数理解析研究所講究録
第 1826 巻 2013 年 7-17

7On local newforms for unramified $U(2,1)$
Michitaka Miyauchi (Osaka Prefecture University)

1

Introduction

Local newforms play an important role in the theory of automorphic representations.
Roughly speaking, a newform for an irreducible generic representation of a -adic groupis a vector which attains the -factor of via Rankin-Selbcrg integral. The existenceof newforms was known only for $GL(n)$ until the work of Roberts and Schmidt [11] forGSp(4). In this note, we study newform theory for unramified $U(2,1)$ .
This note is a survey of the author’s work [6], [7], [9], [8] on newrofms for unramified$U(2,1)$ . Let
denote the unramified unitary group in three variablcs defined over a padic field of odd residual characteristic. Newforms for an irreducible generic representation$(\pi, V)$ ofis defined by using a family of open compact subgroups
of , which
is an analog of paramodular subgroups of GSp(4). For each non-negative integer , wedenote by $V(n)$ the space of -fixed vectors. The smallest integer such that $V(n)$ is nottrivial is called the conductor of . We write
for the conductor of , and callthe space of newforms for . An algebraic structure of $V(n)$ was studied in [6] and [9], forexample, the multiplicity one theorem for newforms and the dimension formula for $V(n)$ ,
$\pi$

$L$

$p$

$\pi$

$G$

$G$

$\{K_{n}\}_{n\geq 0}$

$G$

$n$

$K_{n}$$N_{\pi}$$\pi$$V(N_{\pi})$

$\pi$

$\pi$

$n\geq N_{\pi}.$

Our main concern is the relation of newforms and Rankin-Selberg factors. Gelbart
and Piatetski-Shapiro in [4] attached a family of Rankin-Selberg integrals to an irreducible
generic representation of , and defined and -factors for . In $loc.$ $cit$ . they showedthat the spherical vector attains the -factor of when is an unramified principal series
representation. But there were no results for ramified representations. In this note, weestablish a theory of newforms for Gelbart and Piatetski-Shapiro’s integral. We see that(i) the newform for an irreducible generic representation of attains thc -factor of(Theorem 4.1) (ii) the conductor of coincides with the exponent ofof the -factor,
where denotes the cardinality of the residue field (Theorem 4.3).
We summarize the contents of this paper. In section 2, we recall from [1] the theoryof Rankin-Selberg integral introduced by Gelbart, Piatetski-Shapiro and Baruch. In section 3, we define newforms forand recall their basic properties. In section 4, we show
Theorems 4.1 and 4.3 assuming Lemma 4.2, which is proved in section 5.
$\pi$

$G$

$L$

$L$

$\pi$

$\epsilon$

$\pi$

$\pi$

$\pi$

$\pi$

$q$

$G$

$G$

$L$

$q^{-s}$$\pi$$\epsilon$

82Rankin-Selberg integral

In this section, we recall from [1] the theory of Rankin-Selberg integral for $U(2,1)$ introduced by Gelbart, Piatctski-Shapiro and Baruch.

2.1

Notation

We use the following notation. Let be a non-archimedean local field of characteristic
zero,
its ring of integers, andthe maximal ideal in . We fix a uniformizerin, and denotethc absolute value of normalized so that
, whereis the cardinality of the residue field. Throughout this paper, we assume that thecharacteristic ofis different from two.
Let
be the quadratic unranlified cxtension over , wherc
is a nonsquare unit in . We denote by
the analogous objects for . Thenis auniformizer of , and the cardinality ofis equal to . So we abbreviate
and $q=q_{F}$ . We realize (the group of -points of) the unramified unitary group in threevariables defined over ae$F$$\mathfrak{o}_{F}$$\mathfrak{p}_{F}$$F$$\varpi_{F}$$\mathfrak{o}_{F}$$F$$0y|\cdot|_{F}$$|\varpi_{F}|=q_{F}^{-1}$$q_{F}$$\mathfrak{o}_{F}/\mathfrak{p}_{F}$$\mathfrak{o}_{F}/\mathfrak{p}_{F}$$F$$E=F[\sqrt{\epsilon}]$$\mathfrak{o}_{F}$$\mathfrak{o}_{E},$$E$$\sqrt{\epsilon}$$E$$\mathfrak{p}_{E}$$\mathfrak{o}_{E}/\mathfrak{p}_{E}$$\varpi_{F}$$q_{F}^{2}$$\varpi=\varpi_{F}$

$F$

$F$

$G=U(2,1)=\{g\in GL_{3}(E)|t\overline{g}Jg=J\}.$

Here wc dcnotes b$y^{-}$

the non-trivial element in Gal $(E/F)$ and$J=(\begin{array}{lll}0 0 10 1 01 0 0\end{array})$Lctbe the upper triangular Borel subgroup of $G,$
its unipotent radical, andthe group of thc diagonal elements in . For a non-trivial additive characterof , wealso denote by
the character of defined by $\psi_{E}(u)=\psi_{E}(u_{12})$ , for $u=(u_{ij})\in U$ . Foran irreducible generic representation $(\pi, V)$ of , we write
for the Whittakermodel of associated to
.
$B$

$U$

$T$

$G$

$\psi_{E}$$U$$\psi_{E}$

$G$

$\pi$

2.2

$E$

$\mathcal{W}(\pi, \psi_{E})$$(U, \psi_{E})$Zeta integralsLetbe the space of locally constant, compactly supported functions on . For anirreducible generic representation $(\pi, V)$ of , Gelbart and Piatetski-Shapiro introduceda family of zeta integrals which has the form $Z(s, W, \Phi)(W\in \mathcal{W}(\pi, \psi_{E}), \Phi\in C_{c}^{\infty}(F^{2}))$$F^{2}$$C_{c}^{\infty}(F^{2})$$G$

as follows:

We identify thc subgroup$H=\{(\begin{array}{lll}a 0 b0 1 0c 0 d\end{array})\in G\}$

9
of

$G$

$H$

as

with $U(1,1)$ . Since $SU$ (1, 1) is isomorphic to $SL_{2}(F)$ , we can write any element

(2.1)
where

$h=(0b \frac{0}{b}1)(0\sqrt{\epsilon}01)h_{1}(\sqrt{\epsilon}^{-1}0 01)$

$b\in E^{\cross}$$f(s, h, \Phi)$

on

$C$

and $h_{1}\in SL_{2}(F)$ . For
by$\Phi\in C_{c}^{\infty}(F^{2})$

$h$

in

,

and $h\in H$ , we define a function$f(s, h, \Phi)=|b|_{E}^{s}\int_{F^{\cross}}\Phi((0, r)h_{1})|r|_{E}^{s}d^{\cross}r$by using the decomposition of in (2.1). We note that the definition of $f(s, h, \Phi)$ isindependent of the choices ofand $h_{1}\in SL_{2}(F)$ .
Set $B_{H}=B\cap H$ and $U_{H}=U\cap H$ . Thenis the unipotent radical of the Borel
subgroupof $H=U(1,1)$ . Forand, we define zeta integral$Z(s, W, \Phi)$ by$h$$b\in E^{\cross}$$U_{H}$$B_{H}$$W\in \mathcal{W}(\pi, \psi_{E})$$\Phi\in C_{c}^{\infty}(F^{2})$$Z(s, W, \Phi)=\int_{U_{H}\backslash H}W(h)f(s, h, \Phi)dh.$

Then $Z(s, W, \Phi)$ absolutely converges to a function in $C(q^{-2s})$ if

2.3

$L$

${\rm Re}(s)$is sufficiently large.

and -factors$\epsilon$

We recall the definition of$(\pi, V)$ of
. Set

$L$

and -factors attached to an irreducible generic representation$\epsilon$$G$$I_{\pi}=\langle Z(s, W, \Phi)|W\in \mathcal{W}(\pi, \psi_{E})_{7}\Phi\in C_{c}^{\infty}(F^{2}),$$\psi_{E}$: non-trivial$\rangle.$

Then
is a fractional ideal of $C[q^{-2s}, q^{2s}]$ which contains 1. Thus, there exists a polynomial $P(X)$ in $C[X]$ such that $P(O)=1$ and $I_{\pi}=(1/P(q^{-2s}))$ . We define the -factor$L(s, \pi)$of by$I_{\pi}$

$L$

$\pi$

$L(s, \pi)=\frac{1}{P(q^{-2s})}.$

To define -factor of , we recall the functional equation. Let
be a non-trivial
additive character of . For, we denote by
its Fourier transform withrespect to. Then there exists
which satisfies$F$$\psi_{F}$$\psi_{F}$$\pi$$\epsilon$$\Phi\in C_{c}^{\infty}(F^{2})$$\hat{\Phi}$$\gamma(s, \pi, \psi_{F}, \psi_{E})\in C(q^{-2s})$$\gamma(s, \pi, \psi_{F}, \psi_{E})Z(s, W, \Phi)=Z(1-s, W,\hat{\Phi})$,

10
for all.
and
By using the above functional equation, we define the$\Phi\in C_{c}^{\infty}(F^{2})$$W\in \mathcal{W}(\pi, \psi_{E})$$\epsilon$-factor$\epsilon(s, \pi.

\psi_{F}, \psi_{E})$

of

$\pi$

by

$\epsilon(s, \pi, \psi_{F}, \psi_{E})=\gamma(s, \pi, \psi_{F}, \psi_{E})\frac{L(s,\pi)}{L(1-s,\tilde{\pi})},$arwherc is the contragradient representation of . By [7], we obtain $L(s, \tilde{\pi})=L(s, \pi)$ ,and hence
$\pi$

(2.2)

$\epsilon(s, \pi, \psi_{F}, \psi_{E})=\gamma(s, \pi, \psi_{F}, \psi_{E})\frac{L(s,\pi)}{L(1-s,\pi)}.$Thus, we can show the following proposition by the standard argument:

Proposition 2.3. The$\epsilon$-factor$\epsilon(s, \pi, \psi_{F}, \psi_{E})$is a monomial in$q^{-2s}$of the form$\epsilon(s, \pi, \psi_{F}, \psi_{E})=\pm q^{-2n(s-1/2)},$

with some $n\in Z.$

3

Newforms

In this scction, we introduce a family of open compact subgroups of , and define thenotion of newforms for irrcducible generic representations of . We summarize thc basic
properties of newforms for , which are an analog of those for $GL(n)$ and GSp(4).
$G$

$G$

$G$

3.1

Newforms

Newforms for are defined by thc following open compact subgroups
each non-negative integer , we define an open compact subgroupof$G$$\{K_{n}\}_{n\geq 0}$$K_{n}$

$n$

$G$

of

$G$

. For

by

$K_{n}=(\begin{array}{lll}\mathfrak{o}_{E} \mathfrak{o}_{E} \mathfrak{p}_{E}^{-n}\mathfrak{p}_{E}^{n} 1+\mathfrak{p}_{E}^{n} \mathfrak{o}_{E}\mathfrak{p}_{E}^{n} \mathfrak{p}_{E}^{n} \mathfrak{o}_{E}\end{array})\cap G.$

Remark 3.1. The definition of$K_{n}$

is inspired by the paramodular subgroups of GSp(4),which is used in [11]. We also note that the group$(\begin{array}{lll}\mathfrak{o}_{E} \mathfrak{o}_{E} \mathfrak{p}_{E}^{-n}\mathfrak{p}_{E}^{n} l+\mathfrak{p}_{E}^{n} \mathfrak{o}_{E}\mathfrak{p}_{E}^{n} \mathfrak{p}_{E}^{n} \mathfrak{o}_{E}\end{array})$

is a conjugateof the subgroup of $GL_{3}(E)$ which is used to define newforms for $GL_{3}(E)$ in [5].

11
For an irreducible generic representation $(\pi, V)$ of$G$, we set$V(n)=\{v\in V|\pi(k)v=v, k\in K_{n}\}, n\geq 0.$

Then it follows from [9] that there cxists a non-negative integer$n$such that $V(n)$ is not

zero.
Definition 3.2. We define the conductor of

$\pi$

by

$N_{\pi}= \min\{n\geq 0|V(n)\neq\{0\}\}.$

We call

3.2$V(N_{\pi})$the space of newforms for$\pi$

and $V(n)$ that of oldforms, for $n>N_{\pi}.$

Basic properties of newforms

We recall some basic properties of newforms from [6] and [9]. Firstly, the growth ofdimensions of oldforms for generic representations is independent of , as in the cascs
of $GL(n)$ and GSp(4) (see [2], [10], [11]). The following dimension formula for oldforms
holds:
$\pi$

$\pi$

Proposition 3.3 ([6], [9]). Let $(\pi, V)$ be an irreducible generic representation of G. For, we have$n\geq N_{\pi}$$\dim V(n)=\lfloor\frac{n-N_{\pi}}{2}\rfloor+1.$

In particular,$V(N_{\pi})$and $V(N_{\pi}+1)$ are one-dimensional.

Secondly, newforms for are test vectors for the Whittaker functional. We say that
. The following propositiona function $W$ in
is a newform if $W$ is fixed by
is important to the application to thc theory of zeta integral:$G$$\mathcal{W}(\pi, \psi_{E})$$K_{N_{\pi}}$

Proposition 3.4 ([6]). Suppose that the conductor of, we havenewforms $W$ in$\psi_{E}$is$\mathfrak{o}_{E}$.

Then for all nonzero$\mathcal{W}(\pi, \psi_{E})$$W(1)\neq 0.$

3.3

Zeta integral of newforms

We apply newforms for to the theory of zeta integral. We suppose that the conductorofis . One of the nice properties of the subgroups
is that $K_{n,H}=K_{n}\cap H$ is$G$$\psi_{E}$$\mathfrak{o}_{E}$$\{K_{n}\}_{n\geq 0}$

12
for all . Set $T_{H}=T\cap H$ . Then we have an Iwasawa
decomposition $H=U_{H}T_{H}K_{n,H}$ , for any . There exists an isomorphism

a maximal compact subgroup of

$H$

$n$

$n$

$t:E^{\cross}\simeq T_{H};a\mapsto(a 1 \overline{a}^{-1})$

For$W\in \mathcal{W}(\pi, \psi_{E})$and$\Phi\in C_{c}^{\infty}(F^{2})$, we obtain$Z(s, W, \Phi)=\int_{E^{x}}\int_{K_{n,H}}W(t(a)k)f(s, k, \Phi)|a|_{E}^{s-1}dkd^{\cross}a.$

For $n\geq 0$ , we denote byin, then we have$\Phi_{n}$the characteristic function of$\mathfrak{p}_{F}^{n}\oplus \mathfrak{o}_{F}$

. If

$W$

a newform$\mathcal{W}(\pi, \psi_{E})$(3.5)$Z(s, W, \Phi_{N_{\pi}})=vol(K_{n,H})Z(s, W)L_{E}(s, 1)$ .

Here $L_{E}(s, 1)=1/(1-q^{-2s})$ is the -factor of the trivial represcntation of
$L$

(3.6)

$E^{\cross}$and$Z(s, W)= \int_{E^{\cross}}W(t(a))|a|_{E}^{s-1}d^{\cross}a.$

We note that Proposition 3.4 implies that the integral $Z(s, W)$ docs not vanish for any
non-zero newforms in.
If
has conductor , then we havech’ and hence$\mathcal{W}(\pi, \psi_{E})$$\psi_{F}$$\hat{\Phi}_{N_{\pi}}=q^{-N_{\pi}}$$\mathfrak{o}_{F}$$\mathfrak{o}_{F}\oplus \mathfrak{p}_{F}^{-N_{\pi}}$(3.7)$Z(1-s, W,\hat{\Phi}_{N_{\pi}})=q^{-2N_{\pi}(s-1/2)}Z(1-s, W, \Phi_{N_{\pi}})$by (3.5).

4

Main results

In this section, we show our two main theorems, which describe and -factors of irreducible generic representations of in terms of newforms and conductors.
$L$$\epsilon$

$G$

4.1

$L$

-factors and newforms

We show that zeta integrals of newforms attain -factors. We normalize Haar measuresonandso that the volumes ofand ofare onc respectively. Then thefollowing holds:$L$$E^{\cross}$$K_{n,H}$$\mathfrak{o}_{E}^{\cross}$$K_{n,H}$13
Theorem 4.1 ([8]). Suppose thathas conductor . Let be an irreducible genericrepresentation of and $W$ the newform insuch that $W(1)=1$ . Then we have$\psi_{E}$$\mathfrak{o}_{E}$

$G$

$\pi$

$\mathcal{W}(\pi, \psi_{E})$$Z(s, W, \Phi_{N_{\pi}})=L(s, \pi)$.

Theorem 4.1 is reduced to the following lemma:
Lemma 4.2. With the notation as above, we have$Z(s, W, \Phi_{N_{\pi}})/L(s, \pi)=1$

or $1/L_{E}(s, 1)$ .

We postpone the proof of Lemma 4.2 to the next section.

Proof of Theorem 4.1. We further assume thathas conductor$Z(s, W, \Phi_{N_{\pi}})/L(s, \pi)=1/L_{E}(s, 1)$ . Then by (2.2), we obtain$\psi_{F}$$\mathfrak{o}_{F}$.

Suppose that$\epsilon(s, \pi, \psi_{F)}\psi_{E}) = \gamma(s, \pi, \psi_{F}, \psi_{E})\frac{L(s,\pi)}{L(1-s,\pi)}$$= \frac{Z(1-s,W,\hat{\Phi}_{N_{\pi}})}{Z(s,W,\Phi_{N_{\pi}})}\frac{L(s,\pi)}{L(1-s,\pi)}$$= q^{-2N_{\pi}(s-1/2)} \frac{L_{E}(s,1)}{L_{E}(1-s,1)}.$The last equality follows from (3.7). This contradicts Proposition 2.3 which implics thatis monomial. Thus we get $Z(s, W, \Phi_{N_{\pi}})=L(s, \pi)$ , as required.

$\epsilon(s, \pi, \psi_{F}, \psi_{E})$4.2$\epsilon$$\square$-factors and conductors

We show that the exponent ofof the -factor of an irreducible generic representation ofcoincides with the conductor of . Applying the argument in the proof of
Theorem 4.1, we obtain the following:$q^{-2s}$$\pi$$\epsilon$

$G$

$\pi$

Theorem 4.3 ([8]). Suppose thatand
For any irreducible generic representation$\psi_{E}$$\pi$have conductorsof , we have$\psi_{F}$$G$$\epsilon(s, \pi, \psi_{F}, \psi_{E})=q^{-2N_{\pi}(s-1/2)}.$

5

Proof of Lemma 4.2

In this section, we explain how to prove Lemma 4.2.

$\mathfrak{o}_{E}$and$\mathfrak{o}_{F}$respectively.

14
5.1

Evaluation of -factors$L$

We shall evaluate $L(s, \pi)$ , for each irreduciblc generic representation $(\pi, V)$ of . The-factor $L(s, \pi)$ is defined as the greatest common divisor of the zeta integrals $Z(s, W, \Phi)$ .
For, there exist
andand$(1\leq i\leq m)$ such that
$G$

$L$

$W\in \mathcal{W}(\pi, \psi_{E})$$\Phi\in C_{c}^{\infty}(F^{2})$$W_{i}\in \mathcal{W}(\pi, \psi_{E})$$Z(s, W, \Phi)=\sum_{i=1}^{m}Z(s, W_{i})f(s, 1, \Phi_{i})$$\Phi_{i}\in C_{c}^{\infty}(F^{2})$.

By the theory of zeta integral for $GL$ (1), we have$f(s, 1, \Phi_{i})\in L_{E}(s, 1)C[q^{-2s}, q^{2s}].$Recall that we defined$Z(s, W)= \int_{E^{x}}W(t(a))|a|_{E}^{s-1}d^{\cross}a,$for$W\in \mathcal{W}(\pi, \psi_{E})$

$GL$

(2).

. To estimate $Z(s, W)$ , we can apply thc theory of Kirillov model for

An irreducible gcneric representation of is supercuspidal, or else a subrepresentation of a parabolically induced representation from . The Levi componentofisisomorphic to. For a quasi-characterofand a characterof $U(1)$ , wedenote by
the corresponding parabolically induced representation. According
to the classification of representations of , we have the following evaluation of the shapeof -factors:
$G$

$B$

$E^{\cross}\cross U(1)$

$T$

$B$

$E^{\cross}$$\mu_{1}$$\mu_{2}$$Ind_{B}^{G}\mu_{1}\otimes\mu_{2}$

$G$

$L$

Proposition 5.1. Let be an irreducible generic representation of $G.$(i) If is supercuspidal, then $L(s, \pi)$ divides $L_{E}(s, 1)$ .
(ii) If is a proper submodule of, then $L(s, \pi)$ divides $L_{E}(s, \mu_{1})L_{E}(s, 1)$ .
(iii) If
, then $L(s, \pi)$ divides
.
$\pi$

$\pi$

$Ind_{B}^{G}\mu_{1}\otimes\mu_{2}$$\pi$$\pi=Ind_{B}^{G}\mu_{1}\otimes\mu_{2}$

5.2$L_{E}(s, \mu_{1})L_{E}(s, \overline{\mu}_{1}^{-1})L_{E}(s, 1)$

Calculation of zeta integral of newforms

Lct $W$ be the newform insuch that $W(1)=1$ . We shall compute $Z(s, W, \Phi_{N_{\pi}})$ .
Supposc that has conductor zero. Then
, for some unramified quasicharacterof
In this case, ncwforms inarc just spherical Whittaker
functions. In [4], Gelbart and Piatetski-Shapiro showed that$\mathcal{W}(\pi, \psi_{E})$$\pi=Ind_{B}^{G}(\mu_{1}\otimes 1)$$\pi$$E^{\cross}$$\mu_{1}$$\mathcal{W}(\pi, \psi_{E})$$Z(s, W, \Phi_{0})=L_{E}(s, \mu_{1})L_{E}(s,\overline{\mu}_{1}^{-1})L_{E}(s, 1)$

15
by using Casselman-Shalika’s formula for spherical Whittaker functions in [3]. We therefore obtain $Z(s, W, \Phi_{0})=L(s, \pi)$ because of Proposition 5.1.
From now on, we assume thatis positive. By (3.5), we have
$N_{\pi}$$Z(s, W, \Phi_{N_{\pi}})=Z(s, W)L_{E}(s, 1)$,

and hence it is enough to compute $Z(s, W)$ . One can easily observe that
(5.2)

$Z(s, W)= \int_{E^{\cross}}W(t(a))|a|_{E}^{s-1}d^{\cross}a=\sum_{i=0}^{\infty}W(t(\varpi^{i}))q^{2i(1-s)}.$$i\geq 0$ , in terms of two “Hecke eigenSo we shall give a recursion formula forvalues” andWe abbreviate $N=N_{\pi}$ . Let us define the eigenvalue . We define a level raising
operator $\theta’:V(N)arrow V(N+1)$ by$W(t(\varpi^{i})),$$\lambda$$v.$$\lambda$$\theta’v=\int_{K_{N+1}}\pi(k)vdk, v\in V(N)$and a level lowering operator$\delta$,: $V(N+1)arrow V(N)$ by$\delta w = \int_{K_{N}}\pi(k)wdk, w\in V(N+1)$ .

Since $\dim V(N)=1$ , there exists$\lambda\in C$

such that$\lambda v=\delta\theta’v,$

for all $v\in V(N)$ .
Next, we define the eigenvalue . Put$v$$\zeta=(\varpi 1 \varpi^{-1})\in G.$

We define the Hecke operator$T$

on $V(N+1)$ by$Tv= \int_{K_{N+1}\zeta K_{N+1}}\pi(k)vdk, v\in V(N+1)$ .

Because $\dim V(N+1)=1$ , there exists

$v$

in

$C$

such that$Tv=vv,$

for all $v\in V(N+1)$ .
With the notation as above, we obtain the following recursion formula for $W(t(\varpi^{i}))$ ,$i\geq 0.$

16
Proposition 5.3. Let $(\pi, V)$ be an irreducible generic representation of, we havetor
is positive. For any newform $W$ in$G$whose conduc-$\mathcal{W}(\pi, \psi_{E})$$N_{\pi}$$(\nu+q^{2}-\lambda)q+q(v+q^{2}-q^{3})q_{+1}=q^{5_{C_{i+2}}}, i\geq 0,$$(v-q^{3})c_{0}=q^{4}c_{1},$where $c_{\eta}=W(t(\varpi^{i})),$ $i\geq 0.$
By (5.2) and Proposition 5.3, we can describe thc zeta integral of newforms in tcrms
of and :$\lambda$$\nu$

Proposition 5.4. Let $(\pi, V)$ be an irreducible generic representation of whose conductor
is positive and $W$ its newform insuch that $W(1)=1$ . Then we
have
$G$

$N_{\pi}$$\mathcal{W}(\pi, \psi_{E})$$Z(s, W)= \frac{1-q^{-2s}}{1-\frac{\nu+q^{2}-q^{3}}{q^{2}}q^{-2s}-\frac{v+q^{2}-\lambda}{q}q^{-4s}}.$

In particular,$Z(s, W. \Phi_{N_{\pi}})=\frac{1}{1-\frac{\nu+q^{2}-q^{3}}{q^{2}}q^{-2s}-\frac{v+q^{2}-\lambda}{q}q^{-4s}}.$5.3

Proof of Lemma 4.2

We have seen that Lemma 4.2 holds for the unramified principal series representations.
Let be an irreducible generic representation of . We assume thatis positive.
Proof of Lemma 4.2 is done by comparing Propositions 5.1 and 5.4. Suppose that issupercuspidal or a subrepresentation of, for some ramified quasi-character. Then it follows from Proposition 5.1 that $L(s, \pi)=1$ or $L_{E}(s, 1)$ . By definition,of
we have $Z(s, W, \Phi_{N_{\pi}})/L(s, \pi)\in C[q^{-2s}, q^{2s}]$ . So we get
$G$

$\pi$

$N_{\pi}$$\pi$$Ind_{B}^{G}\mu_{1}\otimes\mu_{2}$$\mu_{1}$$E^{\cross}$$Z(s, W, \Phi_{N_{\pi}})/L(s, \pi)=1or1/L_{E}(s.

1)$ ,by Proposition 5.4.
Suppose that is a subrepresentation of, for unramified . Then we can. Due to [6], non-zero newforms
regard newforms for as functions ininsatisfy $f(1)\neq 0$ . By using this property of newforms, we can compute the eigenvalues
and explicitly, and Lemma 4.2 follows.
$Ind_{B}^{G}\mu_{1}\otimes\mu_{2}$

$\pi$

$\pi$

$Ind_{B}^{G}\mu_{1}\otimes\mu_{2}$$\mu_{1}$

$f$

$\pi$

$\nu$

$\lambda$

17

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$\Pi$

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