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\centerline{Errata for {\bf Automorphic Forms and Representations}}
\centerline{by Daniel Bump 6/12/97}
\medbreak
If you find errors in the book, whether typos or historical or mathematical
mistakes {\bf please}! email me at {\tt bump@math.stanford.edu}. Thanks to
those of you who have already done so.
This list of {\it errata} is available on the world-wide-web at
\centerline{{\tt http://match.stanford.edu/bump/errata.html}}
{\bf Note:} a negative line number is a line number counted upwards from the
bottom of the page.
\medbreak
\item{$\circ$} p.4, line 8. `re' should not be italicized.
\item{$\circ$} p.6, bottom. The hypotheses on f do not imply that F is of
locally bounded variation. Therefore add the hypothesis that f is of bounded
total variation. This is true, for example, if $f(x)$ is monotone for $|x|$
large.
\item{$\circ$} p.9, fourth displayed formula (Fourier Inversion Formula). Omit
subscript of $t$.
\item{$\circ$} p.11, line 7. ``Now substituting (1.19) ....'' It is not (1.19)
which is substituted, but rather the formula at the bottom of p.10.
\item{$\circ$} p.18, line 12: omit space in $Z/ NZ$.
\item{$\circ$} p.19, line $-10$. Missing a closing absolute value sign in
$|\hbox{re}\big(\gamma(z)\big)|\le 1/2$.
\item{$\circ$} p.20, formula (2.3). Comma should be outside the matrix.
\item{$\circ$} p.20, proof of Proposition~1.2.3. The bar is used in two
different ways, which could be confusing. In the second usage,
$\overline{\gamma(F)}$ means the topological closure.
\item{$\circ$} p.29, second displayed formula. The coefficient of $q^5$ should
be 4830, not 2954.
\item{$\circ$} p.32, comments after Proposition 1.3.5. The estimate should be
$a_n\le Cn^{(k-1)/2+\epsilon}$ for any positive $\epsilon$.
\item{$\circ$} p.35, bottom. $H^1$ and $H^2$ should be $H_1$ and $H_2$.
\item{$\circ$} p.41, l. $-10$. In the Fourier expansion, the exponent of
$e$ should be $2\pi inz/t$.
\item{$\circ$} p.49, last line of proof of Theorem~1.4.4, Eq.(4.10) should
be Eq.(4.11).
\item{$\circ$} p.52, Exercise 1.4.13, displayed formula. The last exponent of
$p$ should be $k-1-2s$, not $-2s$.
\item{$\circ$} p.52, bottom. Theorem~1.4.4 should be Theorem~1.4.5.
\item{$\circ$} p.53, line 11. ``theoretic'' should not be repeated.
\item{$\circ$} p.57, last line. (3.14) should be (3.16).
\item{$\circ$} p.58, line 2. Theorem~1.4.3 should be Theorem~1.4.4.
\item{$\circ$} p.60, beginning of paragraph before Theorem 1.5.1,
$f\in\Gamma_0(N)$ should be $f\in S_k(\Gamma_0(N),\psi)$.
\item{$\circ$} p.67, after third displayed formula, omit the unnecessary
``... and the subsequent evaluation of the constant $c$.''
\item{$\circ$} p.69, bottom and p.71, first formula, $\Gamma\infty$ should
be $\Gamma_\infty$.
\item{$\circ$} p.70, statment of Proposition 1.6.1, ``most simple poles''
should be ``at most simple poles.''
\item{$\circ$} p.72, first displayed formula: omit i in $e^{-4\pi iy}$.
\item{$\circ$} p.72, after statement of Theorem 1.6.2, (3.11) should
be (3.13).
\item{$\circ$} p.76, line 2. Do not italicize ``re.''
\item{$\circ$} p.77, line 17. ``resulting the assumption from'' should be
``resulting from the assumption.''
\item{$\circ$} p.74, line 3. Proposition 1.6.3 should be Theorem 1.6.2.
\item{$\circ$} p.69, third line from end of proof,
``follows from Eq.\ (6.5)'' should be ``follows from Eq.\ (6.6).''
\item{$\circ$} p.77, next to last line. The tensor product symbol should not be
there: the index is supposed to be $[o^\times:o^\times_+]$.
\item{$\circ$} p.90, Exercise 1.7.2. Insert space before (7.6).
\item{$\circ$} p.108. It is asserted that the Laplacian is positive
definite. This should be ``semidefinite,'' and the reference should
be to Exercise~2.1.8.
\item{$\circ$} p.129, first displayed formula, the second partial
derivative should be with respect to $\overline z$, not $z$.
\item{$\circ$} p.129, (1.2) at bottom. In the definition of $L_k$, the partial
derivative should be with respect to $\overline z$, not $z$.
\item{$\circ$} p.131 l. $-12$. $\frak H$ should be $\frak h$
\item{$\circ$} p.135, proof of Lemma 2.1.2. Not really an error, but
replace ``closed manifold'' by ``compact manifold.''
\item{$\circ$} p.135, l. $-4$. In $\omega=u+iv=\ldots$, replace $\omega$
by $w$.
\item{$\circ$} p.170, Proposition~2.3.1 (iii). After the backslashes
insert $G$ (twice).
\item{$\circ$} p.188, line 6. The function $\pi(g)\cdot Xf$ is automatically
continuous, so this does not need to be assumed.
\item{$\circ$} p.245, near bottom. Replace $X$ by $D$ in this discussion,
and note that $\pi(D)f$ is defined by (4.1) when $D=X$ is in the Lie algebra
$\frak g$, and extended to $U(\frak g)$ by Proposition 2.2.3.
\item{$\circ$} p.291, line 16. $L^2(\Gamma\backslash G,\chi,\omega)$
should be $L^2_0(\Gamma\backslash G,\chi,\omega)$.
\item{$\circ$} p.310, line $-8$. Amend this to read: ``We will call $\cal H_G$
the Hecke algebra of G.''
\item{$\circ$} p.312, line 3. Amend this to read ``According to the notes in
Knapp and Vogan, Flath had originally ...''
\item{$\circ$} p.317, line $-11$. ``This is a generalization of
Theorem IV.6.6'' should read ``Theorem 4.6.3.'' Any theorem or proposition
with a roman numeral should be suspected of being wrong. Let me know if you
find any others!
\item{$\circ$} p.321, Theorem 3.5.1. The functional is of course only unique
up to constant multiple.
\item{$\circ$} p.322, statement of Theorem 3.5.2. Add the assumption that
$(\pi,V)$ is admissible.
\item{$\circ$} p.375, ``metaplectid'' should be ``metaplectic.''
\item{$\circ$} p.379, table. In the third (L-group) column, $n$ should
be $n+1$ for the first three entries.
\item{$\circ$} p.383, l.21. ``$\hat\pi$ is the Langlands L-function''
should read ``$L(s,\hat\pi)$ is the Langlands L-function.''
\item{$\circ$} p.383, l.$-4$ and $-3$. $GL(2)$ should be $GL(n)$ (twice)
and $GL(8)$ should be $GL(n^2-1)$ (three times).
\item{$\circ$} p.385, Third line from bottom. $(\pi_1,V_0)$ should be
$(\pi_1,V_1)$.
\item{$\circ$} p.426, formula (2.2). Omit parentheses from $d_L(b)$;
similarly, omit parenthesis from $d_L(g)$ in following displayed formula.
\item{$\circ$} p.432. Not a correction, but it is useful to know that a
stronger result than Proposition 4.2.7 is true. If there exists a single open
subgroup $K$ such that $V_1^K$ and $V_2^K$ are nonzero (hence simple $\cal
H_K$ modules by Proposition 4.2.3), and if these are isomorphic as $\cal H_K$
modules, then $V_1$ and $V_2$ are isomorphic. To prove this, adapt the proof
of Theorem 4.6.3 on p.493.
\item{$\circ$} p.436, second sentence of Section 4.3, ``this result'' should be
``these topics.''
\item{$\circ$} p.486, last displayed formula and p.487, top displayed formula.
Domain of integration should be ${\frak p}^{-N}$.
\item{$\circ$} p.488, first displayed formula. The definition of $L_2$ is
slightly wrong. The second term phi(1) should be multiplied by a function h(x)
designed to make the statement that the integral is compactly supported
actually true! For example, we can take:
$$h(x)=\cases{|x|^{-1} (\chi_1^{-1} \chi_2)(x)&\hbox{if $|x|>1$},\cr
0&\hbox{if $|x|<=1$}.}$$
\item{$\circ$} p.493, Theorem 4.6.3. Not a correction, but note that this is a
special case of the generaliza
tion of Proposition 4.2.7 described above on
the note to p.432.
\item{$\circ$} p.540, line 14. Amend this to read ``After partial results
towards Howe's conjecture were obtained by Howe and other authors, the
conjecture was fully proved for local fields of odd residue characteristic by
Waldspurger (1990).''
\item{$\circ$} p.541, Theorem 4.8.6. Since it is assumed here that E is a
field, delete all references to the case $E=F+F$ in the statement and proof of
this theorem! The case where $E=F+F$ is considered separately, later.
\item{$\circ$} p.550. The discussion in the second paragraph switches from
$GL(2)$ to $GL(n)$ in a confusing way. Amend the third and fourth sentences to
read ``The conjecture includes a hypothetical classification of the
irreducible admissible representations of $GL(n,F)$, where $F$ is a local
field, which has been proved in many cases. Over an archimedean field, the
local Langlands conjecture (for an arbitrary reductive group) is a theorem of
Langlands.''
\item{$\circ$} p.557, last paragraph. ``Theorem 4.9.3'' should be
``Proposition 4.9.3,'' and ``Theorem 4.9.4'' should be ``Theorem 4.9.1.''
\item{$\circ$} p.560. The paper of Doi and Naganuma was in vol. 9 of
Inventiones, not vol. 19.
\bye