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Let $\Gamma_n = \operatorname{Sp}_n(\Bbb Z)$ and $\Gamma_{n,0} \subset \Gamma_n$ be a subgroup. We write $$ M = \begin{pmatrix}A & B\\ C & D \end{pmatrix} \in \operatorname{Sp}_n(\mathbb{Z})$$ where $M$ is a $(2n)\times(2n)$ matrix and $A,B,C,D$ are $n\times n$ matrices.

I am reading some function theory script and I encountered the writing (definition of the Eisensteinseries for Siegel modular forms)

$$ E_k(Z) := \sum_{M : \Gamma_{n,0} \backslash \Gamma_n} \det(CZ + D)^{-k} . $$

What does $M : \Gamma_{n,0} \backslash \Gamma_n$ mean? What is $M$?

I know "$\backslash$" as a set-substraction but that doesn't seem to make sense here.

It is probably meant as some form of Quotient space but I'm not sure what exactly. Also, I only know "$/$" for the notation of Quotient spaces.

Also, if $\Gamma_{n,0} \backslash \Gamma_n$ means something (a set), why does the author write "$M:$" and not "$M \in$"? (The same author uses "$\in$" everywhere else.)

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    $\begingroup$ Can you write down the whole sentence? $\endgroup$ May 24, 2013 at 13:53
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    $\begingroup$ @Albert, yes, please write the complete phrase. As it stands, the $\Gamma_{n,0}\backslash \Gamma_n$ means cosets $\Gamma_{n,0}\gamma$ with $\gamma\in\Gamma_n$, but it is impossible to tell what the $M:$ is doing. $\endgroup$ May 24, 2013 at 14:08

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In the theory of automorphic forms, quotients or coset spaces on both sides arise, and $X\backslash Y$ is written for example when $X$ is a subgroup of $Y$ and we want cosets of the form $Xy$ for $y\in Y$.

The group $Sp_n(\mathbb Z)$ is the group of $2n$-by-$2n$ integer matrices $g$ such that $g^t J_n g=J_n$ where $J_n=\pmatrix{0_n&-1_n\cr 1_n&0_n}$, where $1_n$ and $0_n$ are $n$-by-$n$ identity and $0$ matrices. Here, $\Gamma_n$ is another notation for $Sp_n(\mathbb Z)$, and $\Gamma_{n,0}$ is elements $g=\pmatrix{*&*\cr 0&*}$ therein.

The use of "$M:$" is indeed a bit inconsistent, but not surprising as a descriptor of summation. Here, $M$ is meant to range over representatives for that quotient, with lower $n$-by-$n$ block entries $C,D$.

That quotient has representatives uniquely determined by the lower halves $\pmatrix{C&D}$, and the pair $(C,D)$ is left-modulo $GL_n(\mathbb Z)$.

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