## Wednesday, February 15, 2012

### Volume 2


Page 6, Definition of Slash operator does not satisfy
$$\Phi\big|_\rho \gm \gm' = \Phi \big|_\rho\gm \big|_\rho\gm',$$(i.e., does not define a right action). This correction was brought to our attention by Fan Zhou. To correct it, the slash operator should be defined as
$$(\Phi\big|_\rho\gm)(z) = J_\rho( \gm, z) \cdot \Phi(\wt{\gm.z})$$
$$(\Phi\big|_\rho\gm)(z) = J_\rho( \gm, z)^{-1} \cdot \Phi(\wt{\gm.z}).$$
(Delete the $^{-1}$.)

Page 91, two references to Theorem 14.8.5 should actually refer to Theorem 14.8.11 (there is no Theorem 14.8.5).

Page 142, statement of Theorem 15.8.6 when $r=1,$ we get $L_p( s, |\ |_p^{\frac{d-1}2}\cdot \pi),$ not $L_p( s, |\ |_p^{\frac{1-d}2}\cdot \pi).$ (It may also be worth noting that if $r=1,$ then $d=n.$)

Page 142-43, proof of Theorem 15.8.6 in the case r=1 Beginning from equation (15.8.7), explicit formulae are written which are only valid if $\pi$ (which is now a character) is unramified. When $r=1$ and $\pi$ is a ramified character, both (15.8.7) and (15.8.8) become $Z_p(s, \Phi, \beta) = Z_p(1-s, \widehat\Phi, \overset{\vee}\beta).$ Roughly the same argument used in the unramified case shows that in this case $L_p(s, \pi') = 1.$ But $L_p(s, |\ |_p^{\frac{n-1}2} \cdot \pi)$ is also equal to $1,$ so the statement of the theorem is still correct.

Page 143, line 10 The expression $(1-\pi(p) p^{-s-\frac{n-1}2})$ is equal to $L_p(s, |\ |_p^{\frac{n-1}2}\cdot \pi),$ or $L_p(s, |\ |_p^{\frac{d-1}2}\cdot \pi),$ but not $L_p(s, |\ |_p^{\frac{1-d}2}\cdot \pi).$ Also we remind the reader that this holds only if $\pi$ is unramified.

Page 124, Proposition 15.3.2Before one can apply Theorem 13.7.3 to an irreducible cuspidal automorphic representation of $GL(n \A_\Q),$ one must know that it is an admissible $(\mathfrak{g}, K) \times GL(n, \A_{\text{finite}})$-module. The proof is the same as in the $GL(2)$ case, requiring a suitable analogue of theorem 3.12.1. A good reference is Borel, A.; Jacquet, H. Automorphic forms and automorphic representations. With a supplement "On the notion of an automorphic representation'' by R. P. Langlands. Proc. Sympos. Pure Math., XXXIII, Automorphic forms, representations and L-functions (Proc. Sympos. Pure Math., Oregon State Univ., Corvallis, Ore., 1977), Part 1, pp. 189–207, Amer. Math. Soc., Providence, R.I., 1979.