For $g \in \operatorname{SL}_2(\mathbb R)$, and $\mathbb H$ the upper half plane, and $k\geq 1$ an integer, the weight $k$-operator on functions $f: \mathbb H \rightarrow \mathbb C$ is defined by

$$f[g](z) = f(g.z) j(g,z)^{-k}$$

where $j(g,z) = (cz+d)^{-1}$, $g = \begin{pmatrix} a & b \\ c & d\end{pmatrix}$.

In order to define Hecke operators, or to adelize modular forms, or to identify modular forms as functions on $\operatorname{GL}_2(\mathbb R)^+$, it is necessary to extend this definition to $g \in \operatorname{GL}_2^+(\mathbb R)$. In *A First Course in Modular Forms*, in Chapter 5.1 Diamond and Shurman set

$$f[g](z) = f(g.z)j(g,z)^{-k} \det(g)^{k-1}$$

In *Automorphic Forms and Representations*, in Chapter 1.4 Bump sets

$$f[g](z) = f(g.z)j(g,z)^{-k} \det(g)^{k/2}$$

Which exponent of the determinant is better to use, and why? If we adelize a Hecke eigenform for $\operatorname{SL}_2(\mathbb Z)$ and look at the corresponding automorphic representation $\pi = \otimes_p \pi_p$, which normalization is better to define Hecke operators with, if we want the classical Hecke operator $T_p$ to coincide naturally with an action of the spherical Hecke algebra $\mathscr H(\operatorname{GL}_2(\mathbb Q), \operatorname{GL}_2(\mathbb Z_p))$ on the local component $\pi_p$?

Recall that to adelize a modular form $f$ of $\operatorname{SL}_2(\mathbb Z)$ of some given weight, we would first identify $f$ with a function $\phi$ on $\operatorname{GL}_2^+(\mathbb R)$ by setting

$$\phi(g) = f[g](i)$$

and then we would define an automorphic form $\varphi$ on $\operatorname{GL}_2(\mathbb Q) \backslash \operatorname{GL}_2(\mathbb A)$ by using the decomposition $\operatorname{GL}_2(\mathbb A) = \operatorname{GL}_2(\mathbb Q) \operatorname{GL}_2^+(\mathbb R)K$ for $K$ a suitable compact subgroup, writing $g = \alpha g_{\infty}k$, and setting $\varphi(g) = \phi(g_{\infty})$.

whatpower of det? For that matter, how much do you care about multiplying Hecke operators by scalars? The underlying effect is the same, whatever we do, ... $\endgroup$ – paul garrett May 3 '19 at 22:33