Homological finiteness of principal series representations of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msub> <mml:mi mathvariant="normal">GL</mml:mi>

Abstract

<jats:p> Let <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>G</mml:mi> <mml:mo>=</mml:mo> <mml:msub> <mml:mi mathvariant="normal">GL</mml:mi> <mml:mi>n</mml:mi> </mml:msub> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>ℝ</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:mrow> </mml:math> be the general linear group over <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>ℝ</mml:mi> </mml:math> and let <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>B</mml:mi> </mml:math> denote the Borel subgroup of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>G</mml:mi> </mml:math> consisting of nonsingular upper triangular matrices. For every principal series representation <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>π</mml:mi> </mml:math> of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>G</mml:mi> </mml:math> , we show that the Schwartz homology <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msub> <mml:mi mathvariant="normal">H</mml:mi> <mml:mi>i</mml:mi> </mml:msub> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>B</mml:mi> <mml:mo>;</mml:mo> <mml:mi>π</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:mrow> </mml:math> is finite-dimensional for every <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>i</mml:mi> <mml:mo>∈</mml:mo> <mml:mi>ℤ</mml:mi> </mml:mrow> </mml:math> . The proof is based on a simple computation modulo the theory of Nash manifolds. Moreover, by applying Casselman’s subrepresentation theorem, we show that the conclusion still holds with the principal series representation <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>π</mml:mi> </mml:math> replaced by a Casselman–Wallach representation. </jats:p>