The Grothendieck group of unipotent representations: A new basis
<p>Let <inline-formula content-type="math/mathml">
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper G left-parenthesis upper F Subscript q Baseline right-parenthesis">
  <mml:semantics>
    <mml:mrow>
      <mml:mi>G</mml:mi>
      <mml:mo stretchy="false">(</mml:mo>
      <mml:msub>
        <mml:mi>F</mml:mi>
        <mml:mi>q</mml:mi>
      </mml:msub>
      <mml:mo stretchy="false">)</mml:mo>
    </mml:mrow>
    <mml:annotation encoding="application/x-tex">G(F_q)</mml:annotation>
  </mml:semantics>
</mml:math>
</inline-formula> be the group of rational points of a simple algebraic group defined and split over a finite field <inline-formula content-type="math/mathml">
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper F Subscript q">
  <mml:semantics>
    <mml:msub>
      <mml:mi>F</mml:mi>
      <mml:mi>q</mml:mi>
    </mml:msub>
    <mml:annotation encoding="application/x-tex">F_q</mml:annotation>
  </mml:semantics>
</mml:math>
</inline-formula>. In this paper we define a new basis for the Grothendieck group of unipotent representations of <inline-formula content-type="math/mathml">
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper G left-parenthesis upper F Subscript q Baseline right-parenthesis">
  <mml:semantics>
    <mml:mrow>
      <mml:mi>G</mml:mi>
      <mml:mo stretchy="false">(</mml:mo>
      <mml:msub>
        <mml:mi>F</mml:mi>
        <mml:mi>q</mml:mi>
      </mml:msub>
      <mml:mo stretchy="false">)</mml:mo>
    </mml:mrow>
    <mml:annotation encoding="application/x-tex">G(F_q)</mml:annotation>
  </mml:semantics>
</mml:math>
</inline-formula>.</p>