Towards a Goldberg–Shahidi pairing for classical groups

Arnab Mitra, Steven Spallone

Type: Article

Publication Date: 2017-07-08

Citations: 0

DOI: https://doi.org/10.1515/forum-2017-0033

Abstract

Abstract Let <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msup> <m:mi>G</m:mi> <m:mn>1</m:mn> </m:msup> </m:math> {G^{1}} be an orthogonal, symplectic or unitary group over a local field and let <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>P</m:mi> <m:mo>=</m:mo> <m:mrow> <m:mi>M</m:mi> <m:mo>⁢</m:mo> <m:mi>N</m:mi> </m:mrow> </m:mrow> </m:math> {P=MN} be a maximal parabolic subgroup. Then the Levi subgroup M is the product of a group of the same type as <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msup> <m:mi>G</m:mi> <m:mn>1</m:mn> </m:msup> </m:math> {G^{1}} and a general linear group, acting on vector spaces X and W , respectively. In this paper we decompose the unipotent radical N of P under the adjoint action of M , assuming <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mrow> <m:mo>dim</m:mo> <m:mo>⁡</m:mo> <m:mi>W</m:mi> </m:mrow> <m:mo>≤</m:mo> <m:mrow> <m:mo>dim</m:mo> <m:mo>⁡</m:mo> <m:mi>X</m:mi> </m:mrow> </m:mrow> </m:math> {\dim W\leq\dim X} , excluding only the symplectic case with <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mo>dim</m:mo> <m:mo>⁡</m:mo> <m:mi>W</m:mi> </m:mrow> </m:math> {\dim W} odd. The result is a Weyl-type integration formula for N with applications to the theory of intertwining operators for parabolically induced representations of <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msup> <m:mi>G</m:mi> <m:mn>1</m:mn> </m:msup> </m:math> {G^{1}} . Namely, one obtains a bilinear pairing on matrix coefficients, in the spirit of Goldberg–Shahidi, which detects the presence of poles of these operators at 0.

Locations

arXiv (Cornell University) - View - PDF
Forum Mathematicum - View

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