specific case.

Findings

Proved holomorphy of normalized intertwining operators for a family of induced representations.

Identified an intrinsic non-symmetry property of normalization factors.

Suggested the approach could be applicable in more general settings.

Abstract

The theory of intertwining operators plays an important role in the development of the Langlands program. This, in some sense, is a very sophisticated theory, but the basic question of its singularity, in general, is quite unknown. Motivated by its deep connection with the longstanding pursuit of constructing automorphic $L$-functions via the method of integral representations, we prove the holomorphy of normalized local intertwining operators, normalized in the sense of Casselman--Shahidi, for a family of induced representations of quasi-split classical groups as an exercise. Our argument is the outcome of an observation of an intrinsic non-symmetry property of normalization factors appearing in different reduced decompositions of intertwining operators. Such an approach bears the potential to work in general.