Monday, November 26, 2012

Automorphic Forms and L-functions for the group GL(n,R)

$\newcommand{\bpm}{\begin{pmatrix}} \newcommand{\epm}{\end{pmatrix}} \newcommand{\bsm}{\begin{smallmatrix}} \newcommand{\esm}{\end{smallmatrix}} \newcommand{\bspm}{\left(\begin{smallmatrix}} \newcommand{\espm}{\end{smallmatrix}\right)} \newcommand{\bm}{\begin{matrix}} \renewcommand{\em}{\end{matrix}} \newcommand{\bbm}{\begin{bmatrix}} \newcommand{\ebm}{\end{bmatrix}} \newcommand{\bs}{\backslash} \newcommand{\C}{\mathbb{C}} \newcommand{\G}{\mathbb{G}} \newcommand{\A}{\mathbb{A}} \newcommand{\Q}{\mathbb{Q}} \newcommand{\Z}{\mathbb{Z}} \newcommand{\R}{\mathbb{R}} \newcommand{\N}{\mathbb{N}} \renewcommand{\H}{\mathbb{H}} \newcommand{\Tr}{\operatorname{Tr}} \newcommand{\Stab}{\operatorname{Stab}} \newcommand{\Ad}{\operatorname{Ad}} \newcommand{\Invt}{\operatorname{Invt}} \newcommand{\Vol}{\operatorname{Vol}} \newcommand{\Dyn}{\operatorname{Dyn}} \newcommand{\Gal}{\operatorname{Gal}} \newcommand{\diag}{\operatorname{diag}} \newcommand{\pr}{\operatorname{pr}} \newcommand{\der}{\operatorname{der}} \newcommand{\Ind}{\operatorname{Ind}} \newcommand{\End}{\operatorname{End}} \newcommand{\Aut}{\operatorname{Aut}} \newcommand{\Hom}{\operatorname{Hom}} \newcommand{\Inn}{\operatorname{Inn}} \newcommand{\inc}{\operatorname{inc}} \newcommand{\sym}{\operatorname{sym}} \newcommand{\Syl}{\operatorname{Syl}} \newcommand{\Ker}{\operatorname{Ker}} \newcommand{\Spin}{\operatorname{Spin}} \newcommand{\GSpin}{\operatorname{GSpin}} \newcommand{\Span}{\operatorname{Span}} \newcommand{\st}{\text{ s.t. }} \newcommand{\bx}{\boxed} \newcommand{\tx}{\text} \newcommand{\la}{\langle} \newcommand{\ra}{\rangle} \newcommand{\lla}{\left\langle} \newcommand{\rra}{\right\ra} \newcommand{\ptl}[2]{\frac{\partial #1}{\partial #2}} \newcommand{\dd}[2]{\frac{d #1}{d #2}} \newcommand{\ddt}{\dd{}t} \newcommand{\ve}{\varepsilon} \newcommand{\vph}{\varphi} \newcommand{\on}{\operatorname} \newcommand{\ol}{\overline} \newcommand{\ul}{\underline} \renewcommand{\Re}{\on{Re}} \renewcommand{\Im}{\on{Im}} \newcommand{\gm}{\gamma} \newcommand{\Gm}{\Gamma} \newcommand{\sg}{\sigma} \newcommand{\quo}[1]{#1(F)\bs #1(\A)} \newcommand{\il}{\int\limits_} \newcommand{\iq}[1]{\il{\quo{#1}}} \newcommand{\Res}{\on{Res}} \newcommand{\wt}{\widetilde} \newcommand{\Mat}{\on{Mat}} \newcommand{\f}{\mathfrak} \renewcommand{\c}{\mathcal} \newcommand{\heart}{\heartsuit} \newcommand{\cusp}{\on{cusp}}$ p. 44, midpage, it is asserted that "Proposition 2.2.3 shows that the ring of differential operators $\c D^n$ is a realization of the universal enveloping algebra of the Lie algebra $\f{gl}(n, \R).$" What proposition 2.2.3 shows is that the function $\alpha \mapsto D_\alpha$ extends to a ring homomorphism from the universal enveloping algebra to $\c D^n.$ To complete the proof that $\c D^n$ is a realization, one must show that the kernel of this ring homomorphism is trivial. See Lemma 4.5.5 of Goldfeld-Hundley Vol. 1, and its correction below.

p. 44, proposition 2.2.6 is in error. In fact, right invariance by the element $\delta_1$ is not, in general, preserved by the action of the differential operators. To see this, let $f$ be a function $GL(2, \R) \to \C$ which is right-invariant by $\delta_1,$ define $h = D_\alpha f,$ where $\alpha = \bspm 0&1\\0&0\espm,$ and suppose that $h$ is also $\delta_1$-invariant. Then $$ h(g) = h(g\delta_1) = \lim_{t\to 0}\frac 1t (f(g\delta_1 t\alpha)-f(g\delta_1) ) $$ $$= \lim_{t\to 0}\frac 1t (f(g (-t)\alpha\delta_1)-f(g\delta_1) )$$ $$= \lim_{t\to 0}\frac 1t (f(g (-t)\alpha)-f(g) ) =D_{-\alpha}f(g) = -h(g). $$ (We used the fact that $\delta_1 \alpha \delta_1 = -\alpha.$) This shows that if $f$ and $D_\alpha f$ are both fixed by $\delta_1,$ then $D_\alpha f =0.$ It is shown on p. 47 that $D_\alpha$ is essentially $y\ptl{}x,$ so there are many functions which are fixed by $\delta_1$ and not killed by $D_\alpha,$ including many which are fixed by $GL(2, \Z)$ on the left and by the center.

p. 50, proposition 2.3.5 is not correct. In fact, the center of the universal enveloping algebra of $\f{gl}(n, \C)$ is a polynomial algebra in $n$ generators, not $n-1.$ Indeed, proposition 2.3.3 on p. 47 shows that a certain differential operator lies in the center of the universal enveloping algebra for $2 \le m \le n.$ But if $m=1,$ the proof still works. The operator corresponding to $m=1$ is $D_{I_n},$ where $I_n$ is the $n \times n$ identity matrix. This operator is of degree 1, so it is clearly not in the algebra generated by the others.

On p. 105, $\heartsuit$ is described as ``an endomorphism of $\c L^2_{\cusp}= \c L^2_{\cusp}(SL(2, \Z)\bs \f h)$ whose image is purely cuspidal.'' It should be ``an endomorphism of $\c L^2= \c L^2(SL(2, \Z)\bs \f h)$ whose image is purely cuspidal.'' (The key property of $\heart$ is that $\heart f$ is cuspidal even when $f$ is not.)

On p. 108, it says ``it is easy to check that $\heart$ preserves the space $\c L^2_{\cusp,+}$ (in fact, all operators in sight do).'' This may be true but it's not relevant. The key property of $\heart$ is that it maps $\c L^2_+$ into $\c L^2_{\cusp, +}.$

On p. 108 it says "to show that there exist even cusp forms, we must show that $\heart$ is nonzero on $\mathcal{L}^2_{\text{cusp},+}$." It should be ``on $\c L^2_+.$''

On p. 109, the space $C^\infty_+(\f S(Y))$ should be defined as $C^\infty(\f S(Y)) \cap \c L^2_+,$ rather than $C^\infty(\f S(Y)) \cap \c L^2_{\cusp,+}.$

On p. 110, in the last paragraph, the functions $f_{jk}$ are contained in $C^\infty_+(\f S(Y))\subset\c L^2_+,$ but there is no reason to think that they are in $\c L^2_{\cusp,+},$ so one should assume that they are not. Clearly $$ \int_0^1 f_{jk}(x+iy) \, dx = 0 $$ for all $y\ge 1,$ but for smaller values of $y,$ all bets are off. Likewise $W$ is not contained in $\c L^2_{\cusp,+}.$ Only the space $V$ obtained by applying the heart operator to $W$ lies in $\c L^2_{\cusp,+}.$

CORRECTIONS FOR THE PAPERBACK EDITION

The items to be corrected are listed by page number and line number and take the following form: ITEM $--->$ CORRECTED ITEM.

Introduction, Page xii, Line -1
I would like to especially thank Dan Bump, Kevin Broughan, Sol Friedberg, Jeff Hoffstein, Alex Kontorovich, Wenzhi Luo, Carlos Moreno, Yannan Qiu, Ian Florian Sprung, C.J. Mozzochi, Peter Sarnak, Freydoon Shahidi, Meera Thillainatesan, Qiao Zhang, for clarifying and improving various proofs, definitions, and historical remarks in the book.
$---->$
I would like to especially thank Dan Bump, Kevin Broughan, Farrel Brumley, Solomon Friedberg, Gergely Harcos, Jeffrey Hoffstein, Joseph Hundley, Alex Kontorovich, Xiaoqing Li, Min Lee, Wenzhi Luo, Carlos Moreno, Yannan Qiu, Ian Florian Sprung, C.J. Mozzochi, Peter Sarnak, Freydoon Shahidi, Nicolas Templier, Meera Thillainatesan, Qiao Zhang, for pointing out errors and clarifying and improving various proofs, definitions, and historical remarks in the book.

Page 44, Line 11
Proposition 2.2.3 shows that the ring of differential operators $\frak D^n$ is a realization of the universal enveloping algebra of the Lie algebra $\frak g\frak l(n, \mathbb R).$
$---->$
Proposition 2.2.3 shows that the function $\alpha \to \frak D_\alpha$ extends to a ring homomorphism from the universal enveloping algebra to $\frak D^n$. To complete the proof that $\frak D^n$ is a realization, one must show that the kernel of this ring homomorphism is trivial. See Lemma 4.5.4 in [Goldfeld-Hundley, Vol. 1, 2011].

Page 49, Line -5
$D_{1, 2} \circ D_{2, 1} \, f(z)$
$---->$
$D_{2, 1} \circ D_{1, 2} \, f(z)$

Page 50, Line 1
$D_{2, 1} \circ D_{1, 2} \, f(z)$
$---->$
$D_{1, 2} \circ D_{2, 1} \, f(z)$

Page 50, Proposition 2.3.5
Every differential operator which lies in $\frak D^n$ (the center of the universal enveloping algebra of $\frak g\frak l(n, \mathbb R)$) can be expressed as a polynomial (with coefficients in $\mathbb R$) in the Casimir operators defined in proposition 2.3.4. Furthermore, $\frak D^n$ is a polynomial algebra of rank $n-1$.
$---->$
Let $n \ge 2.$ Then $\frak D^n$ (the center of the universal enveloping algebra of $\frak g\frak l(n, \mathbb R)$) is a polynomial algebra of rank $n$. Every differential operator which lies in $\frak D^n$ can be expressed as a polynomial (with coefficients in $\mathbb R$) in the Casimir operators defined in proposition 2.3.3 and the differential operator $D_{I_n}$ where $I_n$ is the $n\times n$ identity matrix. Furthermore, $D_{I_n}$ annihilates any smooth function which is invariant by the center.

Page 52, Line 5
$$D_{i,j}^k\, I_s(z) = \begin{cases} s_{n-i}^k \cdot I_s(z) &\text{if $i = j$}\\ 0 &\text{otherwise,}\end{cases}$$
$---->$
$$D_{i,j}^k\, I_s(z) = \begin{cases} \left(\; \sum\limits_{k=1}^{n-i} ks_{n-k} - \sum\limits_{k=1}^{i-1} k s_k \right)^k \cdot I_s(z) &\text{if $i = j$,}\\ & \\ \;\; 0 &\text{otherwise,}\end{cases}$$

Page 52, Line -10
\begin{align*} D_{i,i}I_s(y) &= \frac{\partial}{\partial t} I_s\big (y + t y\cdot E_{i,i}\big) \, \Big |_{t=0}\\ &= \, \left(y_{n-i}\frac{\partial}{\partial y_{n-i}} - \sum_{\ell=n-i+1}^{n-1} y_{\ell}\frac{\partial}{\partial y_{\ell}}\right)I_s(y)\\ &= s_{n-i}\cdot I_s(y). \end{align*}
$---->$
\begin{multline*} D_{i, i} I_s(y) = \left.\frac{\partial}{\partial t} I_s\left(y+ty\cdot E_{i, i}\right)\right|_{t=0} \\ = \left. \frac{\partial}{\partial t} I_s \left( \begin{pmatrix} y_1 \cdots y_{n-1} & & & & \\ & \ddots & & & \\ & & y_1 \cdots y_{n-j} (1+t) & & \\ & & & \ddots & \\ & & & & 1\end{pmatrix} \right)\right|_{t=0} = \left( \sum_{k=1}^{n-i} ks_{n-k} - \sum_{k=1}^{i-1} k s_k \right) \cdot I_s(y) . \end{multline*}

Page 52, Line -6
$$D_{i,i}^k I_s(y) = \left(\frac{\partial}{\partial t}\right)^k I_s\left(y\cdot e^{t E_{i,i}}\right) \, \Big |_{t=0} = \, s_{n-i}^k I_s(y).$$
$---->$
$$D_{i, i}^k\, I_s(y) = \left(\; \sum_{k=1}^{n-i} ks_{n-k} - \sum_{k=1}^{i-1} k s_k \right)^k \cdot I_s(y).$$

Page 105, Line -10
The key idea of the present approach is to construct an explicit endomorphism $\heartsuit$ of $\mathcal L^2_{\text{cusp}} = \mathcal L^2_{\text{cusp}}\left(SL(2, \mathbb Z\backslash\frak h^2 \right)$ whose image is purely cuspidal.
$---->$
The basic idea of the present approach is to construct an explicit endomorphism $\heartsuit$ of $\mathcal L^2 = \mathcal L^2\left(SL(2, \mathbb Z\backslash\frak h^2 \right)$ whose image is purely cuspidal. (The key property of $\heartsuit$ is that $\heartsuit f$ is cuspidal even when $f$ is not.)

Page 108, Line -15
To show that there exist even cusp forms, we must show that $\heartsuit \ne 0$ on $\mathcal L^2_{\text{cusp},+}$.
$---->$
The key property of $\heartsuit$ is that it maps $\mathcal L^2_+$ into $\mathcal L^2_{\text{cusp},+}$. To show that there exist even cusp forms, we must show that $\heartsuit \ne 0$ on $\mathcal L^2_{+}$.

Page 109, Line -12
$$f \in C^{\infty}_{+}\left(\frak S(Y)\right) := C^{\infty}_c\left(\frak S(Y)\right) \cap \mathcal L^2_{\text{cusp},+}$$
$---->$
$$f \in C^{\infty}_{+}\left(\frak S(Y)\right) := C^{\infty}_c\left(\frak S(Y)\right) \cap \mathcal L^2_{+}$$

Page 110, Line -4
regarded as an element of $C^{\infty}\left(\frak S(Y)\right)_+ \; \subset \; \mathcal L^2_{\text{cusp},+}$.
$---->$
regarded as an element of $C^{\infty}\left(\frak S(Y)\right)_+ \; \subset \; \mathcal L^2_{+}$.

Page 117, Line -4
Cauchy-Schwartz \quad $---->$ \quad Cauchy-Schwarz

Page 118, Line -8
If $\phi$ satisfies conditions (1), (2), but does not satisfy condition (3) of definition 5.1.3, then the Fourier expansion takes the form $$\phi(z) = \sum_{\gamma \; \in \; U_{n-1}(\mathbb Z)\backslash SL(n-1, \mathbb Z)}\;\, \sum_{m_1 = -\infty}^\infty \;\sum_{m_2=0}^\infty \cdots \sum_{m_{n-1}=0}^\infty \; \tilde \phi_{(m_1,\ldots,m_{n-1})}\left( \left(\begin{matrix} \gamma & \\ & 1\end{matrix}\right) z \right).$$
$\Big\uparrow$
PLEASE DELETE ABOVE 3 LINES ON PAGE 118 IN THE PAPERBACK EDITION

Page 137, Lemma 5.6.8
Cauchy-Schwartz \quad $---->$ \quad Cauchy-Schwarz
Please make the above change in 3 places in Lemma 5.6.8 and its proof.

Page 141, Line -7
Cauchy-Schwartz \quad $---->$ \quad Cauchy-Schwarz

Page 163, Line -8
Let $T_\delta=\left(\begin{matrix} -1 & & & \\ & 1& \\ & & 1\end{matrix}\right)$. \qquad $---->$ \qquad $\delta=\left(\begin{matrix} -1 & & & \\ & 1& \\ & & 1\end{matrix}\right)$.

Page 268, Equation (9.3.5)
$$\frac{1}{N^{n-1/2}} \quad ---> \quad \frac{1}{N^{(n-1)/2}}$$

Page 307, Formula 10.7.1
$$E_{P}(z, s) = \sum_{P\backslash\Gamma} \text{Det}(\gamma z)^s, \qquad (\Re(s) > 2/n),$$
$---->$
$$E_{P}(z, s) = \sum_{P\backslash\Gamma} \text{Det}(\gamma z)^s, \qquad (\Re(s) > 1),$$

Page 308, Line 1
While it is not yet clear that (10.7.1) converges absolutely for $\Re(s) > 2/n$,
$---->$
While it is not yet clear that (10.7.1) converges absolutely for $\Re(s) > 1$,

Page 317, Line 1
where $s' \in \mathbb C^{n-1}$ is such that
$---->$
where $s' \in \mathbb C^r$ is such that

Page 317, Line 3
and $\lambda \in \mathbb C^{n-1}$ is such that
$---->$
and $\lambda \in \mathbb C^r$ is such that

Page 318, Line 3
Assume that $\phi$ is a Hecke eigenform, and
$---->$
Assume that $\phi$ is a Hecke eigenform (normalized so that the first Fourier coefficient is 1), and

Page 318, Line 7
$$\lambda_m(s) = \underset {C_1C_2\cdots C_r = m} {\sum_{ 1 \le C_1, C_2, \ldots, C_r \, \in \, \mathbb Z}} A_1(c_1) A_2(c_2) \cdots A_r(c_r)\cdot C_1^{s_1+\eta_1} C_2^{s_2+\eta_2} \cdots C_r^{s_r+\eta_r},$$
$---->$
$$\lambda_m(s) = \underset {C_1C_2\cdots C_r = m} {\sum_{ 1 \le C_1, C_2, \ldots, C_r \, \in \, \mathbb Z}} A_1(c_1) A_2(c_2) \cdots A_r(c_r)\cdot C_1^{s_1+\eta_1+\frac{n_1-n}{2}} C_2^{s_2+\eta_2+\frac{n_2-n}{2}} \cdots C_r^{s_r+\eta_r+\frac{n_r-n}{2}},$$

Page 318, Line 8
where $\eta_1 =0$ and $\eta_i = n_1 +n_2 +\cdots n_{i-1}$ for $i\ge 1.$
$---->$
where $\eta_1 = 0, \; \eta_2 = n_1$, and $\eta_i = n_1+n_2+\cdots +n_{i-1}$ for $3\le i \le r.$

Page 319, Line 7
$$= \; m^{-\frac{n-1}{2}} \hskip -10pt\sum_{C_1 C_2 \cdots C_r = m} \;\; \prod_{i=1}^r A_i(C_i)\, \phi_i\big(\frak m_{n_i}(z)\big) \cdot C_i^{s_i+\eta_i}\cdot \text{Det}\big(\frak m_{n_i}(y)\big)^{s_i}$$
$---->$
$$= \; m^{-\frac{n-1}{2}} \hskip -10pt\sum_{C_1 C_2 \cdots C_r = m} \;\; \prod_{i=1}^r A_i(C_i)\, \phi_i\big(\frak m_{n_i}(z)\big) \cdot C_i^{s_i+\eta_i + \frac{(n_i-1)}{2}}\cdot \text{Det}\big(\frak m_{n_i}(y)\big)^{s_i}$$

Page 320, Line -2
can be expressed in the form $\gamma = w \gamma'$ where $w$ is in the Weyl group and $\gamma' \in B_n(\mathbb Z)\backslash N'(\mathbb Z).$
$---->$
can be expressed in the form $\gamma = \rho w \eta'$ where $w$ is in the Weyl group, $\rho \in P(\mathbb Q)$, and $\eta' \in N'(\mathbb Q).$

Page 321, Line 2
$$\int\limits_{\big(({\gamma'}^{-1}w^{-1}P(\Bbb Z)w\gamma')\,\cap\, N'(\Bbb Z)\big)\big\backslash N'(\Bbb R)} \,\prod_{i=1}^r \phi_i(\frak m_{n_i}(w\gamma' u z)) \cdot \text{Det}(\frak m_{n_i}(w\gamma' u z))^{s_i}\; d^\ast u,$$
$---->$
$$\int\limits_{\big(({\eta'}^{-1}w^{-1}\rho^{-1}P(\Bbb Z)\rho\, w\,\eta')\,\cap\, N'(\Bbb Z)\big)\big\backslash N'(\Bbb R)} \,\prod_{i=1}^r \phi_i(\frak m_{n_i}(\rho w\eta' u z)) \cdot \text{Det}(\frak m_{n_i}(\rho w\eta' u z))^{s_i}\; d^\ast u,$$

Page 321, Line 3
which after changing variables $u \mapsto {\gamma'}^{-1} u \gamma'$ becomes
$---->$
which after changing variables $u \mapsto {\eta'}^{-1} u \eta'$ becomes

Page 321, Line 4
$$\int\limits_{\big((w^{-1}P(\Bbb Z)w)\,\cap\, N'(\Bbb Z)\big)\big\backslash N'(\Bbb R)} \,\prod_{i=1}^r \phi_i(\frak m_{n_i}(wu\gamma' z)) \cdot \text{Det}(\frak m_{n_i}(wu\gamma' z))^{s_i}\; d^\ast u.$$
$---->$
$$\int\limits_{\big((w^{-1}\rho^{-1} P(\Bbb Z) \rho\,w)\,\cap\, N'(\mathbb Z)\big)\big\backslash N'(\mathbb R)} \,\prod_{i=1}^r \phi_i(\frak m_{n_i}(\rho wu\eta' z)) \cdot \text{Det}(\frak m_{n_i}(\rho wu\eta' z))^{s_i}\; d^\ast u.$$

Page 321, Line 6
$$\phantom{}^0N = \left(M/A\right)\cap (wN'w^{-1}).$$
$---->$
$$\phantom{}^0N = \left(M/A\right)\cap (\rho wN'w^{-1} \rho^{-1}).$$

Page 321, Line 8
\begin{align*} &\int\limits_{\big((w^{-1}P(\mathbb Z)w)\,\cap\, N'(\mathbb Z)\big)\big\backslash (w^{-1}N'(\mathbb R)w)\backslash N'(\mathbb R)} \; \int\limits_{\phantom{}^0 N(\mathbb Z)\backslash\phantom{}^0 N(\mathbb R)}\,\prod_{i=1}^r \phi_i(\frak m_{n_i}(u_1 wu z))\\ & \hskip 220pt \cdot \text{Det}(\frak m_{n_i}(u_1 wu z))^{s_i}\; d^\ast u_1\, d^\ast u. \end{align*}
$---->$
\begin{align*} &\int\limits_{\big((w^{-1} \rho^{-1} P(\mathbb Z) \rho\,w)\,\cap\, N'(\mathbb Z)\big)\big\backslash (w^{-1}\rho^{-1}N'(\mathbb R)\, \rho w)\backslash N'(\mathbb R)} \; \int\limits_{\phantom{}^0 N(\mathbb Z)\backslash\phantom{}^0 N(\mathbb R)}\,\prod_{i=1}^r \phi_i(\frak m_{n_i}(u_1\rho wu z))\\ & \hskip 227pt \cdot \text{Det}(\frak m_{n_i}(u_1 \rho wu z))^{s_i}\; d^\ast u_1\, d^\ast u. \end{align*}

Page 353, Line -6
$$J_w(z; \psi_M, \psi_N^v, c) = \int\limits_{\bar U_w(\mathbb R) } I_s(wuz) \,e_{\psi_M}(wcuz) \, \overline{\psi_N^v(u)}\; d^\ast u,$$
$---->$
$$J_w(z; \psi_M, \psi_N^v, c) = \int\limits_{U_w(\mathbb Z)\backslash U_w(\mathbb R) } \; \int\limits_{\overline{ U}_w(\mathbb R) } I_s(wuz) \,e_{\psi_M}(cwuz) \, \overline{\psi_N^v(u)}\; d^\ast u,$$

Page 356, Equation (11.6.8)
$$ e_{\psi_M}(y, z') := \int\limits_{U_n(\mathbb Z)\backslash U_n(\mathbb R)} k(z^{-1} z') \psi_M(x)\, d^\ast x.$$
$---->$
$$ e_{\psi_M}(y, z') := \sum_{\gamma \in U_n(\mathbb Z)\backslash SL(n,\mathbb Z)} \;\int\limits_{U_n(\mathbb R)} k(z^{-1} \gamma z') \psi_M(x)\, d^\ast x.$$

Page 356, Line -7
\begin{align*} e_{\psi_M}(y, uz') &= \int\limits_{U_n(\mathbb Z)\backslash U_n(\mathbb R)} k(z^{-1} u z') \psi_M(x)\, d^\ast x\\ &= \int\limits_{U_n(\mathbb Z)\backslash U_n(\mathbb R)} k(z^{-1} z') \psi_M(ux)\, d^\ast x\\ &= \psi_M(u) \int\limits_{U_n(\mathbb Z)\backslash U_n(\mathbb R)} k(z^{-1} z') \psi_M(x)\, d^\ast x\\ &= \psi_M(u) e_{\psi_M}(y, z'),\end{align*}
$---->$
\begin{align*} e_{\psi_M}(y, uz') &= \int\limits_{U_n(\mathbb R)} k(z^{-1} u z') \psi_M(x)\, d^\ast x\\ &= \int\limits_{ U_n(\mathbb R)} k(z^{-1} z') \psi_M(ux)\, d^\ast x\\ &= \psi_M(u) \int\limits_{ U_n(\mathbb R)} k(z^{-1} z') \psi_M(x)\, d^\ast x\\ &= \psi_M(u) e_{\psi_M}(y, z'),\end{align*}

Page 357, Equation (11.6.10)
\begin{align*} J_w(y,y'; \psi_M, \psi_N^v, c) &= \int\limits_{\bar U_w} e_{\psi_M}(y, wcuy')\overline{\psi_N^{v}(u)}\; d^\ast u\\ &= \int\limits_{\bar U_w} \;\int\limits_{U_n(\mathbb Z)\backslash U_n(\mathbb R)} k(z^{-1} \cdot wcuy') \, \psi_M(x)\, \overline{\psi_N^{v}(u)}\; d^\ast x\, d^\ast u,\end{align*}
$---->$
\begin{align*} J_w(y,y'; \psi_M, \psi_N^v, c) &= \int\limits_{\bar U_w} e_{\psi_M}(y, cwuy')\overline{\psi_N^{v}(u)}\; d^\ast u\\ &= \int\limits_{\bar U_w} \;\int\limits_{ U_n(\mathbb R)} k(z^{-1} \cdot cwuy') \, \psi_M(x)\, \overline{\psi_N^{v}(u)}\; d^\ast x\, d^\ast u,\end{align*}

Page 359, Line -3
$$\sum\limits_{j\geqslant 1}\hat{k} (\nu_j)\phi_j(z^{\prime})\overline{{\phi}_j(z)}$$
$---->$
$$\sum\limits_{j\geqslant 0}\hat{k} (\nu_j)\phi_j(z^{\prime})\overline{{\phi}_j(z)}$$

Page 360
Please change $W_{\operatorname{ Jacquet}}$ to $W^*_{\operatorname{ Jacquet}}$ everywhere on this page (at 6 places).

Page 361
Please change $W_{\operatorname{ Jacquet}}$ to $W^*_{\operatorname{ Jacquet}}$ everywhere on this page (at 6 places).

Page 361, Line -4 $$2\int\limits_{({\Bbb R}^{+})^{n-1}} W_{\text{Jacquet}}(y, \nu)\, W_{\text{Jacquet}}(y, {\nu}^{\prime}) \prod\limits_{j=1}^{n-1} \pi^{(n-j)s}y_j^{(n-j)(s-j)} \;\frac{dy_j}{y_j} $$
$---->$
$$2\int\limits_{({\Bbb R}^{+})^{n-1}} W^*_{\text{Jacquet}}(y, \nu)\, \overline{W^*_{\text{Jacquet}}(y, {\nu}^{\prime})} \; \prod\limits_{j=1}^{n-1} \pi^{(n-j)s}y_j^{(n-j)(s-j)} \;\frac{dy_j}{y_j} $$

Page 362, Line -2 $$H_w\left(\psi_N^v, c\right) = \int\limits_{(\mathbb R^+)^{n-1}} \int\limits_{U_n(\mathbb Z)\backslash U_n(\mathbb R)} \int\limits_{\bar U_w} k\left( t_M^{-1} x^{-1} w cut_N \right) \psi_M(x) \overline{\psi_N^v(u)} \, d^*u \, d^*x$$
$---->$
$$H_w\left(\psi_N^v, c\right) = \int\limits_{(\mathbb R^+)^{n-1}} \; \int\limits_{x\in U_n(\mathbb R)} \; \int\limits_{u\in\overline{U}_w(\mathbb R)} k\left( t_M^{-1} x^{-1} c w u t_N \right) \psi_M(x) \overline{\psi_N^v(u)} \, d^*u \, d^*x$$

Page 363, Line 1
where $t_M = t(M^*)^{-1}, \; t_N = t(N^*)^{-1},$ and
$---->$
where $t_M = t\cdot {M^*}^{-1}, \; t_N = t\cdot {N^*}^{-1},$ and

Page 363, Line 2
$$M^* = \left(\begin{matrix} M_1\cdots |M_{n-1}| & & &\\ & \ddots & &\\ &&&M_1\\ &&&&1 \end{matrix}\right), \quad N^* = \left(\begin{matrix} N_1\cdots |N_{n-1}| & & &\\ & \ddots & &\\ &&&N_1\\ &&&&1 \end{matrix}\right),$$
$---->$

$$M^* = \frac{\left(\begin{matrix} M_1\cdots |M_{n-1}| & & &\\ & \ddots & &\\ &&&M_1\\ &&&&1 \end{matrix}\right)} {\big|\text{Det}(M^*)\big|^{\frac{1}{n}} }, \qquad N^* = \frac{\left(\begin{matrix} N_1\cdots |N_{n-1}| & & &\\ & \ddots & &\\ &&&N_1\\ &&&&1 \end{matrix}\right)} {\big|\text{Det}(N^*)\big|^{\frac{1}{n}} },$$

Page 403, Line -16
We introduce the right regular representation which maps $g \in GL(n, \mathbb R)$ to the endomorphism, $F \to \rho(g) F$ of $\mathcal V_n$, and is defined by $$\rho : GL(n, \mathbb R) \to \text{End}(\mathcal V_n),$$ where $$(\rho(g) F)(z) := F(z\cdot g)$$ for all $F \in \mathcal V_n, \; z \in \frak h^n = GL(n, \mathbb R)/O(n, \mathbb R)\cdot \mathbb R^\times,$ and all $g \in GL(n, \mathbb R).$
$---->$
It is possible to define certain actions on the vector space $ \mathcal V_n$ such as actions by differential operators or actions by translation. These actions determine a very big automorphic representation which decomposes into a direct sum of irreducible automorphic representations. One may think of an irreducible automorphic representation as being generated by a single Maass form $\phi$, i.e., it is given by the $\mathbb C$ vector space generated by all linear combinations of the form $\sum_i c_i \phi_i$ where $c_i \in \mathbb C$ and $\phi_i$ is the function determined by one of the above actions on $\phi$. We have given a vague and imprecise description of automorphic representations. See [Goldfeld-Hundley, 2011] for a rigorous description of this beautiful theory.

Page 403, Remark
Remark: This is a representation into the endomorphisms of an infinite dimensional vector space! The space of Maass forms (cusp forms) is invariant under this representation. It decomposes into an infinite direct sum of irreducible invariant subspaces. If $\pi$ is the representation on one of these invariant subspaces then $\pi$ is termed an automorphic cuspidal representation and corresponds to a Maass form. The L-function associated to $\pi$ is then the L-function associated to the Maass form as in definition 9.4.3, i.e., it is a Godement-Jacquet L-function.
$---->$
Remark: This is a representation into the endomorphisms of an infinite dimensional vector space! It decomposes into an infinite direct sum of irreducible invariant subspaces. If $\pi$ is an irreducible representation on one of these invariant subspaces then $\pi$ is termed an automorphic cuspidal representation and corresponds to a Maass form. The L-function associated to $\pi$ is then the L-function associated to the Maass form as in definition 9.4.3, i.e., it is a Godement-Jacquet L-function. ADDITIONAL REFERENCE
Goldfeld, D.; Hundley J.; Automorphic representations and L-functions for the general linear group, Vols. 1, 2. Cambridge Studies in Advanced Mathematics Vols. 129, 130, Cambridge University Press (2011).

Please add an extra small section at the end of the book entitled Errata as below. This also needs to be added to the table of contents.

ERRATA
Page 44, Proposition 2.2.6 Proposition 2.2.6 is in error. In fact, right invariance by the element $\delta_1$ is not, in general, preserved by the action of the differential operators. To see this, let $f: GL(2, \mathbb R) \to \mathbb C$ be a function which is right invariant by $\delta_1$, and define $h = D_a f$ where $a = \left(\begin{smallmatrix} 0&1\\0&0\end{smallmatrix}\right)$. Suppose that $h$ is also $\delta_1$-invariant. Then \begin{align*} h(g) & = h(g\delta_1) = \lim_{t\to 0} \frac{\big( f(g(\delta_1ta) - f(g\delta_1)\big)}{t}\\ & = \lim_{t\to 0} \frac{\big(f(g(-t) a\delta_1) - f(g\delta_1) \big)}{t}\\ & = \lim_{t\to 0} \frac{f(g(-t)a) - f(g)}{t} = D_{-a} f(g) = -h(g). \end{align*} (We used the fact that $\delta_1a\delta_1 = -a.$) This shows that if $f$ and $D_a f$ are both fixed by $\delta_1$ then $D_a f = 0.$ It is shown on page 48 that $D_a$ is essentially $y\frac{\partial}{\partial x}$, so there are many functions which are fixed by $\delta_1$ and not killed by $D_a$, including many which are fixed by $GL(2, \mathbb Z)$ on the left and by the center.


Page 141, Inequality (5.8.2) The estimate (5.8.2) is not correct. It is claimed that (5.8.2) follows from the Cauchy-Schwarz inequality in Lemma 5.6.8, but this is in error. It is not possible to apply the upper-bound from Lemma 5.6.8. Indeed all the exponents involved in the $I_\nu$ function are negative so the inequality should be reversed. Thus, a lower bound analogue of Lemma 5.6.8 would be required here (but such a lower bound is not valid in general).