Errata for Automorphic Forms and Representations by Daniel Bump 6/12/97

If you find errors in the book, whether typos or historical or mathematical mistakes please! email me at bump@math.stanford.edu. Thanks to those of you who have already done so.

This list of errata is available on the world-wide-web at

              http://match.stanford.edu/bump/errata.html

Note: a negative line number is a line number counted upwards from the bottom of the page.

∘ p.4, line 8. ‘re’ should not be italicized. ∘ 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.

∘ p.9, fourth displayed formula (Fourier Inversion Formula). Omit subscript of t. ∘ 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. ∘ p.18, line 12: omit space in Z/NZ. ∘ p.19, line −10. Missing a closing absolute value sign in |re(γ(z))| ≤ 1/2. ∘ p.20, formula (2.3). Comma should be outside the matrix. ∘ 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{γ(F)} means the topological closure. ∘ p.29, second displayed formula. The coefficient of q^{5} should be 4830, not 2954. ∘ p.32, comments after Proposition 1.3.5. The estimate should be a_{n} ≤ Cn^{(k−1)/2+ϵ} for

any positive ϵ. ∘ p.35, bottom. H^{1} and H^{2} should be H_{1} and H_{2}. ∘ p.41, l. −10. In the Fourier expansion, the exponent of e should be 2πinz/t. ∘ p.49, last line of proof of Theorem 1.4.4, Eq.(4.10) should be Eq.(4.11). ∘ p.52, Exercise 1.4.13, displayed formula. The last exponent of p should be k − 1 − 2s,

not −2s. ∘ p.52, bottom. Theorem 1.4.4 should be Theorem 1.4.5. ∘ p.53, line 11. “theoretic” should not be repeated. ∘ p.57, last line. (3.14) should be (3.16). ∘ p.58, line 2. Theorem 1.4.3 should be Theorem 1.4.4. ∘ p.60, beginning of paragraph before Theorem 1.5.1, f ∈ Γ_{0}(N) should be f ∈

S_{k}(Γ_{0}(N), ψ). ∘ p.67, after third displayed formula, omit the unnecessary “... and the subsequent

evaluation of the constant c.” ∘ p.69, bottom and p.71, first formula, Γ∞ should be Γ_{∞}. ∘ p.70, statment of Proposition 1.6.1, “most simple poles” should be “at most simple

poles.” ∘ p.72, first displayed formula: omit i in e^{−4πiy}. ∘ p.72, after statement of Theorem 1.6.2, (3.11) should be (3.13). ∘ p.76, line 2. Do not italicize “re.”

∘ p.77, line 17. “resulting the assumption from” should be “resulting from the assump-

tion.” ∘ p.74, line 3. Proposition 1.6.3 should be Theorem 1.6.2. ∘ p.69, third line from end of proof, “follows from Eq. (6.5)” should be “follows from

Eq. (6.6).” ∘ p.77, next to last line. The tensor product symbol should not be there: the index is

supposed to be [o^{×} : o_{+}^{×}]. ∘ p.90, Exercise 1.7.2. Insert space before (7.6). ∘ p.108. It is asserted that the Laplacian is positive definite. This should be “semidefi-

nite,” and the reference should be to Exercise 2.1.8. ∘ p.129, first displayed formula, the second partial derivative should be with respect to

\overline{z}, not z. ∘ p.129, (1.2) at bottom. In the definition of L_{k}, the partial derivative should be with

respect to \overline{z}, not z. ∘ p.131 l. −12. ℌ should be 𝔥 ∘ p.135, proof of Lemma 2.1.2. Not really an error, but replace “closed manifold” by

“compact manifold.” ∘ p.135, l. −4. In ω = u + iv = …, replace ω by w. ∘ p.170, Proposition 2.3.1 (iii). After the backslashes insert G (twice). ∘ p.188, line 6. The function π(g) · Xf is automatically continuous, so this does not

need to be assumed. ∘ p.245, near bottom. Replace X by D in this discussion, and note that π(D)f is defined

by (4.1) when D = X is in the Lie algebra 𝔤, and extended to U(𝔤) by Proposition 2.2.3.

∘ p.291, line 16. L^{2}(Γ\G, χ, ω) should be L_{0}^{2}(Γ\G, χ, ω). ∘ p.310, line −8. Amend this to read: “We will call H_{G} the Hecke algebra of G.” ∘ p.312, line 3. Amend this to read “According to the notes in Knapp and Vogan, Flath

had originally ...” ∘ 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!

∘ p.321, Theorem 3.5.1. The functional is of course only unique up to constant multiple. ∘ p.322, statement of Theorem 3.5.2. Add the assumption that (π, V) is admissible. ∘ p.375, “metaplectid” should be “metaplectic.” ∘ p.379, table. In the third (L-group) column, n should be n + 1 for the first three

entries. ∘ p.383, l.21. “π̂ is the Langlands L-function” should read “L(s, π̂) is the Langlands

L-function.” ∘ p.383, l.−4 and −3. GL(2) should be GL(n) (twice) and GL(8) should be GL(n^{2} − 1)

(three times). ∘ p.385, Third line from bottom. (π_{1}, V_{0}) should be (π_{1}, V_{1}). ∘ p.426, formula (2.2). Omit parentheses from d_{L}(b); similarly, omit parenthesis from

d_{L}(g) in following displayed formula.

∘ 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 H_{K} modules by Proposition 4.2.3), and if these are isomorphic as 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.

∘ p.436, second sentence of Section 4.3, “this result” should be “these topics.” ∘ p.486, last displayed formula and p.487, top displayed formula. Domain of integration

should be 𝔭^{−N}. ∘ 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) = \{ \begin{matrix}|x|^{−1}(χ_{1}^{−1}χ_{2})(x) & if |x| > 1, \\ 0 & if |x| <= 1. \end{matrix}

∘ 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. ∘ 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).”

∘ 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.

∘ 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.”

∘ 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.” ∘ p.560. The paper of Doi and Naganuma was in vol. 9 of Inventiones, not vol. 19.