LOCAL COEFFICIENT MATRICES AND THE METAPLECTIC

CORRESPONDENCE

MARK BUDDEN AND GEOFF GOEHLE

Abstract. The local coe�cients of a principal series representation of a meta-plectic group eG are defined in terms of the action of the standard intertwiningoperator on a canonical basis of the space of Whittaker functionals. By analyz-ing the nonsingularity of local coe�cient matrices, we prove that for a certainclass of unramified principal series representations of the metaplectic group,the local metaplectic correspondence preserves irreducibility.

Introduction

In 1973, Shimura [22] described a correspondence between cusp forms of weight 12k,

where k � 3 is an odd integer, and modular forms of weight k� 1. This correspon-

dence was reformulated by Gelbart and Piatetski-Shapiro (see [9], [10], and [11]) in

terms of the representation theory of 2-fold metaplectic coverings of GL2. Begin-

ning in the 1980’s, attention was placed on generalizing Shimura’s correspondence

within its representation-theoretic setting. Flicker [8] generalized the correspon-

dence to n-fold covering groups of GL2 and the works of Kazhdan and Patterson

[14], Kazhdan and Flicker [12], and later, Suzuki [23] and Mezo [17], [18], sought a

complete correspondence between representations on n-fold metaplectic coverings

of GLr

and their nonmetaplectic counterparts. Here, we focus on a generalization of

the local metaplectic correspondence, relating n-fold metaplectic coverings of GLr

to their d-fold counterparts, where d is a divisor of n.

For a certain class of unramified principal series representations, the first author

[5] showed that when r = 2 and n has absolute value 1 in the local field, the local

generalized metaplectic correspondence preserves irreducibility. This was achieved

via a careful consideration of the action of the standard intertwining operator on

a canonical basis of the finite-dimensional space of Whittaker functionals. This

article continues this investigation for arbitrary values of r.

2000 Mathematics Subject Classification. Primary 22D30, 11F32; Secondary 11F70, 11F85.Key words and phrases. principal series, automorphic forms, Shimura’s correspondence.

1

2 M. BUDDEN AND G. GOEHLE

The construction of the n-fold (c-twisted) metaplectic group fGL(c)

r

(F) is brieflyrevisited in Section 1 for the purpose of establishing the notations that will be used

throughout the paper. Section 2 contains a description of the unramified principal

series representations (⇡s

, Vs

)(n,r) that will be the focus of our investigation. For

each w in the Weyl subgroup W ⇢ GLr

(F), the local coe�cients ⌧w

(⇡s

, ⌘, ⌘0) are

defined in Section 3. They are the metaplectic analogues of the local coe�cients

defined by Shahidi in [20] and [21]. In the case of the long element w0 2 W , we

arrange the local coe�cients ⌧w0(⇡s, ⌘, ⌘

0) into an nr ⇥ nr matrix M(n,r)w0 (⇡

s

), and

describe an intimate relationship between the determinant of this matrix and the

irreducibility of (⇡s

, Vs

)(n,r).

In [5], the determinant of M (n,2)w0 (⇡

s

) was computed explicitly. When r > 2,

the direct computation of the determinant of M (n,r)w0 (⇡

s

) proves to be much more

challenging. Instead, we use an indirect method to compute

det(M (n,r)w0

(⇡�s

) ·M (n,r)w0

(⇡s

)),

which is su�cient for determining the irreducibility of our representations, and the

main results in Section 4.

The outcome of our investigation is an explicit description (in Theorem 2) of

the nonsingularity of the local coe�cient matrices. The local metaplectic corre-

spondence is given in Theorem 4, where it is shown that whenever d is a divisor of

n and |n|F = 1,

(⇡s

, Vs

)(n,r) is irreducible () (⇡sn/d

, Vsn/d

)(d,r) is irreducible.

Our analysis leads to another implication. We show in Corollary 5 that if r0 > r � 2

and |n|F = 1, then

(⇡s

, Vs

)(n,r0) is irreducible =) (⇡

s

, Vs

)(n,r) is irreducible .

It should be noted that the correspondence contained here is a local result, and

although it has not been considered in our investigation, it may extend to the global

setting. Such an extension is not trivial as our results assume |n|F = 1. Also, we

focus on a particular class of representations in the unramified principal series, but

it should be possible to apply these techniques to other metaplectic representations

in order to achieve similar criteria for their irreducibility.

1. Metaplectic Groups

In this section, we recapitulate the construction of a metaplectic group and the

main properties of 2-cocycles that allow us to work in this setting. Most of our

LOCAL COEFFICIENT MATRICES AND THE METAPLECTIC CORRESPONDENCE 3

notations and the systematic approach to the construction of metaplectic cocycles

follow closely Section 1 of [2]. These topics are covered here for the sake of complete-

ness and to establish the notations and tools to be used throughout the rest of the

paper. It is also hoped that this brief exposition will give readers unfamiliar with

the general theory of metaplectic groups the background they need to understand

the motivation behind this article.

Let F denote a nonarchimedean local field containing µn

, the full group of nth

roots of unity. We will fix an embedding ✏ : µn

,! C⇥, identifying µn

with the group

of nth roots of unity in C⇥. For any such field, let � : F⇥ ! Z be a normalized

valuation on F. As usual, � is extended to all of F by setting �(0) = 1 (with the

convention that 1 + z = 1 = z + 1 for all z 2 Z). Let O denote the ring of

integers {x 2 F | �(x) � 0} in F and p the maximal ideal {x 2 F | �(x) � 1} in O.

The notation $ will denote a uniformizing element. Let q = pf be the order of the

residue field O/p. Then the absolute value for F is given by |x|�

= q��(x) and for

each s 2 C, the local ⇣-factor is ⇣�

(s) = (1� q�s)�1.

Recall that the local nth-order Hilbert symbol over F is a map

(·, ·)F : F⇥ ⇥ F⇥ �! µn

that satisfies the following properties:

(aa0, b)F = (a, b)F (a0, b)F,

(a, b)F (b, a)F = 1,

(a,�a)F = (a, 1� a)F = 1,

{x 2 F⇥ | (x, y)F = 1, for all y 2 F⇥} = F⇥n,

where F⇥n := {x 2 F⇥ | x = yn, for some y 2 F⇥} (see [24], Chapter VIII sec. 5

for details). Throughout, we will assume that the Hilbert symbol is unramified:

(x, y)F = 1, for all x, y 2 O⇥.

This condition is equivalent to |n|F = 1. These basic properties of the Hilbert

symbol will account for most of the 2-cocycle computations that will come later.

For every positive integer r and every c 2 Z/nZ define the n-fold (c-twisted)

metaplectic covering fGL(c)

r

(F) of GLr

(F) to be a central extension of GLr

(F) by

µn

:

(1) 1 ����! µn

i����! fGL(c)

r

(F) p����! GLr

(F) ����! 1.

4 M. BUDDEN AND G. GOEHLE

As a set, the elements of fGL(c)

r

(F) are of the form (g, ⇣) where g 2 GLr

(F) and

⇣ 2 µn

. Multiplication is given by

(g, ⇣)(g0, ⇣ 0) = (gg0, ⇣⇣ 0 �(c)r

(g, g0))

where �(c)r

is a nontrivial 2-cocycle in Z2�fGL

(c)

r

(F);µn

�that satisfies the cocycle

relation

�(c)r

(g, g0)�(c)r

(gg0, g00) = �(c)r

(g, g0g00)�(c)r

(g0, g00),

for all g, g0, g00 2 GLr

(F) (which is equivalent to associativity of multiplication in

fGL(c)

r

(F)). The twists of �r

:= �(0)r

satisfy

�(c)r

(g, g0) = �r

(g, g0)(detg, detg0)cF.

In the short exact sequence (1), the projection map p : fGL(c)

r

(F) �! GLr

(F) is

given by (g, ⇣) 7! g and the inclusion map i : µn

�! GLr

(F) is given by ⇣ 7! (1, ⇣).

We will also need the preferred section s : GLr

(F) �! fGL(c)

r

(F) where g 7! (g, 1).

The cocycle �2 was first described by Kubota [15] and is defined by

�2(g, g0) =

X(gg0)

X(g),

X(gg0)

X(g0)det(g)

!

Fwhere

X⇣⇣

a bc d

⌘⌘=

⇢c if c 6= 0d if c = 0

.

The cocycle �r

that will be used in this paper was constructed in [3] and agrees

with the Kubota cocycle when r = 2. When r = 1, GL1(F) = F⇥, �1 is trivial, and

the twists are given by �(c)1 (g, g0) = (g, g0)cF (see Corollary 8 in Section 3 of [3]).

To help in the evaluation of �r

when r > 2, we define a standard Levi subgroup

M of GLr

(F) to be a group whose elements are of the form0

B@g1

. . .gk

1

CA

where gi

2 GLri(F), ri � 1, and r = r1 + r2 + · · · + r

k

. The following theorem

concerning the block-compatibility of the cocycle �r

was proved by Banks, Levy,

and Sepanski (Theorem 11 in Section 3 of [3]).

Theorem 1. For every standard Levi subgroup of GLr

(F), the cocycle �r

satisfies

the following block formula:

�r

0

B@

0

B@g1

.

.

.

gk

1

CA ,

0

B@g01

.

.

.

g0k

1

CA

1

CA =kY

i=1

�ri(gi, g

0i

)Y

1i<jk

(det gi

, det g0j

)F.

LOCAL COEFFICIENT MATRICES AND THE METAPLECTIC CORRESPONDENCE 5

We will use the notation T to denote the subgroup of GLr

(F) consisting of

diagonal matrices. If ↵ = (i, j) 2 � denotes a root of GLr

(F) relative to T (1 i, j r and i 6= j), then let

t↵ = ti

/tj

, for all t = diag(ti

).

The set of positive roots (i < j) will be denoted �+ and the set of negative roots

(i > j) will be denoted ��. We will use N to denote the unipotent radical of B,

the Borel subgroup of GLr

(F).

Now assume that r � 2, and consider the r�1 embeddings of GL2(F) along the

diagonal of GLr

(F):

◆i

: GL2(F) ,! GLr

(F), g 7!0

@Ii�1

gIr�1�i

1

A , for all g 2 GL2(F).

For each i, let Gi

denote the image of ◆i

. The group GLr

(F) is then generated by

the subgroups

{Gi

| 1 i r � 1} .Define

wi

= ◆i

✓ �11

◆and s

i

= ◆i

✓1

1

◆

and let `(w) denote the length of an element w in the Weyl subgroup W . In other

words, ` = `(w) is the smallest positive integer such that w can be written as a

product of ` simple reflections: w = si1si2 · · · si` . Let ⌘(w) denote the corresponding

element ⌘(w) = wi1wi2 · · ·wi` (w 7! ⌘(w) is well-defined by Lemma 6.2 of [16]). The

group W acts on T (and on the set of roots) by conjugation:

t(w↵) = (tw)↵ = (w�1tw)↵.

All of the cocycle calculations contained in this paper are computed using �1,

�2, Theorem 1, the properties of the Hilbert symbol given at the beginning of this

section, and the properties of �r

described in Theorem 7 of Section 3 of [3].

2. Unramified Principal Series Representations

In this section, we will describe a class of unramified principal series repre-

sentations (induced from an irreducible representation of a Borel subgroup). Our

description will require several important subgroups of eG = fGL(c)

r

(F). If H is any

subgroup of G = GLr

(F), we will use the notation eH = p�1(H) to denote the

full metaplectic preimage of H. The groups eT and eB are formed in this manner.

Although these groups depend upon the twist c, it will often be suppressed in our

notation.

6 M. BUDDEN AND G. GOEHLE

It is easily checked that the 2-cocycle �(c)r

is trivial on N . Thus, eG splits over

N via the preferred section s. We will denote its image by N⇤ = s(N). The groupeG also splits over the maximal compact subgroup K = GL2(O) of G (see [19]). Let

k : K �! eG

denote the canonical splitting. We will not describe k here as it will be enough to

note that k satisfies the relations

k|T\K

= s|T\K

, k|W

= s|W

, and k|N\K

= s|N\K

(see Proposition 0.1.3 of [13]). Set K⇤ := k(K) and for every m � 0, define

K⇤m

:= k(Km

) where

Km

:= {k 2 K | k ⌘ I (mod pm)}.The collection {K⇤

m

| m � 0} forms a basis of open, compact neighborhoods of the

identity in eG. These groups lead to the decomposition

eG = eBK⇤ = eTN⇤K⇤.

The principal series representations of eG = fGL(c)

r

(F) that we describe in this

section are formed in a similar manner to the representations of fGL(c)

2 (F) that werethe focus of [5]. We begin with a quasicharacter on the center of eT . As was noted

in Section 1.1 of [13], the center of eT is given by eTnZ( eB) where

eTn := p�1(Tn) = p�1

8><

>:

0

B@t1

. . .tr

1

CA

�������t1, . . . , tr 2 F⇥n

9>=

>;

and the center of eB is given by

Z( eB) = p�1

8><

>:

0

B@�

. . .�

1

CA

��������r�1+2rc 2 F⇥n

9>=

>;

by Proposition 0.1.1 of [13].

Let �B

denote the modular quasicharacter of the Borel subgroup B. For any

s 2 C, we will consider �s to be the quasicharacter �sB

on eTnZ( eB) that is genuine.

In other words,

�s

0

B@

0

B@

0

B@t1

. . .tr

1

CA , ⇣

1

CA

1

CA = ⇣(|t1|r�1F |t2|r�3

F · · · |tr

|1�r

F )s.

This quasicharacter can be extended to a maximal abelian subgroup eT⇤ of eT in the

obvious way. We use the maximal abelian subgroup

eT⇤ = ( eT \K⇤) eTnZ( eB)

LOCAL COEFFICIENT MATRICES AND THE METAPLECTIC CORRESPONDENCE 7

and extend the quasicharacter � to eB⇤ := eT⇤N⇤ by making it trivial on N⇤.

For any s 2 C and any unramified quasicharacter

� : F⇥ �! C⇥,

define the (normalized) induced representation

(⇡s

, Vs

)(n,r) := IndeG

eB⇤

((� � det � p) · �s)where V

s

consists of all locally constant functions f : eG �! C that satisfy

f(bg) = �(det(p(b))) · �s(b) · � 12B

(p(b)) · f(g)for all b 2 eB⇤ and g 2 eG and ⇡

s

acts by right translation on Vs

:

(⇡s

(g)f)(g0) = f(g0g).

The Stone-von Neumann Theorem (see [6] or [7]) guarantees that the isomorphism

class of a representation constructed in this way depends only on the central qua-

sicharacter (� � det � p) · �s, not on the choice of maximal abelian subgroup nor on

the quasicharacter’s extension.

Now let w 2 W , ew := s(w), and suppose that that ! : eT⇤ �! C⇥ is an

unramified quasicharacter. Then define w!(t) : eT⇤ �! C⇥ by w!(t) := !( ew�1t ew).Such a character can then be extended to eB⇤ by making it trivial on N⇤ (as was

described above in the case of �). Using this notation, we form

(w⇡s

,w Vs

)(n,r) := IndeG

eB⇤

(w((� � det � p) · �s))and consider the standard intertwining operator I

w

: Vs

�! wVs

given by the

absolutely convergent integral

(Iw

f)(g) =

Z

n2N

⇤(w)

f( ew�1ng) dn

for all f 2 Vs

and g 2 eG. Here, N⇤(w) is the subgroup of N⇤ corresponding to

the roots ↵ 2 �+ such that w�1↵ 2 �� and dn is the unique Haar measure that

satisfies Vol(N⇤ \K⇤; dn) = 1.

By Proposition 1.2.3 of [13], if w1, w2 2 W satisfy `(w1w2) = `(w1) + `(w2),

then

Iw1w2 : V

s

�! w1w2Vs

is given by the composition

Iw1w2f = I

w1(Iw2f).

This allows us to reduce the evaluation of an intertwining operator to simple re-

flections in W .

8 M. BUDDEN AND G. GOEHLE

In general, a principal series representation of a metaplectic group does not

have a unique Whittaker model. However, it does have a finite-dimensional space

of Whittaker functionals. Fix a nontrivial additive unramified (unitary) character

F : F �! C⇥ and extend it to a multiplicative character : N⇤ �! C⇥ by

(en) := F

0

@X

1i<r

ni,i+1

1

A ,

where en := s(n) 2 N⇤ and ni,i+i

is the entry in the (i, i + 1)th position of n 2 N .

A -Whittaker functional is a functional � in the dual space V 0s

of Vs

that satisfies

(2) �(⇡s

(n)f) = (n)�(f)

for all n 2 N⇤ and f 2 Vs

.

For ⌘ 2 eT , let �⌘

2 V 0s

be given by

(3) �⌘

f =

Z

n2N

⇤

f(⌘ ew0n) (n) dn.

Kazhdan and Patterson ([13], Lemma 1.2.3) showed that the integral (3) is abso-

lutely convergent for Re(s) > 0 and extends holomorphically to all of C. One can

check that �⌘

satisfies (2) and that

{�⌘

| ⌘ 2 eT⇤\ eT}is a basis for Wh( , V

s

), the space of -Whittaker functionals of Vs

. By Lemma

1.3.2 of [13],

dim(Wh( , Vs

)) = Card( eT⇤\ eT ) = nr.

Let f 2 (Z/nZ)r, where f = (f1, f2, . . . , fr) (we will always use least residues of Z/nZas coset representatives), and write s($f) := s(diag($fi)). Then we will use the set

⌦ :=�s($f) | f 2 (Z/nZ)r

as coset representatives of eT⇤\ eT .

In the next section, the element

f = (0, 1, . . . , r � 1) 2 (Z/nZ)r

will play an important role in the evaluation of local coe�cients. The corresponding

element in ⌦ will be denoted ⌘ = s($f). If w 2 W , then let w[f] be the element of

(Z/nZ)r given by

$w[f] := w$f�fw�1$f.

This defines a group action of W on (Z/nZ)r. For two elements ⌘ = s($f) and

⌘0 = s($f0) in eT⇤\ eT , we will write ⌘ ⇠ ⌘0 if and only if either f0 ⌘ f (mod n) or

f0 ⌘ s[f] (mod n) for some simple reflection s 2 W . A straight forward computation

shows that for all simple reflections, ⌘ ⇠ ⌘ implies that f ⌘ f (mod n).

LOCAL COEFFICIENT MATRICES AND THE METAPLECTIC CORRESPONDENCE 9

3. Nonsingularity of Local Coefficient Matrices

The local coe�cients of (⇡s

, Vs

)(n,r) are defined by considering the action of

the standard intertwining operators on the space of Whittaker functionals. Using

Kazhdan and Patterson’s canonical basis for Wh(Vs

) described in the previous

section, the local coe�cient ⌧w

(⇡s

, ⌘, ⌘0) is defined by

(4) �⌘

Iw

=X

⌘

02eT⇤\e

T

⌧w

(⇡s

, ⌘, ⌘0)�⌘

0 .

Whenever w,w0 2 W satisfy `(ww0) = `(w) + `(w0), we have

(5) ⌧ww

0(⇡s

, ⌘, ⌘0) =X

⌘

002eT⇤\e

T

⌧w

(w0⇡s

, ⌘, ⌘00)⌧w

0(⇡s

, ⌘00, ⌘0),

thus reducing local coe�cient calculations to the case of simple reflections.

To evaluate the local coe�cients, we follow [13], Section 1.3, starting with an

appropriate set of “test” functions in Vs

. Let

f⌘

0(g) =

(�(det(p(b))) · �s(b) · � 1

2B

(p(b)) if g = b⌘0lw0, b 2 eB⇤, l 2 L0 otherwise

where L is an open, compact subgroup of eG that is taken to be su�ciently small.

As in [13], we let f (m)⌘

0 denote the test function satisfying

(1) Supp(f (m)⌘

0 ) = eB⇤⌘0w0K

⇤m

= eB⇤⌘0w0(K

+m

)⇤, where K+m

= Km

\N,

(2) f(m)⌘

0 is right K⇤m

-invariant,

(3) f(m)⌘

0 (w0) = 1.

By Lemma 1.3.1 of [13], we have

�⌘

f(m)⌘

0 = 0 () ⌘ 6= ⌘0,

and for such functions, (4) gives

⌧w

(⇡s

, ⌘, ⌘0) =�⌘

Iw

f(m)⌘

0

�⌘

0f(m)⌘

0

,

resulting in the following commutative diagram.

Vs

Iw������! wVs

�⌘0

??y??y�⌘

C �������!⌧w(⇡s,⌘,⌘

0)C

For all principal series representations (⇡(!), V (!)) (induced from a genuine un-

ramified quasicharacter ! : eT⇤ �! C⇥ that has been extended to eB⇤ by making it

10 M. BUDDEN AND G. GOEHLE

trivial on N⇤), it was shown in Section 1.3 of [13] that

⌧s

(⇡(!), ⌘, ⌘0) = 0 whenever ⌘ 6⇠ ⌘0.

In particular, this applies to ⇡s

as well as w⇡s

for all w 2 W .

Now we define the local coe�cient matrix

M (n,r)w

(⇡s

) := (⌧w

(⇡s

, ⌘, ⌘0))

by placing ⌧w

(⇡s

, ⌘, ⌘0) in the (nr�1f1 + nr�2f2 + · · ·nfr�1 + f

r

+ 1)th row and

(nr�1f01 + nr�2f02 + · · ·nf0r�1 + f0

r

+ 1)th column. The matrix M(n,r)w

(⇡s

) can be

interpreted as a linear transformation Cn

r �! Cn

r

. It was shown in [1] that

Vs

= Spanhf⌘

0i �Kerh�⌘

0i,and using the linearity of the Whittaker functionals, we obtain the following com-

mutative diagram.

Vs

Iw������! wVs

??y??y

Cn

r �������!M

(n,r)w (⇡s)

Cn

r

Considering the long element w0 2 W , we have

(6)

Vs

Iw0�������! V�s

Iw0�������! Vs

??y??y

??y

Cn

r �������!M

(n,r)w0 (⇡s)

Cn

r ��������!M

(n,r)w0 (⇡�s)

Cn

r

where Iw0 � I

w0 : Vs

�! Vs

is an intertwining operator. By Proposition 1.2.2 of

[13],

Iw0 � Iw0 = c(n,r)(s) · I : V

s

�! Vs

,

where c(n,r)(s) is a constant and I is the identity map. From the diagram (6), it

follows that

M (n,r)w0

(⇡�s

) ·M (n,r)w0

(⇡s

) = c(n,r)(s) · In

r ,

where In

r is the identity matrix in GLn

r (C). Now we turn our attention to the

evaluation of c(n,r)(s).

Theorem 2. If r � 2 and |n|F = 1, then

c(n,r)(s) =

✓⇣�

(2ns)⇣�

(�2ns)

⇣�

(1� 2ns)⇣�

(1 + 2ns)

◆ r2�4r+32 +"

✓⇣�

(4ns)⇣�

(�4ns)

⇣�

(1� 4ns)⇣�

(1 + 4ns)

◆ r�12 �"

·r�1Y

i=1

⇣�

(2ins)⇣�

(�2ins)

⇣�

(1� 2ins)⇣�

(1 + 2ins)

LOCAL COEFFICIENT MATRICES AND THE METAPLECTIC CORRESPONDENCE 11

where

" =

⇢12 if r even

0 if r odd.

Proof. We have seen that

M (n,r)w0

(⇡�s

) ·M (n,r)w0

(⇡s

) = c(n,r)(s) · In

r .

In [5], the determinant of M (n,r)w0 (⇡

s

) was explicitly computed for r = 2. However,

when r > 2, the computation proves to be much more challenging. Rather than

compute the determinant in a straightforward manner, we take advantage of the

structure of the local coe�cient matrices and the abundance of 0 entries. We begin

by decomposing the long element as

w0 = wr�1 · · ·w2w1 where w

j

= s1s2 · · · sj .This allows for the decomposition

M (n,r)w0

(⇡s

) = M (n,r)wr�1

(wr�2···w2w1⇡s

) · · ·M (n,r)w2

(w1⇡s

) ·M (n,r)w1

(⇡s

),

where each M(n,r)w

(w0⇡s

) can be decomposed into local coe�cient matrices associ-

ated with simple reflections. At the end of Section 2, it was noted that, ⌘ ⇠ ⌘

implies that f ⌘ f(mod n). Thus, for each simple reflection si

2 W (1 i < r), the

entry ⌧si(

w⇡s

, ⌘, ⌘) is the only nonzero entry in its row and column in M(n,r)si (w⇡

s

).

This leads to the following decomposition of ⌧w0(⇡s, ⌘, ⌘):

⌧w0(⇡s, ⌘, ⌘) = ⌧

w1(⇡s, ⌘, ⌘) ·r�1Y

j=2

⌧wj (

wj�1···w2w1⇡s

, ⌘, ⌘),

where

⌧wj (

wj�1···w2w1⇡s

, ⌘, ⌘) = ⌧sj (

wj�1···w2w1⇡s

, ⌘, ⌘)

·j�1Y

k=1

⌧sj�k(

sj+1�k···sj�1sjwj�1···w2w1⇡s

, ⌘, ⌘).

Each of the local coe�cients in this product can be computed following Section 1.3

of [13]. Since this process is fairly standard, we omit the details of the computation.

We have

⌧sk(

wk�1···w2w1⇡s

, ⌘, ⌘) = ✏k

(�1,�1)cF⇣�

(2nks✏k

)

⇣�

(1� 2nks),

and if 1 i < k, then

⌧si(

si+1···sk�1skwk�1···w2w1⇡s

, ⌘, ⌘) =

8><

>:

✏i

(�1,�1)cF⇣�(�4ns✏i)⇣�(1�4ns) if k is even, i = k

2

✏i

(�1,�1)cF⇣�(�2ns✏i)⇣�(1�2ns) otherwise,

where

✏i

=

⇢1 if f

i+1 ⌘ 0(mod n)�1 if f

i+1 6⌘ 0(mod n).

12 M. BUDDEN AND G. GOEHLE

By carefully multiplying out each of the simple reflection local coe�cients and using

the property

⌧w0(

w0⇡s

, ⌘, ⌘) · ⌧w0(⇡s, ⌘, ⌘) = c(n,r)(s),

we conclude the statement of the theorem. ⇤

In [5], it was argued that whenever d is any divisor of n and |n|F = 1,

(⇡s

, Vs

)(n,2) is irreducible () (⇡sn/d

, Vsn/d

)(d,2) is irreducible.

This result will be generalized to arbitrary values of r � 2 in this section and is

dependent on the following Lemma. Unfortunately, there was a mistake in [5], so

the following Lemma corrects the mistake for r = 2, in addition to handling the

cases r > 2.

Lemma 3. The principal series representation (⇡s

Vs

)(n,r) is irreducible if and

only if

det(M (n,r)w0

(⇡�s

) ·M (n,r)w0

(⇡s

)) 6= 0

(equivalently, if and only if c(n,r)(s) 6= 0).

Proof. First consider the case in which c(n,r)(s) 6= 0. From Theorem 2, it follows

that 2ins 6= 1 for i = 1, . . . , r � 1. Hence, by Corollary 1.2.8 of [13], it follows that

(⇡s

, Vs

)(n,r) is irreducible. Now suppose that c(n,r)(s) = 0. Then Ker(Iw0) 6= {0}.

Now we follow the approach used in the beginning of the proof of Theorem 1.2.9 of

[13]. Recall that Iw0 : V

s

�! V�s

is defined by

(Iw0f)(g) =

Z

n2N

⇤(w0)

f(w0�1ng) dn.

Let L be a su�ciently small compact open subgroup of eG and let f be the right

L-invariant function in Vs

with support eB⇤L such that f(1) = 1. If we let g = w0,

the integral converges and by choosing L su�ciently small, one has Iw0f(w0) 6= 0.

Hence, Ker(Iw0) 6= V

s

, giving a proper nontrivial eG-stable subspace of Vs

. Thus,

(⇡s

, Vs

) is reducible. ⇤

The main result of this section (and of this paper) is the following theorem,

which follows from a comparison of the zeros of c(n,r)(s) over di↵erent values of n

and r.

Theorem 4. If d is any divisor of n and |n|F = 1, then (⇡s

, Vs

)(n,r) is irreducible

if and only if (⇡sn/d

, Vsn/d

)(d,r) is irreducible.

LOCAL COEFFICIENT MATRICES AND THE METAPLECTIC CORRESPONDENCE 13

Proof. In Lemma 3, it was shown that (⇡s

, Vs

)(n,r) is irreducible if and only if

c(n,r)(s) 6= 0. The zeros of c(n,r)(s) are easily determined from Theorem 2. We see

that c(n,r)(s) = 0 if and only if s has any of the values

±✓

1

2n+

⇡ik

n log q

◆,±1

2

✓1

2n+

⇡ik

n log q

◆, . . . ,± 1

r � 1

✓1

2n+

⇡ik

n log q

◆.

A comparison of these zeros with the corresponding zeros of c(d,r)(sn/d) results in

the correspondence described in Theorem 4 ⇤

Corollary 5. If r0 > r � 2, |n|F = 1, and (⇡s

, Vs

)(n,r0)

is irreducible, then

(⇡s

, Vs

)(n,r) is irreducible.

Proof. This corollary follows from a comparison of the zeros of c(n,r0)(s) and

c(n,r)(s) that were given in the proof of Theorem 4. ⇤

At the present time, only nonarchimedean local fields in which |n|F 6= 1 have

been considered. However, the techniques contained in this article should be ap-

plicable to other metaplectic representations than just the ones considered here.

We leave such investigations for future study, with the eventual goal of further

developing the complete (local and global) metaplectic correspondence using local

coe�cient matrices.

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14 M. BUDDEN AND G. GOEHLE

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Department of Mathematics and Computer Science, Western Carolina University,

Cullowhee, NC 28723 USA

E-mail address: mrbudden@email.wcu.edu

Department of Mathematics and Computer Science, Western Carolina University,

Cullowhee, NC 28723 USA

E-mail address: grgoehle@email.wcu.edu