Unipotent representations and reductive dual pairs over finite fields

Jeffrey Adams, Allen Moy

Type: Article

Publication Date: 1993-01-01

Citations: 37

DOI: https://doi.org/10.1090/s0002-9947-1993-1173855-4

Abstract

Consider the representation correspondence for a reductive dual pair <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="left-parenthesis upper G 1 comma upper G 2 right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msub> <mml:mi>G</mml:mi> <mml:mn>1</mml:mn> </mml:msub> </mml:mrow> <mml:mo>,</mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msub> <mml:mi>G</mml:mi> <mml:mn>2</mml:mn> </mml:msub> </mml:mrow> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">({G_1},{G_2})</mml:annotation> </mml:semantics> </mml:math> </inline-formula> over a finite field. We consider the question of how the correspondence behaves for unipotent representations. In the special case of cuspidal unipotent representations, and a certain fundamental situation, that of "first occurrence", the representation correspondence takes a cuspidal unipotent representation of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper G 1"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msub> <mml:mi>G</mml:mi> <mml:mn>1</mml:mn> </mml:msub> </mml:mrow> <mml:annotation encoding="application/x-tex">{G_1}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> to one of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper G 2"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msub> <mml:mi>G</mml:mi> <mml:mn>2</mml:mn> </mml:msub> </mml:mrow> <mml:annotation encoding="application/x-tex">{G_2}</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. This should serve as a fundamental case in studying the correspondence in general over both finite and local fields.

Locations

Transactions of the American Mathematical Society - View - PDF

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