Normalized intertwining operators and nilpotent elements in the Langlands dual group

TL;DR

This paper constructs explicit unitary isomorphisms between certain $L^2$-spaces associated with parabolic subgroups of a split reductive group over a non-archimedean local field, using nilpotent elements in the Langlands dual group, and applies these to refine $L$-function constructions.

Contribution

It introduces a new approach to intertwining operators involving nilpotent elements in the dual group and defines a novel function space for $L$-functions reformulation.

Findings

Explicit unitary isomorphisms between $L^2$-spaces for parabolics with the same Levi.

A new function space $\\calS(G,M)$ independent of the Levi component.

Reformulation of classical $L$-function constructions using the new framework.

Abstract

Let $F$ be a local non-archimedian field and let $G$ be a group of points of a split reductive group over $F$. For a parabolic subgroup $P$ of $G$ we set $X_P=G/[P,P]$. For any two parabolics $P$ and $Q$ with the same Levi component $M$ we construct an explicit unitary isomorphism $L^2(X_P)\to L^2(X_Q)$ (which depends on a choice of an additive character of $F$). The formula for the above isomorphism involves the action of the principal nilpotent element in the Langlands dual group of $M$ on the unipotent radicals of the corresponding dual parabolics. We use the above isomorphisms to define a new space $\calS(G,M)$ of functions on $X_P$ (which depends only on $P$ and not on $M$). We explain how this space may be applied in order to reformulate in a slightly more elegant way the construction of $L$-functions associated with the standard representation of a classical group due to Gelbart, Piatetski-Shapiro and Rallis.