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Let $G=GL(2, F)$ and $K$ a maximal compact subgroup. Unramified Hecke operators are defined by the action of the double cosets $$T(n) = \bigcup_{\substack{ad=n, a>0 \\ a|d}} K \left( \begin{array}{cc} a & \\ & d \end{array} \right) K$$

There is a decomposition in right cosets $$\bigcup_{\substack{ad=n, a>0}} \bigcup_{b \mod d}K \left( \begin{array}{cc} a & b \\ & d \end{array} \right)$$

I would like to know the volumes of $T(n)$. So here are my questions:

  1. is there a similar standard decomposition in left cosets?

  2. can we directly compute the measure (for a fixed left Haar measure on $G$) of the right cosets?

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    $\begingroup$ For 1, the first definition is manifestly invariant under transpose (if $K$ is the standard maximal compact subgroup) so you can just take the transpose of the second definition, which gives a right coset description. For 2, for $GL_2$, left and right Haar measure are the same so all cosets should have the same measure. $\endgroup$
    – Will Sawin
    Commented Dec 19, 2017 at 21:42
  • $\begingroup$ What is $F$? (And then some more characters.) $\endgroup$
    – LSpice
    Commented Dec 20, 2017 at 2:29

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