1549380232-Automorphic_Forms_and_Applications__Sarnak_

104 J.W. COGDELL, £-FUNCTIONS FOR GLn

Now, by duality theory [96], (kn~-I) c:::'. kn-I. In fact, if we let ( , ) denote

the pairing kn-I x kn-I -+ k by (x, y) = I:; XiYi we have

cp(p) = L 'Px(P)

xEkn- 1

where now we write

'Px(P) = 1 n-l\An-l cp(yp)'l/J-I((x,y)) dy.

GLn-I(k) acts on kn-I with two orbits: {O} and kn-I_ {O} = GLn-I(k) .ten-I

where en-I= (0, ... , 0, 1). The stabili zer often-I in GLn-I(k) is tpn-I· Therefore,

we may write

cp(p) = cpo(P) + 'P"f·'en-1 (p).

"(EGLn-1 (k)/'Pn-1 (k)

Since cp(p) is cuspidal and Y n is a standard unipotent subgroup of GLn, we see

that

cpo(P) = ( cp(yp) dy = 0.

}Yn(k)\ Yn(A)

On the other hand an elementary calculation as before gives

'P"f·'en- 1 (p) = 'P'en-1 ( Cci ~) P) ·

Hence we have

cp(p) = :L 'P'e,,_1 ( ( 6 ~) P)

'"YEPn-1(k)\ GLn-dk)

and the convergence is still absolute and uniform on compact subsets.

Note that if n = 2 this is exactly the fact we used previously for GL 2. This

then begins our induction.

Next, let us write the above in a form more suitable for induction. Structurally,

we have Pn = GLn-I IX Y n and Nn = Nn-I IX Y n· Therefore, Nn-I \ GLn-I c:::'.

Nn \ Pn. Furthermore, if we let Pn-I = Pn-I IX Yn c Pn, then Pn-I \ GLn-I c:::'.

P n-I\ P n. Next, note that the function 'P'en-l (p) satisfies, for y E Y n (A) c:::'. An-I,

'P'en-1 (yp) = 'l/J(Yn-i)cpten-1 (p).

Since 'ljJ is trivial on k we see that 'P'en-l (p) is left invariant under Y n ( k). Hence

cp(p) = L 'P'en-1 ( ( 6 ~) P) = L 'P'en-1 (lip).

"(EPn-1(k)\ GLn-1(k) 8EPn-1(k)\ Pn(k)

To proceed with the induction, fix p E P n(A) and consider the function cp' (p') =

cp~(p') on P n-I (A) given by

cp'(p') = 'P'en-1 ( (~ ~) P) ·

cp' is a smooth function on P n-I (A) since cp was smooth. One checks that cp' is left
invariant by Pn-I(k) and cuspidal on Pn-I(A). Then we may apply our inductive