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This is certainly not a research level question, but I didn't get an answer to my question on MSE, so here goes:

The holomorphic Eisenstein series can be given as $$G_{2k}(z)=\sum_{(c,d)\in{\bf Z}^2\backslash(0,0)}\frac{1}{(cz+d)^{2k}}$$ while the real-analytic Eisenstein series is $$E(z,s)=\frac{1}{2}\sum_{(c,d)=1}\frac{y^s}{|cz+d|^{2s}}$$ satisfying the relation $$2\zeta(2s)E(s,z)=\sum_{(c,d)\in{\bf Z}^2\backslash(0,0)}\frac{\text{Im}(z)^s}{|cz+d|^{2s}}.$$ They look to be very similar, but have different analytic behaviours. I haven't found a discussions on how they are related, so I'm hoping to find some answers.

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  • $\begingroup$ Thanks! Your comment was enough for me to understand the references I am using. $\endgroup$
    – Tian An
    Commented Oct 23, 2015 at 2:17
  • $\begingroup$ I'm glad my comment was helpful -- I've reposted it as an answer. $\endgroup$
    – David Loeffler
    Commented Oct 23, 2015 at 6:39

1 Answer 1

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[Comment reposted as an answer]

Up to the scaling by $2 \zeta(2s)$, which is just a matter of conventions, both are special cases of a single more general object: the series $$ E_k(z, s) = \sum_{(c, d) \in \mathbf{Z}^2 \setminus (0,0)} \frac{y^s}{(cz + d)^k |cz + d|^{2s}}, $$ which converges for $Re(s) \gg 0$ (actually for $k + 2 Re(s) > 2$ if I remember correctly) and has meromorphic continuation to all $s \in \mathbf{C}$ (analytic if $k \ne 0$).

For a fixed value of $s$, the function $E_k(-, s)$ transforms like a modular form of weight $k$ under the modular group (although it is not usually holomorphic in the $z$ variable).

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  • $\begingroup$ Just to clarify - the $E_-(z,s)$, for a fixed $s$, do they generate the same principal series representation for $SL_2$, except that $E_k$ corresponds to the weight $k$ part of the $SO(2)$ action in real place? $\endgroup$
    – Pig
    Commented Oct 23, 2015 at 7:07
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    $\begingroup$ @user31814, for generic $s$, they all generate the same irreducible principal series of $SL_2(\mathbb R)$, but the holomorphic/anti-holomorphic discrete series appear as proper subrepresentations of certain principal series, so the vectors inside those holo discrete series obviously cannot generate the whole thing. $\endgroup$
    – paul garrett
    Commented Oct 23, 2015 at 14:16
  • $\begingroup$ @paulgarrett, thanks! Somehow I have never seen this "combined" object before. $\endgroup$
    – Pig
    Commented Oct 23, 2015 at 14:33
  • $\begingroup$ I have a clarification in the last statement. When you say for a fixed value of $s$, the function $E_k(-,s)$ transforms like a modular form of weight $k$, do you require $Re(s) \gg 0$? or does this statement hold for example $s=1/2$ ? $\endgroup$
    – Krishnarjun
    Commented Feb 22, 2023 at 7:12
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    $\begingroup$ Yes, this statement holds for all $s$ (except $s = 0$ and $s = 1$ when $k = 0$, in which case the series is divergent). This follows easily from analytic continuation -- the functions $E_k(\gamma z, s)$ and $j(\gamma, z)^k E_k(z, s)$ agree for $Re(s) \gg 0$, so their analytic continuations must agree everwhere they are both defined. $\endgroup$
    – David Loeffler
    Commented Feb 22, 2023 at 7:26

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